The coefficients $\beta_{0}$ and $\beta_{X}$ are proportional to the relative
amount of time that the trajectory $\gamma_{n}$ stays respectively in a
neighborhood of $0$ and $X$, as $n\to+\infty$.

Until the end of this proof, let us no longer think of points on the circle
$\mathbb{T}^{1}$ but by lifting to $\mathbb{R}$, but keeping the same
notations. Hence the function $V$ is now a $1$-periodic function on
$\mathbb{R}$. Integrating (4.3), one computes that if $x<y$, the time it takes
$\gamma_{n}$ to go from $x$ to $y$ is

$t_{x,y}=\int_{x}^{y}\frac{{\rm
d}s}{\sqrt{2\big{(}\alpha(c_{n})-V(s)\big{)}}}.$ (4.4)

In particular, $T_{c_{n}}=\int_{0}^{1}\frac{{\rm
d}s}{\sqrt{2\textrm{$\big{(}$}\alpha(c_{n})-V(s)\textrm{$\big{)}$}}}$.

Let $0<\varepsilon<\max\\{-V^{\prime\prime}(0),-V^{\prime\prime}(X)\\}$. Let
us consider $\eta>0$ such that,

$\left[|x|<\eta\right]\Longrightarrow\left[\left|V(x)-V^{\prime\prime}(0)\frac{x^{2}}{2}\right|<\varepsilon\frac{x^{2}}{2}\right],$

and

$\left[|x-X|<\eta\right]\Longrightarrow\left[\left|V(x)-V^{\prime\prime}(X)\frac{(x-X)^{2}}{2}\right|<\varepsilon\frac{(x-X)^{2}}{2}\right].$

We now split the integral defining $T_{c_{n}}$ into the $4$ following pieces:

$T_{c_{n}}=\int_{0}^{1}\frac{{\rm
d}s}{\sqrt{2\left(\alpha(c_{n})-V(s)\right)}}\\\
=\underbrace{\int_{-\eta}^{\eta}\frac{{\rm
d}s}{\sqrt{2\left(\alpha(c_{n})-V(s)\right)}}}_{\raisebox{-0.5pt}{\fontsize{7pt}{0}\fontfamily{phv}\selectfont1⃝}}+\underbrace{\int_{X-\eta}^{X+\eta}\frac{{\rm
d}s}{\sqrt{2\left(\alpha(c_{n})-V(s)\right)}}}_{\raisebox{-0.5pt}{\fontsize{7pt}{0}\fontfamily{phv}\selectfont2⃝}}\\\
+\underbrace{\int_{\eta}^{X-\eta}\frac{{\rm
d}s}{\sqrt{2\left(\alpha(c_{n})-V(s)\right)}}}_{\raisebox{-0.5pt}{\fontsize{7pt}{0}\fontfamily{phv}\selectfont3⃝}}+\underbrace{\int_{X+\eta}^{1-\eta}\frac{{\rm
d}s}{\sqrt{2\left(\alpha(c_{n})-V(s)\right)}}}_{\raisebox{-0.5pt}{\fontsize{7pt}{0}\fontfamily{phv}\selectfont4⃝}}.$
(4.5)

Let $M>0$ be a constant independent of $n$ such that
$|{\raisebox{-0.5pt}{\fontsize{7pt}{0}\fontfamily{phv}\selectfont3⃝}}|+|{\raisebox{-0.5pt}{\fontsize{7pt}{0}\fontfamily{phv}\selectfont4⃝}}|<M$
for all $n>0$. Such an $M$ exists as the denominators appearing in the
integrals are uniformly positive.

Let us now study and estimate \fontsize{7pt}{0}\fontfamily{phv}\selectfont1⃝.
From the definition of $\eta$ the following inequalities are infered:

$\underbrace{\int_{-\eta}^{\eta}\dfrac{{\rm
d}s}{\sqrt{2\left(\alpha(c_{n})+\big{(}\varepsilon-V^{\prime\prime}(0)\big{)}\dfrac{s^{2}}{2}\right)}}}_{\raisebox{-0.5pt}{\fontsize{7pt}{0}\fontfamily{phv}\selectfont5⃝}}\\\
\leqslant\int_{-\eta}^{\eta}\dfrac{{\rm
d}s}{\sqrt{2\left(\alpha(c_{n})-V(s)\right)}}\\\
\leqslant\underbrace{\int_{-\eta}^{\eta}\dfrac{{\rm
d}s}{\sqrt{2\left(\alpha(c_{n})+\big{(}-V^{\prime\prime}(0)-\varepsilon\big{)}\dfrac{s^{2}}{2}\right)}}}_{\raisebox{-0.5pt}{\fontsize{7pt}{0}\fontfamily{phv}\selectfont6⃝}}.$
(4.6)

Terms \fontsize{7pt}{0}\fontfamily{phv}\selectfont5⃝ and
\fontsize{7pt}{0}\fontfamily{phv}\selectfont6⃝ can be integrated explicitly,
let us deal with \fontsize{7pt}{0}\fontfamily{phv}\selectfont5⃝.

$\displaystyle\int_{-\eta}^{\eta}$ $\displaystyle\frac{{\rm
d}s}{\sqrt{2\left(\alpha(c_{n})+\big{(}\varepsilon-V^{\prime\prime}(0)\big{)}\dfrac{s^{2}}{2}\right)}}=\frac{\sqrt{\frac{2\alpha(c_{n})}{\varepsilon-V^{\prime\prime}(0)}}}{\sqrt{2\alpha(c_{n})}}\int_{-\eta\sqrt{\frac{\varepsilon-V^{\prime\prime}(0)}{2\alpha(c_{n})}}}^{\eta\sqrt{\frac{\varepsilon-V^{\prime\prime}(0)}{2\alpha(c_{n})}}}\dfrac{{\rm
d}t}{\sqrt{1+t^{2}}}$
$\displaystyle=\frac{1}{\sqrt{\varepsilon-V^{\prime\prime}(0)}}\left[\ln\left(\sqrt{1+t^{2}}+t\right)\right]_{-\eta\sqrt{\frac{\varepsilon-V^{\prime\prime}(0)}{2\alpha(c_{n})}}}^{\eta\sqrt{\frac{\varepsilon-V^{\prime\prime}(0)}{2\alpha(c_{n})}}}$
$\displaystyle=\frac{1}{\sqrt{\varepsilon-V^{\prime\prime}(0)}}\ln\left[\dfrac{\sqrt{\dfrac{\big{(}\varepsilon-V^{\prime\prime}(0)\big{)}\eta^{2}}{2\alpha(c_{n})}+1}+\eta\sqrt{\dfrac{\varepsilon-V^{\prime\prime}(0)}{2\alpha(c_{n})}}}{\sqrt{\dfrac{\big{(}\varepsilon-V^{\prime\prime}(0)\big{)}\eta^{2}}{2\alpha(c_{n})}+1}-\eta\sqrt{\dfrac{\varepsilon-V^{\prime\prime}(0)}{2\alpha(c_{n})}}}\right]$
$\displaystyle=\frac{1}{\sqrt{\varepsilon-V^{\prime\prime}(0)}}\ln\left[\left(\sqrt{\dfrac{\big{(}\varepsilon-V^{\prime\prime}(0)\big{)}\eta^{2}}{2\alpha(c_{n})}+1}+\eta\sqrt{\dfrac{\varepsilon-V^{\prime\prime}(0)}{2\alpha(c_{n})}}\right)^{2}\right]$
$\displaystyle=\frac{1}{\sqrt{\varepsilon-V^{\prime\prime}(0)}}\left\\{-\ln(2)-\ln\big{(}\alpha(c_{n})\big{)}+\ln\left[\sqrt{\big{(}\varepsilon-V^{\prime\prime}(0)\big{)}\eta^{2}+2\alpha(c_{n})}+\eta\sqrt{\varepsilon-V^{\prime\prime}(0)}\right]\right\\}$
$\displaystyle\underset{n\to+\infty}{\sim}\dfrac{-\ln\big{(}\alpha(c_{n})\big{)}}{\sqrt{\varepsilon-V^{\prime\prime}(0)}}.$

The last relation uses the continuity of $\alpha$ and the consecutive limit:
$\lim\limits_{n\to+\infty}\alpha(c_{n})=0$.

The same computation for \fontsize{7pt}{0}\fontfamily{phv}\selectfont6⃝ yields

$\int_{-\eta}^{\eta}\dfrac{{\rm
d}s}{\sqrt{2\left(\alpha(c_{n})+\big{(}-V^{\prime\prime}(0)-\varepsilon\big{)}\dfrac{s^{2}}{2}\right)}}\underset{n\to+\infty}{\sim}\dfrac{-\ln\big{(}\alpha(c_{n})\big{)}}{\sqrt{-\varepsilon-V^{\prime\prime}(0)}}.$

As for \fontsize{7pt}{0}\fontfamily{phv}\selectfont2⃝ the same strategy is
adopted:

$\underbrace{\int_{X-\eta}^{X+\eta}\dfrac{{\rm
d}s}{\sqrt{2\left(\alpha(c_{n})+\big{(}\varepsilon-V^{\prime\prime}(X)\big{)}\dfrac{s^{2}}{2}\right)}}}_{\raisebox{-0.5pt}{\fontsize{7pt}{0}\fontfamily{phv}\selectfont7⃝}}\\\
\leqslant\int_{X-\eta}^{X+\eta}\dfrac{{\rm
d}s}{\sqrt{2\left(\alpha(c_{n})-V(s)\right)}}\\\
\leqslant\underbrace{\int_{X-\eta}^{X+\eta}\dfrac{{\rm
d}s}{\sqrt{2\left(\alpha(c_{n})+\big{(}-V^{\prime\prime}(X)-\varepsilon\big{)}\dfrac{s^{2}}{2}\right)}}}_{\raisebox{-0.5pt}{\fontsize{7pt}{0}\fontfamily{phv}\selectfont8⃝}}.$
(4.7)

Similar computations yield

$\int_{X-\eta}^{X+\eta}\dfrac{{\rm
d}s}{\sqrt{2\left(\alpha(c_{n})+\big{(}\varepsilon-V^{\prime\prime}(X)\big{)}\dfrac{s^{2}}{2}\right)}}\underset{n\to+\infty}{\sim}\dfrac{-\ln\big{(}\alpha(c_{n})\big{)}}{\sqrt{\varepsilon-V^{\prime\prime}(X)}},$

and

$\int_{X-\eta}^{X+\eta}\dfrac{{\rm
d}s}{\sqrt{2\left(\alpha(c_{n})+\big{(}-V^{\prime\prime}(X)-\varepsilon\big{)}\dfrac{s^{2}}{2}\right)}}\underset{n\to+\infty}{\sim}\dfrac{-\ln\big{(}\alpha(c_{n})\big{)}}{\sqrt{-\varepsilon-V^{\prime\prime}(X)}}.$

Let now $f:\mathbb{R}\to\mathbb{R}$ be a continuous $1$–periodic function that
is constant on $[-\eta,\eta]$ and on $[X-\eta,X+\eta]$, with
$0\leqslant\min\big{(}f(0),f(X)\big{)}$. We know that

$\int_{[0,1]}f(s){\rm
d}\pi_{1*}\mu_{c_{n}}(s)\underset{n\to+\infty}{\longrightarrow}\beta_{0}f(0)+\beta_{X}f(X).$

Gathering the previous computations we infer that

$\frac{f(0)\raisebox{-0.5pt}{\fontsize{7pt}{0}\fontfamily{phv}\selectfont1⃝}+f(X)\raisebox{-0.5pt}{\fontsize{7pt}{0}\fontfamily{phv}\selectfont2⃝}-\|f\|_{\infty}M}{T_{c_{n}}}\leqslant\int_{[0,1]}f(s){\rm
d}\pi_{1*}\mu_{c_{n}}(s)\leqslant\dfrac{f(0)\raisebox{-0.5pt}{\fontsize{7pt}{0}\fontfamily{phv}\selectfont1⃝}+f(X)\raisebox{-0.5pt}{\fontsize{7pt}{0}\fontfamily{phv}\selectfont2⃝}+\|f\|_{\infty}M}{T_{c_{n}}}.$

And letting $n\to+\infty$ one discovers that

$\dfrac{\dfrac{f(0)}{\sqrt{\varepsilon-V^{\prime\prime}(0)}}+\dfrac{f(X)}{\sqrt{\varepsilon-V^{\prime\prime}(X)}}}{\dfrac{1}{\sqrt{-\varepsilon-V^{\prime\prime}(0)}}+\dfrac{1}{\sqrt{-\varepsilon-V^{\prime\prime}(X)}}}\\\
\leqslant\beta_{0}f(0)+\beta_{X}f(X)\leqslant\\\
\dfrac{\dfrac{f(0)}{\sqrt{-\varepsilon-V^{\prime\prime}(0)}}+\dfrac{f(X)}{\sqrt{-\varepsilon-V^{\prime\prime}(X)}}}{\dfrac{1}{\sqrt{\varepsilon-V^{\prime\prime}(0)}}+\dfrac{1}{\sqrt{-\varepsilon-V^{\prime\prime}(X)}}}.$

This being true for all $\varepsilon>0$ and all non–negative $f(0)$ and
$f(X)$, the lemma is proved. ∎

At last, we can deduce the following:

######  Proposition 4.5.2.

Assume that $H_{0}:(x,p)\mapsto\frac{1}{2}p^{2}+V(x)$ is a Hamiltonian such
that $V:\mathbb{T}^{1}\to\mathbb{R}$ is smooth, non–positive, and verifies
$V^{-1}\\{0\\}=\\{0,X\\}$ for some $X\in\mathbb{T}^{1}\setminus\\{0\\}$.
Assume moreover that $V^{\prime\prime}(0)\neq V^{\prime\prime}(X)$ are both
negative and again

$\alpha:=\frac{\int_{0}^{X}f^{+}(x){\rm d}x}{X}<\frac{\int_{X}^{1}f^{+}(x){\rm
d}x}{1-X}:=\beta.$

Then, using the previous notations, if $(c_{n})_{n>0}$ is a decreasing
sequence converging to $c_{0}$ such that $\rho(c_{n})$ is irrational for all
$n>0$, the family of functions $(u_{1}^{c_{n}})_{n>0}$ does not converge to
$u_{1}^{c_{0}}$.

###### Proof.

Using that a limit of weak KAM solutions is a weak KAM solution and the
previous Proposition 4.5.1, one proves that $(u_{1}^{c_{n}})_{n>0}$ converges
to the unique weak KAM solution $u$, for $H_{c_{0}}$ such that

$\frac{\big{(}\sqrt{-V^{\prime\prime}(0)}\big{)}^{-1}}{\big{(}\sqrt{-V^{\prime\prime}(0)}\big{)}^{-1}+\big{(}\sqrt{-V^{\prime\prime}(X)}\big{)}^{-1}}u(0)+\frac{\big{(}\sqrt{-V^{\prime\prime}(X)}\big{)}^{-1}}{\big{(}\sqrt{-V^{\prime\prime}(0)}\big{)}^{-1}+\big{(}\sqrt{-V^{\prime\prime}(X)}\big{)}^{-1}}u(X)=0.$

Recall that at cohomology $c_{0}$, there is a unique weak KAM solution, up to
constants. Moreover, the condition $\alpha<\beta$ ensures that it is not
possible to have $u(0)=u(X)=0$. Hence $u(0)u(X)<0$. On the contrary, as
$u_{1}^{c_{0}}$ is given by the formula (4.2):

$\forall x\in[0,1],\quad u_{1}^{c_{0}}(x)=\int_{0}^{x}f^{+}(s){\rm
d}s-{c_{0}}x,$

we have $u_{1}^{c_{0}}(0)u_{1}^{c_{0}}(X)=0$.

∎

####  4.5.2 A situation where $u_{1}^{c}\neq U_{0}^{c}$

We come back to Proposition 3.6.13. More precisely, we answer by the negative
the natural question: does the discounted procedure select the same weak KAM
solution in the discrete setting and in the continuous setting?

We now focus our attention on the unique real number
$c_{\frac{1}{2}}\in\mathbb{R}$ such that $\rho(c_{\frac{1}{2}})=\frac{1}{2}$.
Note that $c_{\frac{1}{2}}>0$ and that
${\mathcal{A}}^{*}_{c_{\frac{1}{2}}}\subset
H^{-1}(\\{\alpha(c_{\frac{1}{2}})\\}$. It is actually the upper connected
component of this level set of $H$. Moreover, it can be characterized as
follows:

######  Proposition 4.5.3.

Let $\Phi_{H_{c_{\frac{1}{2}}}}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ denote the
lift of $\varphi_{H_{c_{\frac{1}{2}}}}$ that fixes the point
$(0,-{c_{\frac{1}{2}}})$ and
$P:\mathbb{R}^{2}\to\mathbb{T}^{1}\times\mathbb{R}$ the canonical projection.
Then

${\mathcal{A}}^{*}_{c_{\frac{1}{2}}}=P\Big{(}\big{\\{}(x,p)\in\mathbb{R}^{2},\quad\Phi_{H_{c_{\frac{1}{2}}}}^{2}(x,p)=(x+1,p)\big{\\}}\Big{)}.$

###### Sketch of Proof.

There are many possible ways to tackle this Proposition according to the
property of the system used. The proof is essentially given in [5, Proposition
15] and very much related to [3, Proposition 2] that proves a version of the
result, in arbitrary dimension, by using $C^{0}$–integrability in a
neighborhood of ${\mathcal{A}}^{*}_{c_{\textrm{$\frac{1}{2}$}}}$. It can also
be deduced from the Implicit function Theorem, proving that the right hand
side of the above equality is a manifold.

Let us sketch a proof using a stronger integrability, reminiscent of the
Arnol’d–Liouville Theorem [84]. Let us call $B$ the right hand side of the
equality to prove. A first step is that $B$ is a (potentially partial) graph
over $\mathbb{T}^{1}$. We distinguish three cases remembering that Hamiltonian
orbits are included in level sets of $H$.

  * •

if $H(x,p)<\max V=0$ then the orbit
$\\{\varphi_{H_{c_{\frac{1}{2}}}}^{s}(x,p),\quad s\in\mathbb{R}\\}$ projects
on an interval strictly included in $\mathbb{T}^{1}$ hence $(x,p)\notin B$.

  * •

if $H(x,p)\geqslant\max V$ with $p\leqslant-c_{\frac{1}{2}}$. Let
$(\tilde{x},p)\in\mathbb{R}^{2}$ such that $P(\tilde{x},p)=(x,p)$. Then if we
define for $s\in\mathbb{R}$,
$\Phi_{H_{c_{\frac{1}{2}}}}^{s}(\tilde{x},p)=(\tilde{x}_{s},p_{s})$, one
computes that $s\mapsto\tilde{x}_{s}$ is non–increasing, hence $(x,p)\notin
B$.

  * •

if $H(x,p)\geqslant\max V$ with $p\geqslant-c_{\frac{1}{2}}$, meaning that
$p\geqslant\sqrt{-2V(x)}-c_{\frac{1}{2}}$. We fix here
$\tilde{x}\in\mathbb{R}$ and let $p$ vary. Looking at the Hamiltonian
equations and more precisely computing the time $t_{p}$ such that
$\Phi_{H_{c_{\frac{1}{2}}}}^{t_{p}}(\tilde{x},p)=(\tilde{x}+1,p+1)$ with
equation (4.4), it can be seen that $p\mapsto t_{p}$ is decreasing. Hence
there is at most one $p$ such that $t_{p}=2$.

Now recall that
${\mathcal{A}}^{*}_{c_{\frac{1}{2}}}=\big{\\{}\big{(}x,f_{\alpha(c_{\frac{1}{2}})}^{+}(x)-c_{\frac{1}{2}}\big{)},\
\ x\in\mathbb{T}^{1}\big{\\}}$. We denote by $\mathfrak{A}$ the lift of
${\mathcal{A}}^{*}_{c_{\frac{1}{2}}}$ to $\mathbb{R}^{2}$, that is invariant
under $\Phi_{H_{c_{\frac{1}{2}}}}$. We define the map
$g:\mathbb{R}\to\mathbb{R}$ by

$\forall
x\in\mathbb{R},\quad\Phi_{H_{c_{\frac{1}{2}}}}^{1}\big{(}x,f_{\alpha(c_{\frac{1}{2}})}^{+}(x)-c_{\frac{1}{2}}\big{)}=\big{(}g(x),f_{\alpha(c_{\frac{1}{2}})}^{+}\circ
g(x)-c_{\frac{1}{2}}\big{)}.$

The function $g$ is the lift of a circle diffeomorphism and its rotation
number here is $\rho(c_{\frac{1}{2}})=\frac{1}{2}$. It follows from Poincaré’s
theory of rotation numbers that there exists a real number
$x_{0}\in\mathbb{R}$ such that $g^{2}(x_{0})=x_{0}+1$. Let now
$x\in\mathbb{R}$ be any real number. There is a time $t\in\mathbb{R}$ such
that
$\big{(}x,f_{\alpha(c_{\frac{1}{2}})}^{+}(x)-c_{\frac{1}{2}}\big{)}=\Phi_{H_{c_{\frac{1}{2}}}}^{t}\big{(}x_{0},f_{\alpha(c_{\frac{1}{2}})}^{+}(x_{0})-c_{\frac{1}{2}}\big{)}$.
It follows that

$\Phi_{H_{c_{\frac{1}{2}}}}^{2}\big{(}x,f_{\alpha(c_{\frac{1}{2}})}^{+}(x)-c_{\frac{1}{2}}\big{)}=\Phi^{2}\circ\Phi_{H_{c_{\frac{1}{2}}}}^{t}\big{(}x_{0},f_{\alpha(c_{\frac{1}{2}})}^{+}(x_{0})-c_{\frac{1}{2}}\big{)}\\\
=\Phi^{t}\circ\Phi_{H_{c_{\frac{1}{2}}}}^{2}\big{(}x_{0},f_{\alpha(c_{\frac{1}{2}})}^{+}(x_{0})-c_{\frac{1}{2}}\big{)}\\\
=\Phi^{t}\circ\Phi_{H_{c_{\frac{1}{2}}}}^{2}\big{(}x_{0}+1,f_{\alpha(c_{\frac{1}{2}})}^{+}(x_{0})-c_{\frac{1}{2}}\big{)}=\big{(}x+1,f_{\alpha(c_{\frac{1}{2}})}^{+}(x)-c_{\frac{1}{2}}\big{)}.$

Thus it has been proven that $\mathfrak{A}\subset B$ and that the right hand
side is a partial graph while the left hand side is a full graph. Hence both
terms are equal. ∎

Let us define the map $h:\mathbb{T}^{1}\to\mathbb{R}$ by

$\forall
x\in\mathbb{T}^{1},\quad\varphi_{H_{c_{\frac{1}{2}}}}^{1}\big{(}x,f_{\alpha(c_{\frac{1}{2}})}^{+}(x)-c_{\frac{1}{2}}\big{)}=\big{(}h(x),f_{\alpha(c_{\frac{1}{2}})}^{+}\circ
h(x)-c_{\frac{1}{2}}\big{)}.$

The function $h$ has $g:\mathbb{R}\to\mathbb{R}$ as a lift. It follows from
the previous result and Theorem 3.1.3 that for all $x\in\mathbb{T}^{1}$, the
measure
$\mu_{x}=\frac{1}{2}\Big{(}\delta_{\textrm{$\big{(}x,h(x)\big{)}$}}+\delta_{\textrm{$\big{(}h(x),x\big{)}$}}\Big{)}$
is a discrete Mather measure.

######  Lemma 4.5.4.

Assume that $u_{1}^{c_{\frac{1}{2}}}=U_{0}^{c_{\frac{1}{2}}}$, then for all
$x\in\mathbb{T}^{1}$,
$u_{1}^{c_{\frac{1}{2}}}(x)=-u_{1}^{c_{\frac{1}{2}}}\circ g(x)$.

###### Proof.

From the definitions of $u_{1}^{c_{\frac{1}{2}}}$ and
$U_{0}^{c_{\frac{1}{2}}}$, and from the previous discussions result that

$\forall
x\in\mathbb{T}^{1},\quad\int_{\mathbb{T}^{1}}u_{1}^{c_{\frac{1}{2}}}(s){\rm
d}\pi_{1*}\mu_{x}(s)=\frac{1}{2}\Big{(}u_{1}^{c_{\frac{1}{2}}}(x)+u_{1}^{c_{\frac{1}{2}}}\circ
g(x)\Big{)}\leqslant 0.$ (4.8)

Moreover, using Proposition 3.6.12 and the fact that there is a single
classical Mather measure, we obtain that

$\int_{\mathbb{T}^{1}}u_{1}^{c_{\frac{1}{2}}}(x){\rm
d}\pi_{*}\mu^{*}_{c_{\frac{1}{2}}}(x)=0.$

Recalling the definition of $\mu^{*}_{c_{\frac{1}{2}}}$ we find:

$0=\int_{\mathbb{T}^{1}}u_{1}^{c_{\frac{1}{2}}}(x){\rm
d}\pi_{*}\mu^{*}_{c_{\frac{1}{2}}}(x)=\int_{0}^{2}u_{1}^{c_{\frac{1}{2}}}\Big{(}\pi\circ\varphi_{H_{c_{\frac{1}{2}}}}^{s}\big{(}0,f_{\alpha(c_{\frac{1}{2}})}^{+}(0)-c_{\frac{1}{2}}\big{)}\Big{)}{\rm
d}s\\\
=\int_{0}^{1}\Big{[}u_{1}^{c_{\frac{1}{2}}}\Big{(}\pi\circ\varphi_{H_{c_{\frac{1}{2}}}}^{s}\big{(}0,f_{\alpha(c_{\frac{1}{2}})}^{+}(0)-c_{\frac{1}{2}}\big{)}\Big{)}+u_{1}^{c_{\frac{1}{2}}}\Big{(}\pi\circ\varphi_{H_{c_{\frac{1}{2}}}}^{s+1}\big{(}0,f_{\alpha(c_{\frac{1}{2}})}^{+}(0)-c_{\frac{1}{2}}\big{)}\Big{)}\Big{]}{\rm
d}s\\\ =\int_{0}^{1}2\int_{\mathbb{T}^{1}}u_{1}^{c_{\frac{1}{2}}}(x){\rm
d}\pi_{1*}\mu_{\pi\circ\varphi_{H_{c_{\frac{1}{2}}}}^{s}\textrm{$\big{(}0,f_{\alpha(c_{\frac{1}{2}})}^{+}(0)-c_{\frac{1}{2}}\big{)}$}}(x){\rm
d}s\leqslant 0.$

It follows that all inequalities in (4.8) are equalities, hence the result.

∎

It is deduced that under the hypotheses of the previous Lemma,
$u_{1}^{c_{\frac{1}{2}}}(x)$ and $u_{1}^{c_{\frac{1}{2}}}\circ g(x)$ must have
opposite signs for all $x$. An example in which it is not the case is provided
in [14, Appendix A.2.]. We give below a different simple situation where this
cannot happen:

######  Proposition 4.5.5.

Let $V:\mathbb{T}^{1}\to\mathbb{R}$ be a non–constant $\frac{1}{2}$–periodic
function. Then for the associated Hamiltonian $H_{c_{\frac{1}{2}}}$, it holds
$u_{1}^{c_{\frac{1}{2}}}\neq U_{0}^{c_{\frac{1}{2}}}$.

###### Proof.

Indeed, in this case, both functions $u_{1}^{c_{\frac{1}{2}}}$ and
$U_{0}^{c_{\frac{1}{2}}}$ are also $\frac{1}{2}$–periodic (this follows from
the uniqueness of the solutions to the discounted equations that hence must be
$\frac{1}{2}$–periodic). Consequently if they coincide, the previous lemma
tells us that they must be identically $0$. This is clearly not the case. ∎

###  4.6 Concluding example

Due to the simple structure of the Aubry set and of the set of minimizing
measures in all previous examples, one can check that $u_{1}^{c}$ and
$v_{1}^{c}$ form a conjugate pair “up to a constant”. More precisely, the
modified pair $\big{(}u_{1}^{c}-u_{1}^{c}(0),v_{1}^{c}-v_{1}^{c}(0)\big{)}$ is
a conjugate pair in all the previous examples. One may wonder if this is
always the case.

As a matter of fact, the answer is again negative. We propose here a slightly
(but not too much) sophisticated example shedding light on this fact. The
computations are not carried on fully and left to the reader. We hope the
previous examples give enough insight to make what follows quite
straightforward.

Once again we consider that $H_{0}:(x,p)\mapsto\frac{1}{2}p^{2}+V(x)$ is a
Hamiltonian such that $V:\mathbb{T}^{1}\to\mathbb{R}$ is smooth, non–positive,
and verifies $V^{-1}\\{0\\}=\\{0,X_{1},X_{2}\\}$ for some
$0<X_{1}<X_{2}<1\in\mathbb{T}^{1}$. Assume again that

$\underbrace{\frac{\int_{0}^{X_{1}}f^{+}(x){\rm
d}x}{X_{1}}}_{=\alpha}<\underbrace{\frac{\int_{X_{1}}^{X_{2}}f^{+}(x){\rm
d}x}{X_{2}-X_{1}}}_{=\beta}<\underbrace{\frac{\int_{X_{2}}^{1}f^{+}(x){\rm
d}x}{1-X_{2}}}_{=\gamma}.$

If $c\in\mathbb{R}$ we again denote by
$H_{c}:(x,p)\mapsto\frac{1}{2}(p+c)^{2}+V(x)$ and respectively by $u_{1}^{c}$
and $v_{1}^{c}$ the negative and positive weak KAM solutions selected by the
discounted procedure for the time–$1$ minimal action functional associated to
$H_{c}$. Reasoning as in the previous sections, one checks that for
$\alpha<c<\min(\beta,c_{0})$, the function $u_{1}^{c}$ verifies
$u_{1}^{c}(0)=u_{1}^{c}(X_{2})=0$ and $u_{1}^{c}(X_{1})<0$ while the function
$v_{1}^{c}$ verifies $v_{1}^{c}(X_{1})=v_{1}^{c}(X_{2})=0$ and
$v_{1}^{c}(0)>0$. Hence $u_{1}^{c}$ and $v_{1}^{c}$ are not conjugated “up to
a constant”.

##  Chapter 5 Twist maps

The results of discrete weak KAM theory will now be applied to the particular
and founding case of Exact Conservative Twist Maps of the annulus. Excellent
surveys on the subject are [12, 21, 148] and we refer to those references for
classical results that we leave without proof. The aim of this section is to
present some of the results of [14, 15, 16] obtained in collaboration with
Marie–Claude Arnaud, shedding light on the structure of weak KAM solutions and
minimizing orbits for exact conservative twist maps. Before giving the precise
definition of an exact conservative twist map, let us emphasize that the
examples of the previous section are closely related to such transformations.
Indeed, for a Tonelli Hamiltonian on $T^{*}\mathbb{T}^{1}$, the Hamiltonian
flow $\varphi^{s}_{H}$ is an exact conservative twist map for small times
$s>0$. Hence the time $1$ map $\varphi^{1}_{H}$ is a composition of a finite
number of conservative twist maps.

###  5.1 Definitions and variational structure

In the rest of this section $\mathbb{T}^{1}=\mathbb{R}/\mathbb{Z}$ is the
circle, the $2$–dimensional annulus is denoted by
$\mathbb{A}=T^{*}\mathbb{T}^{1}=\mathbb{T}^{1}\times\mathbb{R}$. The points of
that annulus are denoted by $(\theta,r)\in\mathbb{A}$. Throughout this section
we will often consider objects coming from $\mathbb{A}$ lifted to
$\mathbb{R}^{2}$, its universal cover, or from $\mathbb{T}^{1}$, lifted to
$\mathbb{R}$. When done so, a $\sim$ will be added to the original name. For
example if $g:\mathbb{T}^{1}\to\mathbb{R}$ is any function then
$\tilde{g}:\mathbb{R}\to\mathbb{R}$ is the lift of $g$.

When dealing with products, $\mathbb{T}^{1}\times\mathbb{T}^{1}$,
$\mathbb{T}^{1}\times\mathbb{R}$, or $\mathbb{R}\times\mathbb{R}$, the
notations $\pi_{1}$ and $\pi_{2}$ stand for the projections on the first and
second variable.

####  5.1.1 Definition and Birkhoff’s theorem

######  Definition 5.1.1.

An exact conservative twist map of the annulus (abbreviated ECTM) is a
$C^{1}$–diffeomorphism $f:\mathbb{A}\to\mathbb{A}$ such that

  1. 1.

$f$ is isotopic to the identity map.

  2. 2.

$f$ is exact symplectic: by denoting
$f(\theta,r)=\big{(}\Theta(\theta,r),R(\theta,r)\big{)}$ the $1$–form $R{\rm
d}\Theta-r{\rm d}\theta$ is exact.

  3. 3.

$f$ twists verticals to the right: if
$\tilde{f}=(\widetilde{\Theta},R):\mathbb{R}^{2}\to\mathbb{R}^{2}$ is a lift
of $f$ to the universal cover of $\mathbb{A}$ then for all
$\tilde{\theta}\in\mathbb{R}$, the map
$r\mapsto\widetilde{\Theta}(\tilde{\theta},r)$ is an increasing
$C^{1}$–diffeomorphism of $\mathbb{R}$.

######  Remark 5.1.2.

  1. 1.

The first property means that there is a continuous path of diffeomorphisms of
the annulus $(f_{s})_{s\in[0,1]}$ such that $f_{0}$ is the identity map and
$f_{1}=f$. In other topological words, $f$ preserves both ends of the annulus
in the sense that uniformly in $\theta$,
$\lim\limits_{r\to+\infty}R(\theta,r)=+\infty$ and
$\lim\limits_{r\to-\infty}R(\theta,r)=-\infty$.

  2. 2.

The second point means that there is a function $S:\mathbb{A}\to\mathbb{R}$,
called generating function, such that

${\rm d}S=R{\rm d}\Theta-r{\rm
d}\theta=R\big{(}\frac{\partial\Theta}{\partial\theta}{\rm
d}\theta+\frac{\partial\Theta}{\partial r}{\rm d}r\big{)}-r{\rm d}\theta.$
(5.1)

The generating function is defined up to a constant.

Another way of formulating this is to say that if we denote by $\lambda=r{\rm
d}\theta$ (the $1$–form called Liouville form) then $f^{*}\lambda-\lambda$ is
exact:

This has two major implications.

    1. (i)

(proved by using Stokes’ furmula) if $\mathcal{C}$ is a $C^{1}$ essential
circle, meaning an injective $C^{1}$ closed curve going around the cylinder,
then the algebraic area between $\mathcal{C}$ and $f(\mathcal{C})$ is $0$.
This means that points are not globally shifted up or down by $f$ in the
annulus and that it is worth looking for invariant compact sets.

    2. (ii)

The second is that $f$ preserves the canonical symplectic $2$–form which is
here the Lebesgue area form. Indeed, as this symplectic form is ${\rm
d}\lambda={\rm d}r\wedge{\rm d}\theta$, one finds that

$0={\rm d}{\rm d}S={\rm d}R\wedge{\rm d}\Theta-{\rm d}r\wedge{\rm
d}\theta=f^{*}({\rm d}r\wedge{\rm d}\theta)-{\rm d}r\wedge{\rm d}\theta.$

  3. 3.

The last point implies that for $\tilde{\theta}_{0}\in\mathbb{R}$ and
$\widetilde{\Theta}_{0}\in\mathbb{R}$ there is a unique $r\in\mathbb{R}$ such
that $\widetilde{\Theta}(\tilde{\theta}_{0},r)=\widetilde{\Theta}_{0}$. It
means that
$(\tilde{\theta},r)\mapsto\big{(}\tilde{\theta},\widetilde{\Theta}(\tilde{\theta},r)\big{)}$
is a $C^{1}$–diffeomorphism that plays the role of Legendre transform and
$(\tilde{\theta},\widetilde{\Theta})$ will serve as coordinates of
$\mathbb{R}^{2}$. The latter powerful idea is behind all the variational
structure of twist maps.

The twist condition implies that

$\forall(\theta_{0},r_{0})\in\mathbb{A},\quad\frac{\partial\Theta}{\partial
r}(\theta_{0},r_{0})>0.$ (5.2)

In fact, it is sometimes replaced by the stronger uniform condition that there
exists $\varepsilon>0$ such that $\frac{\partial\Theta}{\partial
r}(\theta_{0},r_{0})>\varepsilon$ for all $(\theta_{0},r_{0})\in\mathbb{A}$.

One can at this stage state an important dynamical property of ECTM on
invariant curves (see [50]):

######  Theorem 5.1.3 (Birkhoff).

Let ${\mathcal{C}}\subset\mathbb{A}$ be a continuous essential circle
invariant by $f$, i.e. a continuous embedding of $\mathbb{T}^{1}$ that is not
homotopic to a point and such that $f({\mathcal{C}})={\mathcal{C}}$. Then
there exists a Lipschitz function $g:\mathbb{T}^{1}\to\mathbb{R}$ such that
${\mathcal{C}}=\big{\\{}\big{(}\theta,g(\theta)\big{)},\ \
\theta\in\mathbb{T}^{1}\big{\\}}$ is the graph of $g$.

This important Theorem uses all properties of the ECTM. Indeed in [130,
Proposition 5.13] are examples of non conservative twist maps leaving
invariant essential circles that are not graphs. It was generalized a few
decades later to higher dimensional settings by Arnaud [6].

####  5.1.2 The generating function, properties and consequences

Until the end of this section, we choose once and for all a generating
function $S:\mathbb{A}\to\mathbb{R}$ (see Remark 5.1.2),
${\widetilde{S}}:\mathbb{R}^{2}\to\mathbb{R}$ is the lift of $S$ and let us
choose ${\tilde{f}}=(\widetilde{\Theta},R):\mathbb{R}^{2}\to\mathbb{R}^{2}$ a
lift of $f$. By the twist condition (Definition 5.1.1 point $3$), the map
${\mathcal{L}}:(\tilde{\theta},r)\mapsto\big{(}\tilde{\theta},\widetilde{\Theta}(\tilde{\theta},r)\big{)}$
is a $C^{1}$–diffeomorphsim of $\mathbb{R}^{2}$ (see the links with Definition
2.5.2). In the sequel, $(\tilde{\theta},\widetilde{\Theta})$ are
systematically used as coordinates, meaning that the function
$\widetilde{S}\circ{\mathcal{L}}^{-1}$ is considered instead of
$\widetilde{S}$. For readability issues, it is still written
$\widetilde{S}(\tilde{\theta},\widetilde{\Theta})$.

The properties of $f$ translate into the following features of $S$:

######  Proposition 5.1.4.

The function $S$, through $\widetilde{S}$, verifies the following:

  1. 1.

The function $\widetilde{S}$ is $C^{2}$ and periodic:

$\forall(\tilde{\theta},\widetilde{\Theta})\in\mathbb{R}^{2},\quad\widetilde{S}(\tilde{\theta}+1,\widetilde{\Theta}+1)=\widetilde{S}(\tilde{\theta},\widetilde{\Theta}).$

  2. 2.

For all $(\tilde{\theta},\widetilde{\Theta},r,R)\in{\mathbb{R}}^{4}$,

$\tilde{f}(\tilde{\theta},r)=(\widetilde{\Theta},R)\iff\begin{cases}r=-\dfrac{\partial\widetilde{S}}{\partial\tilde{\theta}}(\tilde{\theta},\widetilde{\Theta}),\vspace{1mm}\\\
R=\dfrac{\partial\widetilde{S}}{\partial\widetilde{\Theta}}(\tilde{\theta},\widetilde{\Theta}).\end{cases}$
(5.3)

  3. 3.

The twist condition translates as follows:

     * •

for $\tilde{\theta}\in\mathbb{R}$ fixed, the map
$\widetilde{\Theta}\mapsto\frac{\partial\widetilde{S}}{\partial\tilde{\theta}}(\tilde{\theta},\widetilde{\Theta})$
is a decreasing $C^{1}$–diffeomorphism of $\mathbb{R}$;

     * •

for $\widetilde{\Theta}\in\mathbb{R}$ fixed, the map
$\tilde{\theta}\mapsto\frac{\partial\widetilde{S}}{\partial\widetilde{\Theta}}(\tilde{\theta},\widetilde{\Theta})$
is a decreasing $C^{1}$–diffeomorphism of $\mathbb{R}$.

Let us comment on the previous proposition.

######  Remark 5.1.5.

  1. 1.

It can actually be established that given a function $\widetilde{S}$
satisfying the three points of the previous proposition the associated
function $\tilde{f}$ is a lift of an ECTM.

  2. 2.

The second point of Proposition 5.1.4 can be directly read on equation (5.1).

  3. 3.

The two items of point $3$ are equivalent. The first one results from the fact
that the map
$\Theta\mapsto-\frac{\partial\widetilde{S}}{\partial\tilde{\theta}}(\tilde{\theta},\widetilde{\Theta})$
is the inverse of the map $r\mapsto\widetilde{\Theta}(\tilde{\theta},r)$.

The second one is an emanation of the fact that if $f$ is an ECTM that twists
verticals to the right then $f^{-1}$ is an ECTM that twists verticals to the
left. A direct consequence of any of these two facts is that

$\forall(\tilde{\theta}_{0},\widetilde{\Theta}_{0})\in\mathbb{R}^{2},\quad\frac{\partial^{2}\widetilde{S}}{\partial\tilde{\theta}\partial\widetilde{\Theta}}(\tilde{\theta}_{0},\widetilde{\Theta}_{0})<0.$
(5.4)

Another important aftermath is that $\widetilde{S}$ is superlinear, meaning
that

$\lim_{|\tilde{\theta}-\widetilde{\Theta}|\to+\infty}\frac{\widetilde{S}(\tilde{\theta},\widetilde{\Theta})}{|\tilde{\theta}-\widetilde{\Theta}|}=+\infty.$
(5.5)

It is proven in [140] (see also [141, 21]) that finite compositions of ECTM
also possess a similar generating function. Hence all variational results
relying solely on the generating function and its minimizers also apply to
such finite compositions of ECTM.

Let us now define the notion of minimizing chains and sequences.

######  Definition 5.1.6.

  1. 1.

Let $a<b$ be two integers such that $b-a>1$ and
$(\tilde{\theta}_{i})_{a\leqslant i\leqslant b}\in\mathbb{R}^{b-a+1}$. The
chain $(\tilde{\theta}_{i})_{a\leqslant i\leqslant b}$ is termed minimizing if

$\sum_{i=a}^{b-1}\widetilde{S}(\tilde{\theta}_{i},\tilde{\theta}_{i+1})=\min_{\begin{subarray}{c}\\{x_{i}\in\mathbb{R},\
a\leqslant i\leqslant b\\}\\\
x_{a}=\tilde{\theta}_{a},x_{b}=\tilde{\theta}_{b}\end{subarray}}\sum_{i=a}^{b-1}\widetilde{S}(x_{i},x_{i+1}).$

  2. 2.

An infinite sequence $(\tilde{\theta}_{i})_{-\infty<i\leqslant b}$ or
$(\tilde{\theta}_{i})_{a\leqslant i<+\infty}$ or
$(\tilde{\theta}_{i})_{i\in\mathbb{Z}}$ is minimizing if all its finite
subchains are minimizing.

Note that if $(\tilde{\theta}_{i})_{a\leqslant i\leqslant
b}\in\mathbb{R}^{b-a+1}$ is minimizing between $\tilde{\theta}_{a}$ and
$\tilde{\theta}_{b}$ and if $a\leqslant
a^{\prime}<b^{\prime}-1<b^{\prime}\leqslant b$ then the subchain
$(\tilde{\theta}_{i})_{a^{\prime}\leqslant i\leqslant
b^{\prime}}\in\mathbb{R}^{b^{\prime}-a^{\prime}+1}$ is minimizing between
$\tilde{\theta}_{a^{\prime}}$ and $\tilde{\theta}_{b^{\prime}}$ which
justifies the second definition.

The first order necessary condition for a chain to be minimizing translates as
follows: let $(\tilde{\theta}_{i})_{i\in I}$ be a minimizing chain or sequence
and $i_{0}\in I$ not be an extremity of $I$, then

$\frac{\partial\widetilde{S}}{\partial\widetilde{\Theta}}(\tilde{\theta}_{i_{0}-1},\tilde{\theta}_{i_{0}})+\frac{\partial\widetilde{S}}{\partial\tilde{\theta}}(\tilde{\theta}_{i_{0}},\tilde{\theta}_{i_{0}+1})=0.$
(5.6)

This, together with equation (5.3), yields the following proposition:

######  Proposition 5.1.7.

Let $(\tilde{\theta}_{i})_{i\in I}$ be a minimizing chain or sequence and for
all $i\in I$, let

$r_{i}=-\frac{\partial\widetilde{S}}{\partial\tilde{\theta}}(\tilde{\theta}_{i},\tilde{\theta}_{i+1})=\frac{\partial\widetilde{S}}{\partial\widetilde{\Theta}}(\tilde{\theta}_{i-1},\tilde{\theta}_{i}),$

(where only the well defined term is taken if $i$ is an extremity of $I$).
Then $(\tilde{\theta}_{i},r_{i})_{i\in I}$ is a piece of orbit of $\tilde{f}$
in the sense that for all $i\in I$ such that $i+1\in I$, then
$\tilde{f}(\tilde{\theta}_{i},r_{i})=(\tilde{\theta}_{i+1},r_{i+1})$.

In particular, a minimizing chain is uniquely determined by two consecutive
terms $(\tilde{\theta}_{i},\tilde{\theta}_{i+1})$.

Let us then introduce the notion of crossing and its consequences:

######  Definition 5.1.8.

Two chains $(\tilde{\theta}_{i})_{i\in I}$ and
$(\tilde{\theta}^{\prime}_{i})_{i\in I}$ are said to cross

  * •

at some index $i_{0}\in I$ if
$\tilde{\theta}_{i_{0}}=\tilde{\theta}^{\prime}_{i_{0}}$;

  * •

between two indices $i_{0}\in I$ and $i_{0}+1\in I$ if
$(\tilde{\theta}_{i_{0}}-\tilde{\theta}^{\prime}_{i_{0}})(\tilde{\theta}_{i_{0}+1}-\tilde{\theta}^{\prime}_{i_{0}+1})<0$.

######  Remark 5.1.9.

If $(\tilde{\theta}_{i})_{i\in I}$ and $(\tilde{\theta}^{\prime}_{i})_{i\in
I}$ are distinct minimizing chains that cross at some index $i_{0}\in I$ that
is an interior index, then the twist condition implies that

$(\tilde{\theta}_{i_{0}-1}-\tilde{\theta}^{\prime}_{i_{0}-1})(\tilde{\theta}_{i_{0}+1}-\tilde{\theta}^{\prime}_{i_{0}+1})<0.$
(5.7)

Indeed, let
$r_{i_{0}}=-\frac{\partial\widetilde{S}}{\partial\tilde{\theta}}(\tilde{\theta}_{i_{0}},\tilde{\theta}_{i_{0}+1})$
and
$r^{\prime}_{i_{0}}=-\frac{\partial\widetilde{S}}{\partial\tilde{\theta}}(\tilde{\theta}_{i_{0}}^{\prime},\tilde{\theta}_{i_{0}+1}^{\prime})$.
Necessarily $r_{i_{0}}\neq r_{i_{0}}^{\prime}$ because otherwise
$(\tilde{\theta}_{i})_{i\in I}$ and $(\tilde{\theta}^{\prime}_{i})_{i\in I}$
are the projections of the same orbit of $\tilde{f}$ (Proposition 5.1.7).
Assume then for example that $r_{i_{0}}>r^{\prime}_{i_{0}}$. We deduce from
the twist condition that
$\tilde{\theta}_{i_{0}+1}>\tilde{\theta}^{\prime}_{i_{0}+1}$ and that
$\tilde{\theta}_{i_{0}-1}<\tilde{\theta}_{i_{0}-1}^{\prime}$.

Inequality (5.7) is often taken as the definition of crossing at $i_{0}$.

One can then state Aubry & Le Daeron’s Fundamental Lemma:

######  Lemma 5.1.10.

Let
$(\tilde{\theta},\tilde{\theta}^{\prime},\widetilde{\Theta},\widetilde{\Theta}^{\prime})\in\mathbb{R}^{4}$
such that
$(\tilde{\theta}-\tilde{\theta}^{\prime})(\widetilde{\Theta}-\widetilde{\Theta}^{\prime})<0$.
Then

$\widetilde{S}(\tilde{\theta},\widetilde{\Theta})+\widetilde{S}(\tilde{\theta}^{\prime},\widetilde{\Theta}^{\prime})>\widetilde{S}(\tilde{\theta},\widetilde{\Theta}^{\prime})+\widetilde{S}(\tilde{\theta}^{\prime},\widetilde{\Theta}).$

###### Proof.

It follows from the chain of equalities (in which we use the notations
$\widetilde{\Theta}_{t}=t\widetilde{\Theta}^{\prime}+(1-t)\widetilde{\Theta}$
and $\tilde{\theta}_{t}=t\tilde{\theta}^{\prime}+(1-t)\tilde{\theta}$),

$\widetilde{S}(\tilde{\theta},\widetilde{\Theta}^{\prime})-\widetilde{S}(\tilde{\theta},\widetilde{\Theta})+\widetilde{S}(\tilde{\theta}^{\prime},\widetilde{\Theta})-\widetilde{S}(\tilde{\theta}^{\prime},\widetilde{\Theta}^{\prime})\\\
=\int_{0}^{1}\Big{[}\frac{\partial\widetilde{S}}{\partial\widetilde{\Theta}}(\tilde{\theta},\widetilde{\Theta}_{t})-\frac{\partial\widetilde{S}}{\partial\widetilde{\Theta}}(\tilde{\theta}^{\prime},\widetilde{\Theta}_{t})\Big{]}{\rm
d}t\cdot(\widetilde{\Theta}^{\prime}-\widetilde{\Theta})\\\
=\int_{0}^{1}\int_{0}^{1}\frac{\partial^{2}\widetilde{S}}{\partial\tilde{\theta}\partial\widetilde{\Theta}}(\tilde{\theta}_{s},\widetilde{\Theta}_{t}){\rm
d}s{\rm
d}t\cdot(\tilde{\theta}-\tilde{\theta}^{\prime})(\widetilde{\Theta}^{\prime}-\widetilde{\Theta})<0.$

The last inequality is a consequence of the crossing hypothesis and of (5.4).
∎

We may now state Aubry’s Non–crossing Lemma:

######  Proposition 5.1.11.

Let $(\tilde{\theta}_{i})_{i\in I}$ and $(\tilde{\theta}^{\prime}_{i})_{i\in
I}$ be two distinct minimizing chains. Then one of the following holds

  * •

$(\tilde{\theta}_{i})_{i\in I}$ and $(\tilde{\theta}^{\prime}_{i})_{i\in I}$
don’t cross,

  * •

$(\tilde{\theta}_{i})_{i\in I}$ and $(\tilde{\theta}^{\prime}_{i})_{i\in I}$
cross exactly once,

  * •

$I=[a,b]$ is a finite interval and $(\tilde{\theta}_{i})_{i\in I}$ and
$(\tilde{\theta}^{\prime}_{i})_{i\in I}$ cross exactly twice, at $a$ and $b$.

In the last case, both $(\tilde{\theta}_{i})_{i\in I}$ and
$(\tilde{\theta}^{\prime}_{i})_{i\in I}$ are maximal in the sense that neither
one is a strict subchain of a minimizing chain.

###### Proof.

To prove this proposition, let us assume that $(\tilde{\theta}_{i})_{i\in I}$
and $(\tilde{\theta}^{\prime}_{i})_{i\in I}$ cross (at least) twice and that
one of those crossing is not an extremity of $I$. There are several cases to
deal with, we only cover two of them and let the other ones as an exercise as
the ideas are the same.

First case: there are $\alpha<\beta$ such that $(\tilde{\theta}_{i})_{i\in I}$
and $(\tilde{\theta}^{\prime}_{i})_{i\in I}$ cross between $\alpha$ and
$\alpha+1$ and between $\beta$ and $\beta+1$. Then define two chains
$(\hat{\theta}_{i})_{i\in[\alpha,\beta+1]}$ and
$(\hat{\theta}^{\prime}_{i})_{i\in[\alpha,\beta+1]}$ as follows:

$\hat{\theta}_{i}=\begin{cases}\tilde{\theta}_{i}^{\prime}\ \ \textrm{if}\ \
i\in[\alpha+1,\beta]\\\ \tilde{\theta}_{i}\ \ \textrm{if}\ \
i\in\\{\alpha,\beta+1\\};\end{cases}$
$\hat{\theta}^{\prime}_{i}=\begin{cases}\tilde{\theta}_{i}\ \ \textrm{if}\ \
i\in[\alpha+1,\beta]\\\ \tilde{\theta}_{i}^{\prime}\ \ \textrm{if}\ \
i\in\\{\alpha,\beta+1\\}.\end{cases}$

Note that $(\tilde{\theta}_{i})_{i\in[\alpha,\beta+1]}$ and
$(\hat{\theta}_{i})_{i\in[\alpha,\beta+1]}$ have same endpoints and so do
$(\tilde{\theta}^{\prime}_{i})_{i\in[\alpha,\beta+1]}$ and
$(\hat{\theta}^{\prime}_{i})_{i\in[\alpha,\beta+1]}$ hence

$\sum_{i=\alpha}^{\beta}\widetilde{S}(\tilde{\theta}_{i},\tilde{\theta}_{i+1})\leqslant\sum_{i=\alpha}^{\beta}\widetilde{S}(\hat{\theta}_{i},\hat{\theta}_{i+1});\quad\textrm{and}\quad\sum_{i=\alpha}^{\beta}\widetilde{S}(\tilde{\theta}^{\prime}_{i},\tilde{\theta}^{\prime}_{i+1})\leqslant\sum_{i=\alpha}^{\beta}\widetilde{S}(\hat{\theta}^{\prime}_{i},\hat{\theta}^{\prime}_{i+1}).$

Moreover

$\sum_{i=\alpha}^{\beta}\widetilde{S}(\tilde{\theta}_{i},\tilde{\theta}_{i+1})+\sum_{i=\alpha}^{\beta}\widetilde{S}(\tilde{\theta}^{\prime}_{i},\tilde{\theta}^{\prime}_{i+1})-\sum_{i=\alpha}^{\beta}\widetilde{S}(\hat{\theta}_{i},\hat{\theta}_{i+1})-\sum_{i=\alpha}^{\beta}\widetilde{S}(\hat{\theta}^{\prime}_{i},\hat{\theta}^{\prime}_{i+1})\\\
=\widetilde{S}(\tilde{\theta}_{\alpha},\tilde{\theta}_{\alpha+1})+\widetilde{S}(\tilde{\theta}_{\beta},\tilde{\theta}_{\beta+1})+\widetilde{S}(\tilde{\theta}^{\prime}_{\alpha},\tilde{\theta}^{\prime}_{\alpha+1})+\widetilde{S}(\tilde{\theta}^{\prime}_{\beta},\tilde{\theta}^{\prime}_{\beta+1})\\\
-\widetilde{S}(\tilde{\theta}_{\alpha},\tilde{\theta}^{\prime}_{\alpha+1})-\widetilde{S}(\tilde{\theta}_{\beta},\tilde{\theta}^{\prime}_{\beta+1})-\widetilde{S}(\tilde{\theta}^{\prime}_{\alpha},\tilde{\theta}_{\alpha+1})-\widetilde{S}(\tilde{\theta}^{\prime}_{\beta},\tilde{\theta}_{\beta+1})>0;$
(5.8)

where the last inequality is obtained by two applications of Aubry’s
Fundamental Lemma 5.1.10. This contradicts either the fact that
$(\tilde{\theta}_{i})_{i\in[\alpha,\beta+1]}$ is minimizing or that
$(\tilde{\theta}^{\prime}_{i})_{i\in[\alpha,\beta+1]}$ is minimizing.

Second case: there are $\alpha<\beta$ such that $(\tilde{\theta}_{i})_{i\in
I}$ and $(\tilde{\theta}^{\prime}_{i})_{i\in I}$ cross at $\alpha$ and at
$\beta$ and such that $\alpha-1\in I$.

Then define two chains $(\hat{\theta}_{i})_{i\in[\alpha-1,\beta]}$ and
$(\hat{\theta}^{\prime}_{i})_{i\in[\alpha-1,\beta]}$ as follows:

$\hat{\theta}_{i}=\begin{cases}\tilde{\theta}_{i}^{\prime}\ \ \textrm{if}\ \
i\in[\alpha,\beta]\\\ \tilde{\theta}_{i}\ \ \textrm{if}\ \
i\in\\{\alpha-1,\alpha\\};\end{cases}$
$\hat{\theta}^{\prime}_{i}=\begin{cases}\tilde{\theta}_{i}\ \ \textrm{if}\ \
i\in[\alpha,\beta]\\\ \tilde{\theta}_{i}^{\prime}\ \ \textrm{if}\ \
i\in\\{\alpha-1,\alpha\\}.\end{cases}$

We purposely insist on the fact that
$\hat{\theta}_{\alpha}=\hat{\theta}^{\prime}_{\alpha}=\tilde{\theta}_{\alpha}=\tilde{\theta}^{\prime}_{\alpha}$.
Note that $(\tilde{\theta}_{i})_{i\in[\alpha-1,\beta]}$ and
$(\hat{\theta}_{i})_{i\in[\alpha-1,\beta]}$ have same endpoints and so do
$(\tilde{\theta}^{\prime}_{i})_{i\in[\alpha-1,\beta]}$ and
$(\hat{\theta}^{\prime}_{i})_{i\in[\alpha-1,\beta]}$. Hence

$\sum_{i=\alpha-1}^{\beta-1}\widetilde{S}(\tilde{\theta}_{i},\tilde{\theta}_{i+1})\leqslant\sum_{i=\alpha-1}^{\beta-1}\widetilde{S}(\hat{\theta}_{i},\hat{\theta}_{i+1});\quad\textrm{and}\quad\sum_{i=\alpha-1}^{\beta-1}\widetilde{S}(\tilde{\theta}^{\prime}_{i},\tilde{\theta}^{\prime}_{i+1})\leqslant\sum_{i=\alpha-1}^{\beta-1}\widetilde{S}(\hat{\theta}^{\prime}_{i},\hat{\theta}^{\prime}_{i+1}).$

Moreover the definitions of $(\hat{\theta}_{i})_{i\in[\alpha-1,\beta]}$ and
$(\hat{\theta}^{\prime}_{i})_{i\in[\alpha-1,\beta]}$ yield

$\sum_{i=\alpha}^{\beta}\widetilde{S}(\tilde{\theta}_{i},\tilde{\theta}_{i+1})+\sum_{i=\alpha}^{\beta}\widetilde{S}(\tilde{\theta}^{\prime}_{i},\tilde{\theta}^{\prime}_{i+1})-\sum_{i=\alpha}^{\beta}\widetilde{S}(\hat{\theta}_{i},\hat{\theta}_{i+1})-\sum_{i=\alpha}^{\beta}\widetilde{S}(\hat{\theta}^{\prime}_{i},\hat{\theta}^{\prime}_{i+1})=0.$

It follows that both inequalities above are equalities and that
$(\tilde{\theta}_{i})_{i\in[\alpha-1,\beta]}$ and
$(\hat{\theta}_{i})_{i\in[\alpha-1,\beta]}$ are both minimizing. As they
coincide for both indices $\alpha-1$ and $\alpha$, they are equal (see
Proposition 5.1.7). The same argument shows that
$(\tilde{\theta}^{\prime}_{i})_{i\in[\alpha-1,\beta]}$ and
$(\hat{\theta}^{\prime}_{i})_{i\in[\alpha-1,\beta]}$ are equal, and finally we
have proved that $(\tilde{\theta}_{i})_{i\in[\alpha-1,\beta]}$ and
$(\tilde{\theta}^{\prime}_{i})_{i\in[\alpha-1,\beta]}$ are equal. This in turn
implies that $(\tilde{\theta}_{i})_{i\in
I}=(\tilde{\theta}^{\prime}_{i})_{i\in I}$ which is a contradiction.

The last assertion of the Proposition remains to be proven. The previous
argument could be adapted here. Let us though propose another one. Assume by
contradiction that $(\tilde{\theta}_{i})_{i\in[a,b]}$ and
$(\tilde{\theta}^{\prime}_{i})_{i\in[a-1,b]}$ are minimizing and verify
$\tilde{\theta}_{a}=\tilde{\theta}^{\prime}_{a}$,
$\tilde{\theta}_{b}=\tilde{\theta}^{\prime}_{b}$ (the other cases are treated
the same way). Then by the minimization hypothesis it is inferred that

$\sum_{i=a}^{b-1}\widetilde{S}(\tilde{\theta}_{i},\tilde{\theta}_{i+1})=\sum_{i=a}^{b-1}\widetilde{S}(\tilde{\theta}^{\prime}_{i},\tilde{\theta}^{\prime}_{i+1}).$

Moreover, note that the chain
$(\tilde{\theta}_{a-1}^{\prime},\tilde{\theta}_{a},\tilde{\theta}_{a+1})$ is
not minimizing, otherwise there would be equality
$\tilde{\theta}_{a+1}=\tilde{\theta}_{a+1}^{\prime}$ contradicting the
beginning of the Proposition. It follows there exists
$\hat{\theta}\in\mathbb{R}$ such that

$\widetilde{S}(\tilde{\theta}_{a-1}^{\prime},\hat{\theta})+\widetilde{S}(\hat{\theta},\tilde{\theta}_{a+1})<\widetilde{S}(\tilde{\theta}_{a-1}^{\prime},\tilde{\theta}_{a})+\widetilde{S}(\tilde{\theta}_{a},\tilde{\theta}_{a+1}).$

The chain $(\hat{\theta}_{i})_{i\in[a-1,b]}$ is then defined as follows:

$\hat{\theta}_{i}=\begin{cases}\tilde{\theta}_{a-1}^{\prime}\ \ \textrm{if}\ \
i=a-1\\\ \hat{\theta}\ \ \textrm{if}\ \ i=a\\\ \tilde{\theta}_{i}\ \
\textrm{if}\ \ i\in[a+1,b];\end{cases}$

inducing that

$\sum_{i=a-1}^{b-1}\widetilde{S}(\hat{\theta}_{i},\hat{\theta}_{i+1})=\widetilde{S}(\tilde{\theta}_{a-1}^{\prime},\hat{\theta})+\widetilde{S}(\hat{\theta},\tilde{\theta}_{a+1})+\sum_{i=a+1}^{b-1}\widetilde{S}(\tilde{\theta}_{i},\tilde{\theta}_{i+1})\\\
<\widetilde{S}(\tilde{\theta}_{a-1}^{\prime},\tilde{\theta}_{a})+\widetilde{S}(\tilde{\theta}_{a},\tilde{\theta}_{a+1})+\sum_{i=a+1}^{b-1}\widetilde{S}(\tilde{\theta}_{i},\tilde{\theta}_{i+1})=\sum_{i=a-1}^{b-1}\widetilde{S}(\tilde{\theta}^{\prime}_{i},\tilde{\theta}^{\prime}_{i+1}).$

This contradicts the fact that $(\tilde{\theta}^{\prime}_{i})_{i\in[a-1,b]}$
is minimizing. ∎

The previous non–crossing Proposition can be enforced.

######  Definition 5.1.12.

When $I$ is suitably infinite, one says that two chains
$(\tilde{\theta}_{i})_{i\in I}$ and $(\tilde{\theta}^{\prime}_{i})_{i\in I}$
are respectively

  * •

$\alpha$–asymptotic if
$\lim\limits_{i\to-\infty}|\tilde{\theta}_{i}-\tilde{\theta}^{\prime}_{i}|=0$;
we then say $(\tilde{\theta}_{i})_{i\in I}$ and
$(\tilde{\theta}^{\prime}_{i})_{i\in I}$ cross at $-\infty$;

  * •

$\omega$–asymptotic if
$\lim\limits_{i\to+\infty}|\tilde{\theta}_{i}-\tilde{\theta}^{\prime}_{i}|=0$;
we then say $(\tilde{\theta}_{i})_{i\in I}$ and
$(\tilde{\theta}^{\prime}_{i})_{i\in I}$ cross at $+\infty$.

Using this terminology, we state without proof (see for instance [21, Lemma
3.9]) the following strengthening of Proposition 5.1.11:

######  Proposition 5.1.13.

Let $(\tilde{\theta}_{i})_{i\in I}$ and $(\tilde{\theta}^{\prime}_{i})_{i\in
I}$ be two distinct minimizing chains. Assume furthermore that the sequence
$|\tilde{\theta}_{i+1}-\tilde{\theta}_{i}|$ is bounded. Then
$(\tilde{\theta}_{i})_{i\in I}$ and $(\tilde{\theta}^{\prime}_{i})_{i\in I}$
cross at most once, except possibly at both ends of $I$.

In the latter case, both chains are maximal minimizing chains.

######  Remark 5.1.14.

A consequence of our forthcoming analysis will be that the hypothesis
concerning the boundedness of $|\tilde{\theta}_{i+1}-\tilde{\theta}_{i}|$ is
in fact automatically verified.

This section ends with a fundamental property of minimizing sequences. It is
by no means a direct consequence of the previous stated facts. The proof is
quite tricky and we refer the interested reader to [21, Theorem 3.15] for
details.

######  Theorem 5.1.15 (Aubry–Mather).

Let $(\tilde{\theta}_{i})_{i\in\mathbb{Z}}$ be a minimizing sequence. Then the
following hold:

  1. 1.

For all $(a,b)\in\mathbb{Z}^{2}$, $(\tilde{\theta}_{i})_{i\in\mathbb{Z}}$ and
$(\tilde{\theta}_{i-a}+b)_{i\in\mathbb{Z}}$ do not cross.

  2. 2.

There exists a homeomorphism $\tilde{g}:\mathbb{R}\to\mathbb{R}$ verifying
$\tilde{g}(x+1)=\tilde{g}(x)+1$ for all $x\in\mathbb{R}$ and such that111$g$
is the lift of an orientation preserving circle homeomorphism.

$\forall
i\in\mathbb{Z},\quad\tilde{g}(\tilde{\theta}_{i})=\tilde{\theta}_{i+1}.$

  3. 3.

It follows then from Poincaré theory that there exists a real number
$\rho\in\mathbb{R}$ called rotation number such that

$\forall i\in\mathbb{Z},\quad|\tilde{\theta}_{i}-\tilde{\theta}_{0}-i\rho|<1.$

In particular,
$\lim\limits_{|i|\to+\infty}\dfrac{\tilde{\theta}_{i}}{i}=\rho$.

###  5.2 Examples and Moser’s Theorem

####  5.2.1 Notions of integrability

Our first family of examples are called integrable ECTMs. They are of the
following form:

######  Example 5.2.1.

Let $\rho:\mathbb{R}\to\mathbb{R}$ be an increasing diffeomorphism. Then the
map $f_{\rho}:(\theta,r)\mapsto(\theta+\rho(r),r)$ is an ECTM.

Such maps have a particularly simple dynamics. Indeed, for all
$r\in\mathbb{R}$ if we denote by ${\mathcal{C}}_{r}=\\{(\theta,r),\ \
\theta\in\mathbb{T}^{1}\\}$ the canonical circle of height $r$, then
${\mathcal{C}}_{r}$ is invariant by $f_{\rho}$ and the dynamics of $f_{\rho}$
restricted to ${\mathcal{C}}_{r}$ is a rotation of angle $\rho(r)$ (mod $1$).
One checks that $f_{\rho}$ is the time $1$ map of the Tonelli Hamiltonian flow
of $H_{\rho}:(\theta,r)\mapsto\varrho(r)$ where $\varrho$ is a primitive of
$\rho$.

By extension, let us introduce several classes of ECTM exhibiting similar
dynamical features.

######  Definition 5.2.2.

Let $f:\mathbb{A}\to\mathbb{A}$ be an ECTM. We say $f$ is integrable if there
exists a $C^{1}$ area preserving diffeomorphism $\Phi:\mathbb{A}\to\mathbb{A}$
and $\rho:\mathbb{R}\to\mathbb{R}$ such that
$f=\Phi^{-1}\circ{\tilde{f}}_{\rho}\circ\Phi$.

######  Remark 5.2.3.

In the case of an integrable ECTM, the function $\rho$ will automatically be
an increasing diffeomorphism. However, note that the previous definition can
be extended trivially to any area preserving transformation of the annulus.

In the previous definition, the transformation $f$ is the time $1$ map of the
Hamiltonian flow222This is a general fact and proves that the group of
Hamiltonian diffeomorphisms is a normal subgroup of the group of
symplectomorphisms (see [19, Exercise 7 page 471]). of $H_{\rho}\circ\Phi$.

The following notions are obtained by considering twist maps for which the
annulus $\mathbb{A}$ is foliated by invariant circles. Bare in mind that by
Birkhoff’s Theorem 5.1.3, such circles are automatically Lipschitz graphs. The
integrable case corresponds to a classical foliation. The next notions are
obtained by weakening the regularity of the invariant foliation. We therefore
start by defining what are those non–regular foliations.

######  Definition 5.2.4.

A continuous foliation ${\mathcal{F}}=\\{{\mathcal{F}}_{c},\
c\in\mathbb{R}\\}$, (otherwise called lamination) of $\mathbb{A}$ by graphs is
defined through a continuous function $\eta:(\theta,c)\mapsto\eta_{c}(\theta)$
from $\mathbb{A}$ to $\mathbb{R}$ such that

  * •

for all $\theta\in\mathbb{R}$, the map $c\mapsto\eta_{c}(\theta)$ is a
(increasing) homeomorphism of $\mathbb{R}$,

  * •

for all $c\in\mathbb{R}$, $\int_{\mathbb{T}^{1}}\eta_{c}(\theta){\rm
d}\theta=c$.

For $c\in\mathbb{R}$ the circle
${\mathcal{F}}_{c}=\big{\\{}\big{(}\theta,\eta_{c}(\theta)\big{)},\ \
\theta\in\mathbb{T}^{1}\big{\\}}$ is the leaf of the continuous foliation at
cohomology $c$. It then follows that $\mathbb{A}$ is the disjoint union of the
leaves of the foliation.

Reciprocally, any such function $\eta:(\theta,c)\mapsto\eta_{c}(\theta)$ from
$\mathbb{A}$ to $\mathbb{R}$ defines a continuous foliation of $\mathbb{A}$ by
graphs.

######  Remark 5.2.5.

The second point in the definition of $\eta$ is a normalization condition. It
could be dropped to give an equivalent notion. However, it is so convenient we
prefer to include it directly in the definition.

For an intermediate regularity, arises naturally the notion of Lipschitz
foliation:

######  Definition 5.2.6.

A Lipschitz foliation ${\mathcal{F}}=\\{{\mathcal{F}}_{c},\ c\in\mathbb{R}\\}$
of $\mathbb{A}$ by graphs is the data of a continuous foliation with
associated function $\eta:(\theta,c)\mapsto\eta_{c}(\theta)$ from $\mathbb{A}$
to $\mathbb{R}$ such that

$\exists
K>0,\forall(c_{1},c_{2})\in\mathbb{R}^{2},\forall\theta\in\mathbb{T}^{1},\quad\frac{1}{K}|c_{1}-c_{2}|\leqslant|\eta_{c_{1}}(\theta)-\eta_{c_{2}}(\theta)|\leqslant
K|c_{1}-c_{2}|.$

It is now at grasp to define weaker notions of integrability for twist maps:

######  Definition 5.2.7.

  * •

An ECTM $f$ of the annulus is $C^{0}$–integrable if there exists a continuous
foliation, ${\mathcal{F}}=\\{{\mathcal{F}}_{c},\ c\in\mathbb{R}\\}$, of
$\mathbb{A}$ by graphs, such that for all $c\in\mathbb{R}$,
$f({\mathcal{F}}_{c})={\mathcal{F}}_{c}$.

  * •

An ECTM $f$ of the annulus is Lipschitz integrable if there exists a Lipschitz
foliation, ${\mathcal{F}}=\\{{\mathcal{F}}_{c},\ c\in\mathbb{R}\\}$, of
$\mathbb{A}$ by graphs, such that for all $c\in\mathbb{R}$,
$f({\mathcal{F}}_{c})={\mathcal{F}}_{c}$.

Those various notions of integrability each have dynamical consequences on the
underlying ECTM. Such results are presented without proof in the last Section
5.7. Research on those notions is quite frustrating though, starting from the
fact that it is still conjectural whether those three notions of integrability
are different. For instance, there are no known examples of ECTM that are
$C^{0}$–integrable but not integrable (meaning that all the leaves are
smooth).

On the bright side, it is proved in [14, 15] that continuous foliations by
graphs that are invariant by an ECTM must satisfy some particular properties.
This is used to exhibit foliations that cannot be invariant by an ECTM.

######  Theorem 5.2.8.

Let ${\mathcal{F}}=\\{{\mathcal{F}}_{c},\ c\in\mathbb{R}\\}$ be the foliation
associated to the function
$\eta_{c}(\theta)=c+\varepsilon(c)\cos(2\pi\theta)$, where
$\varepsilon:\mathbb{R}\to\mathbb{R}$ is a non $C^{1}$, Lipschitz, function
with Lipschitz constant less than $(2\pi)^{-1}$. Then ${\mathcal{F}}$ is not
invariant by any ECTM.

Though the full proof of this Theorem goes beyond the scope of the present
text, some explanations will be provided at the very end, in Section 5.7.

####  5.2.2 The standard family

According to Remark 5.1.5 it is very easy to construct twist maps with no
particular property. However, let us mention a historically very important
family of examples.

######  Example 5.2.9.

For all $\varepsilon\in\mathbb{R}$ let us define the map
$F_{\varepsilon}:\mathbb{A}\to\mathbb{A}$ by

$\forall(\theta,r)\in\mathbb{A},\quad
F_{\varepsilon}(\theta,r)=\Big{(}\theta+r-\frac{\varepsilon}{2\pi}\sin(2\pi\theta),r-\frac{\varepsilon}{2\pi}\sin(2\pi\theta)\Big{)}.$

This family serves as a test for the state–of–the–art research on twist maps.
On certain aspects, the picture is not glorious. For instance, a conjecture of
Sinai is that $F_{\varepsilon}$ has positive metric entropy for all
$\varepsilon\neq 0$ (with respect to the Lebesgue measure) even though nobody
knows how to prove it even for a single parameter. More on such questions is
discussed in works of Berger and Berger, Turaev [31, 30]. A picture of the
dynamics for some arbitrary value of $\varepsilon$ is presented in the
introduction of [108] and clearly shows that such a dynamics is very rich. On
the one hand, for $\varepsilon=0$ the map $F_{0}$ is the most basic example
among integrable ECTMs. For $\varepsilon$ small, KAM theory applies and many
invariant circles with diophantine rotation numbers persist (see [111, 112]
and references therein). On the other hand, a theorem of Mather [139] states
that for $\varepsilon>\frac{4}{3}$, $F_{\varepsilon}$ has no invariant
essential circle.

What happens to the circles when they disappear is a challenging question:
what is their regularity, the dynamics on them at the last parameter… After
they disappear Aubry–Mather theory provides an answer as to what they become.

####  5.2.3 General twist maps and a Theorem of Moser

Amongst other examples and open questions, the nature of possible invariant
circles of twist maps is still not well understood. In the recent [20], the
authors construct a $C^{1}$ ECTM having an invariant circle that is not
everywhere differentiable with a minimal (irrational) restricted dynamics on
it333Recall that a homeomorphism of the circle having an irrational rotation
number is either minimal (all orbits are dense) or not. In the first case, it
is conjugated to the irrational rotation. In the second case, it is only
semi–conjugated to the irrational rotation as there are wandering intervals.
We then speak of a Denjoy counterexample. A Theorem of Denjoy states that
Denjoy counterexamples cannot be of class $C^{2}$.. This answers partially a
question of Arnaud [7] asking whether such $C^{2}$ maps exist.

Concerning invariant essential circles on which the dynamics is irrational and
not minimal (that of a Denjoy counterexample) examples have been constructed
in [111] where a $C^{1}$ invariant circle is constructed, and in [11] where a
non–differentiable invariant curve is constructed (thus answering a question
of Mather). By Denjoy’s Theorem, such an invariant circle cannot be $C^{2}$.

Last, concerning the inverse problem, in [5] are provided examples of
essential circles that are graphs of Lipschitz maps but cannot be invariant by
any ECTM.

Before turning back to weak KAM theory, let us mention an important Theorem of
Moser [153] stating that any regular ECTM can be represented by a Hamiltonian
function. The drawback is that the latter is not autonomous (i.e. it is
time–dependent).

######  Theorem 5.2.10 (Moser).

Let $f:\mathbb{A}\to\mathbb{A}$ be an ECTM. There exists a $C^{2}$
time–dependent Hamiltonian $H:\mathbb{R}\times\mathbb{A}\to\mathbb{R}$ such
that

  * •

for all $t\in\mathbb{R}$, $H(t,\cdot,\cdot):\mathbb{A}\to\mathbb{R}$ is a
Tonelli Hamiltonian;

  * •

for all $(t,\theta,r)\in\mathbb{R}\times\mathbb{A}$, then
$H(t+1,\theta,r)=H(t,\theta,r)$;

  * •

the ECTM $f$ is the time $1$ map $\varphi_{0}^{1}$ of the Hamiltonian flow of
$H$ generated by the equations

$\begin{cases}\dot{\theta}(t)=\partial_{r}H\big{(}t,\theta(t),r(t)\big{)},\\\
\dot{r}(t)=-\partial_{\theta}H\big{(}t,\theta(t),r(t)\big{)}.\end{cases}$
(5.9)

Note that in Moser’s original article, only the case of a $C^{\infty}$ ECTM is
treated. However, it is stated in the paper that the proof adapts to less
regular functions. Moreover, it can be checked that Moser’s construction
allows to interpolate between Identity and $f$ by twist maps using the family
$(\varphi_{0}^{t})_{t\in[0,1]}$. The generating functions
$(\widetilde{S}_{t})_{t\in(0,1]}$ of those twist maps are given by the
relations

$\tilde{\varphi}_{0}^{t}(\tilde{\theta},r)=(\widetilde{\Theta},R)\
\iff\widetilde{S}_{t}(\tilde{\theta},\widetilde{\Theta})=\int_{0}^{t}\widetilde{L}\big{(}s,\gamma_{(\tilde{\theta},r)}(s),\dot{\gamma}_{(\tilde{\theta},r)}(s)\big{)},$

where the Lagrangian function $L:\mathbb{R}\times\mathbb{A}\to\mathbb{R}$ is
defined as previously by

$\forall(t,\theta,v)\in\mathbb{R}\times\mathbb{A},\quad
L(t,\theta,v)=\sup_{r\in\mathbb{R}}rv-H(t,\theta,r),$

and the curve $\gamma_{(\tilde{\theta},r)}$ is defined by

$\forall
s\in\mathbb{R},\quad\gamma_{(\tilde{\theta},r)}(s)=\pi_{1}\circ\tilde{\varphi}_{0}^{s}(\tilde{\theta},r).$

Here, the generating function is given by the Lagrangian (minimal) action.

###  5.3 Weak KAM for twist maps

We now turn to the study of the Lax–Oleinik semigroup and weak KAM solutions
for an ECTM, $f:\mathbb{A}\to\mathbb{R}$. As was already apparent in the
previous Chapter dedicated to examples, there is actually a one parameter
family of semigroups, indexed by the cohomology. The underlying metric space
here is $X=\mathbb{T}^{1}$ that is obviously compact. We will precisely build
a $1$–parameter family of cost functions using that the first cohomology group
$H^{1}(\mathbb{T}^{1},\mathbb{R})$ is isomorphic to $\mathbb{R}$. This appears
in the early works of Mather on twist maps that later led him to define the
$\alpha$ function in a higher dimensional Lagrangian setting ([143]). The
latter idea was also adapted in [173] on more general metric spaces.

As previously, $S$ is a generating function of $f$,
${\widetilde{S}}:\mathbb{R}^{2}\to\mathbb{R}$ is the lift of $S$ and we choose
${\tilde{f}}=(\widetilde{\Theta},R):\mathbb{R}^{2}\to\mathbb{R}^{2}$ a lift of
$f$. The canonical projection is denoted by $\pi:\mathbb{R}\to\mathbb{T}^{1}$.

######  Definition 5.3.1.

Let $c\in\mathbb{R}$ that we will refer to as a cohomology class. The cost
function $S^{c}:\mathbb{T}^{1}\times\mathbb{T}^{1}\to\mathbb{R}$ is defined by

$\forall(\theta,\Theta)\in\mathbb{T}^{1}\times\mathbb{T}^{1},\quad
S^{c}(\theta,\Theta)=\inf_{\begin{subarray}{c}\pi(\tilde{\theta})=\theta\\\
\pi(\widetilde{\Theta})=\Theta\end{subarray}}\widetilde{S}(\tilde{\theta},\widetilde{\Theta})+c(\tilde{\theta}-\widetilde{\Theta}).$

######  Remark 5.3.2.

  * •

In the previous definition, using the translation invariance of
$\widetilde{S}$, it is also possible to fix a $\tilde{\theta}\in[-1,1]$ such
that $\pi(\tilde{\theta})=\theta$ and take the infimum solely on
$\widetilde{\Theta}$.

  * •

As $\widetilde{S}$ is superlinear (5.5) it is easily seen that the previous
infimum is actually a minimum. Moreover, if as asserted, we restrict to
$\tilde{\theta}\in[-1,1]$ then there exists $K>0$ (depending on $c$) such that
the minimum on $\widetilde{\Theta}$ can be restricted to
$\widetilde{\Theta}\in[-K,K]$.

  * •

Given $\theta\in\mathbb{T}^{1}$ and $\tilde{\theta}\in\mathbb{R}$ such that
$\pi(\tilde{\theta})=\theta$, the derivative
$\partial_{2}S^{c}(\theta,\Theta)$ exists if and only if there exists a unique
$\widetilde{\Theta}_{0}$ realizing the minimum $\
S^{c}(\theta,\Theta)=\widetilde{S}(\tilde{\theta},\widetilde{\Theta}_{0})+c(\tilde{\theta}-\widetilde{\Theta}_{0})$.
In such case,
$\partial_{2}S^{c}(\theta,\Theta)=\partial_{2}S(\tilde{\theta},\widetilde{\Theta}_{0})-c$.

  * •

Similarly, given $\Theta\in\mathbb{T}^{1}$ and
$\widetilde{\Theta}\in\mathbb{R}$ such that $\pi(\widetilde{\Theta})=\Theta$,
the derivative $\partial_{1}S^{c}(\theta,\Theta)$ exists if and only if there
exists a unique $\tilde{\theta}_{0}$ realizing the minimum $\
S^{c}(\theta,\Theta)=\widetilde{S}(\tilde{\theta}_{0},\widetilde{\Theta})+c(\tilde{\theta}_{0}-\widetilde{\Theta})$.
In such case,
$\partial_{1}S^{c}(\theta,\Theta)=\partial_{1}\widetilde{S}(\tilde{\theta}_{0},\widetilde{\Theta})+c$.

  * •

The previous points and the fact that $f$ is a twist map yields that $S^{c}$
verifies the left and right twist conditions as defined in Definition 2.5.2
(see also [173, Proposition 6.4]).

  * •

From the previous points, the same strange regularity property observed for
costs coming from Tonelli Lagrangians (see discussion following Proposition
2.6.14) is brought to light: given
$(\theta,\Theta)\in\mathbb{T}^{1}\times\mathbb{T}^{1}$, the following are
equivalent

    * –

$\partial_{1}S^{c}(\theta,\Theta)$ exists;

    * –

$\partial_{2}S^{c}(\theta,\Theta)$ exists.

With those facts and bearing in mind that an infimum of a compact family of
$C^{2}$ functions is semiconcave, it follows that

######  Proposition 5.3.3.

The function $S^{c}$ is locally semiconcave. More precisely, any of its lifts
to $\mathbb{R}^{2}$ is semiconcave.

In particular, if $S^{c}$ admits any partial derivative at points
$(\theta,\Theta)$ then $S^{c}$ is differentiable at $(\theta,\Theta)$.

From the previous points and the semiconcavity of $S^{c}$ (see the proof of
Proposition 2.4.11 and the following Remark 2.4.12), it follows that:

######  Proposition 5.3.4.

If $(\theta_{k})_{k\in I}$ is a minimizing sequence or chain (with at least
$3$ points) for $S^{c}$ then all the derivatives
$\partial_{i}S^{c}(\theta_{k},\theta_{k+1})$, $i\in\\{1,2\\}$ and $(k,k+1)\in
I\times I$, exist. If $k_{0}\in I$ and $\tilde{\theta}_{k_{0}}$ is a lift of
$\theta_{k_{0}}$ then there exists a unique (minimizing) chain
$(\tilde{\theta}_{k})_{k\in I}$ such that

$\forall
k<k^{\prime},\quad\sum_{\ell=k}^{k^{\prime}-1}S^{c}(\theta_{\ell},\theta_{\ell+1})=\sum_{\ell=k}^{k^{\prime}-1}\widetilde{S}(\tilde{\theta}_{\ell},\tilde{\theta}_{\ell+1})+c(\tilde{\theta}_{k}-\tilde{\theta}_{k^{\prime}}).$

As is now customary, setting then

$r_{\ell}=-\partial_{1}\widetilde{S}(\tilde{\theta}_{\ell},\tilde{\theta}_{\ell+1})=\partial_{2}\widetilde{S}(\tilde{\theta}_{\ell-1},\tilde{\theta}_{\ell})=c-\partial_{1}S^{c}(\theta_{\ell},\theta_{\ell+1})=c+\partial_{2}S^{c}(\theta_{\ell-1},\theta_{\ell}),$

then $(\theta_{k},r_{k})_{k\in I}$ is a piece of orbit of $f$ and
$(\tilde{\theta}_{k},r_{k})_{k\in I}$ is a piece of orbit of $\tilde{f}$.

We will now focus our attention on the negative Lax–Oleinik semigroups
associated to $S^{c}$ and gather results previously proven.

######  Definition 5.3.5.

Given $c\in\mathbb{R}$, we define the operator $T^{c}$ which, to a bounded
function $u:\mathbb{T}^{1}\to\mathbb{R}$, associates the function

$T^{c}u:\Theta\in\mathbb{T}^{1}\mapsto\inf_{\theta\in\mathbb{T}^{1}}u(\theta)+S^{c}(\theta,\Theta).$

Equivalently, recalling that two infimums commute, if
$\tilde{u}:\mathbb{R}\to\mathbb{R}$ is the lift of $u$ then the lift of
$T^{c}u$ is given by the relation

$\forall\widetilde{\Theta}\in\mathbb{R},\quad\widetilde{T}^{c}\tilde{u}(\widetilde{\Theta})=\inf_{\tilde{\theta}\in\mathbb{R}}\tilde{u}(\tilde{\theta})+\widetilde{S}(\tilde{\theta},\widetilde{\Theta})+c(\tilde{\theta}-\widetilde{\Theta}).$

The weak KAM Theorem then states:

######  Theorem 5.3.6.

For all $c\in\mathbb{R}$ there exists a unique constant $\alpha(c)$ for which
the equation

$u=T^{c}u+\alpha(c)$ (5.10)

admits solutions $u^{c}:\mathbb{T}^{1}\to\mathbb{R}$. Such a solution (or its
lift $\tilde{u}_{c}:\mathbb{R}\to\mathbb{R}$) will be called a weak KAM
solution at cohomology $c$.

The function $\alpha:\mathbb{R}\to\mathbb{R}$ is called Mather’s $\alpha$
function.

######  Proposition 5.3.7.

Mather’s $\alpha$ function is convex and superlinear.

###### Proof.

One easily reconstructs from the definition of $S^{c}$ and Proposition 1.2.7
that if $c\in\mathbb{R}$, then $\alpha(c)$ is the least constant
$a\in\mathbb{R}$ such that there exists a $1$–periodic function
$\tilde{v}:\mathbb{R}\to\mathbb{R}$ verifying

$\forall(\tilde{\theta},\tilde{\theta}^{\prime})\in\mathbb{R}\times\mathbb{R},\quad\tilde{v}(\tilde{\theta}^{\prime})-\tilde{v}(\tilde{\theta})\leqslant\widetilde{S}(\tilde{\theta},\tilde{\theta}^{\prime})+c(\tilde{\theta}-\tilde{\theta}^{\prime})+a.$

If now $c_{1}$ and $c_{2}$ are real numbers and $\tilde{u}_{1}$ and
$\tilde{u}_{2}$ are lifts of weak KAM solutions at the corresponding
cohomology classes, if $t\in(0,1)$ one infers that, setting
$\tilde{u}_{t}=t\tilde{u}_{1}+(1-t)\tilde{u}_{2}$,

$\forall(\tilde{\theta},\tilde{\theta}^{\prime})\in\mathbb{R}\times\mathbb{R},\quad\tilde{u}_{t}(\tilde{\theta}^{\prime})-\tilde{u}_{t}(\tilde{\theta})\leqslant\widetilde{S}(\tilde{\theta},\tilde{\theta}^{\prime})+(tc_{1}+(1-t)c_{2})(\tilde{\theta}-\tilde{\theta}^{\prime})+t\alpha(c_{1})+(1-t)\alpha(c_{2}).$

It follows that $\alpha(tc_{1}+(1-t)c_{2})\leqslant
t\alpha(c_{1})+(1-t)\alpha(c_{2})$ and the convexity is proved.

For superlinearity, let $k>0$ be an integer. If $c\in\mathbb{R}$, let
$u_{c}:\mathbb{T}^{1}\to\mathbb{R}$ be a weak KAM solution at cohomology $c$.
If $\tilde{u}_{c}$ is its lift, by periodicity, we infer that

$0=\tilde{u}_{c}(k)-\tilde{u}_{c}(0)\leqslant\widetilde{S}(0,k)-ck+\alpha(c),$
$0=\tilde{u}_{c}(-k)-\tilde{u}_{c}(0)\leqslant\widetilde{S}(0,-k)+ck+\alpha(c).$

From those inequalities, we infer that, setting
$C_{k}=\max\big{(}\widetilde{S}(0,k),\widetilde{S}(0,-k)\big{)}$,

$\forall c\in\mathbb{R},\quad\alpha(c)\geqslant k|c|-C_{k},$

from which it follows that
$\lim\limits_{|c|\to+\infty}\dfrac{\alpha(c)}{|c|}=+\infty$.

∎

Informations obtained from Proposition 2.4.11, Proposition 5.1.7 and Remark
5.3.2 are gathered in the next Theorem (keeping in mind that a weak KAM
solution is locally semiconcave):

######  Theorem 5.3.8.

Let $c\in\mathbb{R}$, $u_{c}:\mathbb{T}^{1}\to\mathbb{R}$ be a weak KAM
solution at cohomology $c$ and $\tilde{u}_{c}:\mathbb{R}\to\mathbb{R}$ its
lift.

  1. 1.

For all $\theta_{0}\in\mathbb{T}^{1}$, there exists
$(\theta_{k}^{c})_{k\leqslant 0}$ such that $\theta_{0}=\theta_{0}^{c}$ and

$\forall k<0,\ \
u_{c}(\theta_{0})=u_{c}(\theta_{k}^{c})+\sum_{i=k}^{-1}S^{c}(\theta_{i}^{c},\theta_{i+1}^{c})+|k|\alpha(c).$

  2. 2.

The previous sequence may not be unique but it is uniquely determined by
$(\theta^{c}_{-1},\theta_{0})$. Moreover, setting for $k\leqslant 0$,
$r_{k}=c+\partial_{2}S^{c}(\theta_{k-1}^{c},\theta_{k}^{c})$ (that exists)
then $(\theta_{k}^{c},r_{k})_{k\leqslant 0}$ is a piece of orbit of $f$.

  3. 3.

For all $k<0$, $u_{c}$ is derivable at $\theta_{k}^{c}$ and
$c+u_{c}^{\prime}(\theta_{k}^{c})=r_{k}$. Moreover,
$r_{0}\in\partial^{+}u_{c}(\theta_{0})$ and $u_{c}$ is derivable at
$\theta_{0}$ if and only if the sequence $(\theta_{k}^{c})_{k\leqslant 0}$ is
unique.

  4. 4.

It follows that
$f^{-1}\left(\overline{\mathcal{G}(c+u^{\prime}_{c})}\right)\subset\mathcal{G}(c+u^{\prime}_{c})$
where $\mathcal{G}(c+u^{\prime}_{c})$ is the set of
$\big{(}\theta,c+u^{\prime}_{c}(\theta)\big{)}$ for $\theta\in\mathbb{T}^{1}$
such that $u^{\prime}_{c}(\theta)$ exists.444In the previous inclusion, the
fact that we can take a closure on the left hand side is because a limit of
calibrating sequences for $u_{c}$ is still calibrating. Moreover, if
$(\theta_{0},r_{0})\in\overline{\mathcal{G}(c+u^{\prime}_{c})}$ and for
$k\leqslant 0$, $(\theta_{k},r_{k})=f^{k}(\theta_{0},r_{0})$, then

$\forall k<0,\ \
u_{c}(\theta_{0})=u_{c}(\theta_{k})+\sum_{i=k}^{-1}S^{c}(\theta_{i},\theta_{i+1})+|k|\alpha(c).$

  5. 5.

Given a sequence $(\theta_{k}^{c})_{k\leqslant 0}$ as above and
$\tilde{\theta}_{0}\in\mathbb{R}$ a lift of $\theta_{0}$, there exists a
unique $(\tilde{\theta}_{k}^{c})_{k\leqslant 0}$ that projects on
$(\theta_{k}^{c})_{k\leqslant 0}$ and such that

$\forall k<0,\ \
\tilde{u}_{c}(\tilde{\theta}_{0})=\tilde{u}_{c}(\tilde{\theta}_{k}^{c})+\sum_{i=k}^{-1}\widetilde{S}(\tilde{\theta}_{i}^{c},\tilde{\theta}_{i+1}^{c})+c(\tilde{\theta}_{k}^{c}-\tilde{\theta}_{0}^{c})+|k|\alpha(c).$

  6. 6.

With the previous notations for $k\leqslant 0$,
$r_{k}=\partial_{2}\widetilde{S}(\tilde{\theta}_{k-1}^{c},\tilde{\theta}_{k}^{c})$
and $(\tilde{\theta}_{k}^{c},r_{k})_{k\leqslant 0}$ is a piece of orbit of
$\tilde{f}$.

  7. 7.

For all $k<0$, $\tilde{u}_{c}$ is derivable at $\tilde{\theta}_{k}^{c}$ and
$c+\tilde{u}_{c}^{\prime}(\tilde{\theta}_{k}^{c})=r_{k}$. Moreover,
$r_{0}\in\partial^{+}\tilde{u}_{c}(\tilde{\theta}_{0})$ and $\tilde{u}_{c}$ is
derivable at $\tilde{\theta}_{0}$ if and only if the sequence
$(\tilde{\theta}_{k}^{c})_{k\leqslant 0}$ is unique.

  8. 8.

It follows that
$\tilde{f}^{-1}\left(\overline{\mathcal{G}(c+\tilde{u}^{\prime}_{c})}\right)\subset\mathcal{G}(c+\tilde{u}^{\prime}_{c})$
where $\mathcal{G}(c+\tilde{u}^{\prime}_{c})$ is the set of
$\big{(}\theta,c+\tilde{u}^{\prime}_{c}(\theta)\big{)}$ for
$\tilde{\theta}\in\mathbb{R}$ such that
$\tilde{u}^{\prime}_{c}(\tilde{\theta})$ exists. Finally, if
$(\tilde{\theta}_{0},r_{0})\in\overline{\mathcal{G}(c+\tilde{u}^{\prime}_{c})}$
and for $k\leqslant 0$,
$(\tilde{\theta}_{k},r_{k})=\tilde{f}^{k}(\theta_{0},r_{0})$, then

$\forall k<0,\ \
\tilde{u}_{c}(\tilde{\theta}_{0})=\tilde{u}_{c}(\tilde{\theta}_{k})+\sum_{i=k}^{-1}\widetilde{S}(\tilde{\theta}_{i},\tilde{\theta}_{i+1})+c(\tilde{\theta}_{k}-\tilde{\theta}_{0})+|k|\alpha(c).$

######  Remark 5.3.9.

Being in dimension $1$, concave functions have very strong derivability
properties that semiconcave functions inherit from. It follows that if
$v:\mathbb{T}^{1}\to\mathbb{R}$ is a semiconcave function, then it admits at
all $\theta\in\mathbb{T}^{1}$ a left derivative $v^{\prime}_{-}(\theta)$ and a
right derivative $v^{\prime}_{+}(\theta)$ that verify
$v^{\prime}_{-}(\theta)\geqslant v^{\prime}_{+}(\theta)$. Its
superdifferential at $\theta$ is then the segment
$\partial^{+}v(\theta)=[v^{\prime}_{+}(\theta),v^{\prime}_{-}(\theta)]$.
Finally, using the previous notation, if $c\in\mathbb{R}$ then

$\overline{\mathcal{G}(c+v^{\prime})}=\bigcup_{\theta\in\mathbb{R}}\\{\theta\\}\times\\{c+v^{\prime}_{+}(\theta),c+v^{\prime}_{-}(\theta)\\}.$

We introduce the notion of full pseudograph that is the graph of the
superdifferential of a semiconcave function and appears in various weak KAM
related works ([34, 9]):

######  Definition 5.3.10.

Let $v:\mathbb{T}^{1}\to\mathbb{R}$ be a semiconcave function. Then its full
pseudograph is

$\mathcal{PG}(v^{\prime})=\bigcup_{\theta\in\mathbb{T}^{1}}\\{\theta\\}\times\partial^{+}v(\theta)=\bigcup_{\theta\in\mathbb{T}^{1}}\\{\theta\\}\times[v^{\prime}_{+}(\theta),v^{\prime}_{-}(\theta)].$

A related notation: for $c\in\mathbb{R}$,

$\mathcal{PG}(c+v^{\prime})=\bigcup_{\theta\in\mathbb{T}^{1}}\\{\theta\\}\times\big{(}c+\partial^{+}v(\theta)\big{)}=\bigcup_{\theta\in\mathbb{T}^{1}}\\{\theta\\}\times[c+v^{\prime}_{+}(\theta),c+v^{\prime}_{-}(\theta)].$

We will use similar notations for semiconcave functions on $\mathbb{R}$.

A beautiful theorem due to Marie-Claude Arnaud (using weak KAM methods)
yields:

######  Proposition 5.3.11.

Let $v:\mathbb{T}^{1}\to\mathbb{R}$ be a semiconcave function. Then its full
pseudograph $\mathcal{PG}(v^{\prime})$ is a Lipschitz manifold. In the present
case it is a Lipschitz essential circle (meaning it separates the annulus in
two unbounded connected components).

We are now ready to state a first result on the interplay between Lax–Oleinik
and the non–crossing lemma:

######  Lemma 5.3.12.

Let $v:\mathbb{T}^{1}\to\mathbb{R}$ be a continuous function, $c\in\mathbb{R}$
and $\tilde{\theta}_{1}<\tilde{\theta}_{2}$ two real numbers.

  1. 1.

Assume that $\tilde{\theta}^{\prime}_{1}$ and $\tilde{\theta}^{\prime}_{2}$
verify for $i\in\\{1,2\\}$,

$\widetilde{T}^{c}\tilde{v}(\tilde{\theta}_{i})=\min_{\tilde{\theta}^{\prime}\in\mathbb{R}}\big{(}\tilde{v}(\tilde{\theta}^{\prime})+\widetilde{S}(\tilde{\theta}^{\prime},\tilde{\theta}_{i})+c(\tilde{\theta}^{\prime}-\tilde{\theta}_{i})\big{)}=\tilde{v}(\tilde{\theta}^{\prime}_{i})+\widetilde{S}(\tilde{\theta}^{\prime}_{i},\tilde{\theta}_{i})+c(\tilde{\theta}^{\prime}_{i}-\tilde{\theta}_{i}).$

Then $\tilde{\theta}^{\prime}_{1}\leqslant\tilde{\theta}^{\prime}_{2}$.

  2. 2.

If moreover $v$ is semiconcave, then
$\tilde{\theta}^{\prime}_{1}<\tilde{\theta}^{\prime}_{2}$.

###### Proof.

Let us argue by contradiction. Then by Proposition 5.1.10 the following holds:

$\tilde{v}(\tilde{\theta}^{\prime}_{1})+\widetilde{S}(\tilde{\theta}^{\prime}_{1},\tilde{\theta}_{1})+c(\tilde{\theta}^{\prime}_{1}-\tilde{\theta}_{1})+\tilde{v}(\tilde{\theta}^{\prime}_{2})+\widetilde{S}(\tilde{\theta}^{\prime}_{2},\tilde{\theta}_{2})+c(\tilde{\theta}^{\prime}_{2}-\tilde{\theta}_{2})>\\\
>\tilde{v}(\tilde{\theta}^{\prime}_{2})+\widetilde{S}(\tilde{\theta}^{\prime}_{2},\tilde{\theta}_{1})+c(\tilde{\theta}^{\prime}_{2}-\tilde{\theta}_{1})+\tilde{v}(\tilde{\theta}^{\prime}_{1})+\widetilde{S}(\tilde{\theta}^{\prime}_{1},\tilde{\theta}_{2})+c(\tilde{\theta}^{\prime}_{1}-\tilde{\theta}_{2}).$

We infer that at least one of the two inequalities

$\tilde{v}(\tilde{\theta}^{\prime}_{1})+\widetilde{S}(\tilde{\theta}^{\prime}_{1},\tilde{\theta}_{1})+c(\tilde{\theta}^{\prime}_{1}-\tilde{\theta}_{1})>\tilde{v}(\tilde{\theta}^{\prime}_{2})+\widetilde{S}(\tilde{\theta}^{\prime}_{2},\tilde{\theta}_{1})+c(\tilde{\theta}^{\prime}_{2}-\tilde{\theta}_{1}),$
$\tilde{v}(\tilde{\theta}^{\prime}_{2})+\widetilde{S}(\tilde{\theta}^{\prime}_{2},\tilde{\theta}_{2})+c(\tilde{\theta}^{\prime}_{2}-\tilde{\theta}_{2})>\tilde{v}(\tilde{\theta}^{\prime}_{1})+\widetilde{S}(\tilde{\theta}^{\prime}_{1},\tilde{\theta}_{2})+c(\tilde{\theta}^{\prime}_{1}-\tilde{\theta}_{2}),$

is valid that is a contradiction.

To prove the second item, we recall that thanks to Proposition 2.4.11, if
$\tilde{v}$ is semiconcave, then it is derivable both at
$\tilde{\theta}^{\prime}_{1}$ and $\tilde{\theta}^{\prime}_{2}$ and
$\pi_{1}\circ\tilde{f}\big{(}\tilde{\theta}^{\prime}_{i},c+\tilde{v}^{\prime}(\tilde{\theta}_{i}^{\prime})\big{)}=\tilde{\theta}_{i}$.
As $\tilde{\theta}_{1}\neq\tilde{\theta}_{2}$, necessarily
$\tilde{\theta}^{\prime}_{1}\neq\tilde{\theta}^{\prime}_{2}$.

∎

An interesting Corollary, reminiscent of Bernard’s Theorem 2.6.11 and that
will be needed later is:

######  Corollary 5.3.13.

Let $c\in\mathbb{R}$ and $u_{c}:\mathbb{T}^{1}\to\mathbb{R}$ be a weak KAM
solution at cohomology $c$. Then $T^{c+}u_{c}$ is a $C^{1}$ function (where
$T^{c+}$ is the positive Lax–Oleinik semigroup given by Definition 1.3.1,
associated to $S^{c}$) and
$\mathcal{G}\big{(}c+(T^{c+}u_{c})^{\prime}\big{)}=f^{-1}\big{(}\mathcal{PG}(c+u^{\prime}_{c})\big{)}$.

###### Proof.

We start by proving that $f^{-1}\big{(}\mathcal{PG}(c+u^{\prime}_{c})\big{)}$
is the graph of a continuous function. By Proposition 5.3.11, there exists a
Lipschitz embedding $\gamma:\mathbb{T}^{1}\to{\mathbb{A}}$ such that
$\gamma(\mathbb{T}^{1})=\mathcal{PG}(c+u^{\prime}_{c})$. Denote by
$\tilde{\gamma}:\mathbb{R}\to\mathbb{R}\times\mathbb{R}$ a lift of $\gamma$
and set $\tilde{\gamma}=(\tilde{\gamma}_{1},\tilde{\gamma}_{2})$ the
coordinates of $\tilde{\gamma}$. As $u^{\prime}_{c}$ is semiconcave, up to
reversing the time parametrization of $\gamma$ we may assume that
$\tilde{\gamma}_{1}$ is non–decreasing and it follows that
$\tilde{\gamma}_{2}$ is decreasing on intervals where $\tilde{\gamma}_{1}$ is
constant. We now define for all $t\in\mathbb{R}$,
$\big{(}\Gamma_{1}(t),\Gamma_{2}(t)\big{)}=f^{-1}\big{(}\tilde{\gamma}_{1}(t),\tilde{\gamma}_{2}(t)\big{)}$.
Let us establish that $\Gamma_{1}$ is increasing that will imply our point.

Let $t_{1}<t_{2}$.

  * •

Assume for a start that $\tilde{\gamma}_{1}(t_{1})=\tilde{\gamma}_{1}(t_{2})$.
It follows that $\tilde{\gamma}_{2}(t_{1})>\tilde{\gamma}_{2}(t_{2})$ and by
the twist condition, we deduce that $\Gamma_{1}(t_{1})<\Gamma_{1}(t_{2})$.

  * •

For the remaining case, $\tilde{\gamma}_{1}(t_{1})<\tilde{\gamma}_{1}(t_{2})$
, define now $T_{1}=\max\big{(}t\geqslant t_{1},\
\tilde{\gamma}_{1}(t_{1})=\tilde{\gamma}_{1}(t)\big{)}$ and
$T_{2}=\min\big{(}t\leqslant t_{2},\
\tilde{\gamma}_{1}(t_{2})=\tilde{\gamma}_{1}(t)\big{)}$. It follows that
$t_{1}\leqslant T_{1}<T_{2}\leqslant t_{2}$ and that

$\tilde{\gamma}_{2}(T_{1})=c+u^{\prime}_{c+}\big{(}\tilde{\gamma}_{1}(t_{1})\big{)},\quad\tilde{\gamma}_{2}(T_{2})=c+u^{\prime}_{c-}\big{(}\tilde{\gamma}_{1}(t_{2})\big{)}.$
(5.11)

As $\tilde{\gamma}_{1}(t_{1})=\tilde{\gamma}_{1}(T_{1})$ and
$\tilde{\gamma}_{1}(t_{2})=\tilde{\gamma}_{1}(T_{2})$ we obtain from the first
case that $\Gamma_{1}(t_{1})\leqslant\Gamma_{1}(T_{1})$ and
$\Gamma_{1}(T_{2})\leqslant\Gamma_{1}(t_{2})$.

Then by (5.11) and Theorem 5.3.8, we deduce that for $i\in\\{1,2\\}$,

$\widetilde{T}^{c}\tilde{u}_{c}\big{(}\tilde{\gamma}_{1}(t_{i})\big{)}=\tilde{u}_{c}\big{(}\Gamma_{1}(T_{i})\big{)}+\widetilde{S}\big{(}\Gamma_{1}(T_{i}),\tilde{\gamma}_{1}(t_{i})\big{)}+c\big{(}\Gamma_{1}(T_{i})-\tilde{\gamma}_{1}(t_{i})\big{)}.$

By Lemma 5.3.12 we obtain that $\Gamma_{1}(T_{1})<\Gamma_{1}(T_{2})$ and
finally

$\Gamma_{1}(t_{1})\leqslant\Gamma_{1}(T_{1})<\Gamma_{1}(T_{2})\leqslant\Gamma_{1}(t_{2}).$

We now turn to the interpretation in terms of positive Lax–Oleinik semigroup.
To this end, we use the analogues for $T^{c+}$ of the results established for
$T^{c}$, without proofs. Note that $T^{c+}u_{c}$ is a semiconvex function. Let
$\theta\in\mathbb{T}^{1}$ and $\Theta\in\mathbb{T}^{1}$ such that
$T^{c+}u_{c}(\theta)=u_{c}(\Theta)-S^{c}(\theta,\Theta)$. Then $S^{c}$ is
differentiable at $(\theta,\Theta)$. By setting
$R=c+\partial_{2}S^{c}(\theta,\Theta)$ and
$r=c-\partial_{1}S^{c}(\theta,\Theta)$,

  * •

$f(\theta,r)=(\Theta,R)$,

  * •

$(\Theta,R)\in\mathcal{PG}(c+u^{\prime}_{c})$,

  * •

$r-c\in\partial^{-}T^{c+}u_{c}(\theta)$.

As we have established that
$f^{-1}\big{(}\mathcal{PG}(c+u^{\prime}_{c})\big{)}$ is the graph of a
continuous function, there is a unique
$(\Theta,R)\in\mathcal{PG}(c+u^{\prime}_{c})$ such that $\pi_{1}\circ
f^{-1}(\Theta,R)=\theta$. It follows that $\Theta$ (realizing equality in the
definition of $T^{c+}u_{c}(\theta)$) is unique and that $T^{c+}u_{c}$ is
derivable at $\theta$. As this holds for all $\theta$ and by semiconvexity,
$T^{c+}u_{c}$ is indeed $C^{1}$. Finally, as
$\mathcal{G}\big{(}c+(T^{c+}u_{c})^{\prime}\big{)}\subset
f^{-1}\big{(}\mathcal{PG}(c+u^{\prime}_{c})\big{)}$ and since both sets are
graphs, they are equal.

∎

######  Definition 5.3.14.

Given $c\in\mathbb{R}$ we will denote by

  * •

${\mathcal{A}}_{c}\subset\mathbb{T}^{1}$ the projected Aubry set,

  * •

$\widehat{\mathcal{A}}_{c}\subset\mathbb{T}^{1}\times\mathbb{T}^{1}$ the
$2$-Aubry set,

  * •

$\widetilde{\mathcal{A}}_{c}\subset(\mathbb{T}^{1})^{\mathbb{Z}}$ the Aubry
set,

all three associated to the cost $\widetilde{S}_{c}$.

We similarly denote by
${\mathcal{A}}^{*}_{c}\subset{\mathbb{A}}=\mathbb{T}^{1}\times\mathbb{R}$ the
set given by Proposition 2.5.1 associated to the cost $\widetilde{S}_{c}$ that
is also refered to as Aubry set.

We will denote by $\mathfrak{A}_{c}$ and $\mathfrak{A}^{*}_{c}$ the lifts of
${\mathcal{A}}_{c}$ and ${\mathcal{A}}_{c}^{*}$ to respectively $\mathbb{R}$
and $\mathbb{R}\times\mathbb{R}$ that we will also refer to as projected Aubry
set and Aubry set.

As sequences in $\widetilde{\mathcal{A}}_{c}$ are minimizing for $S^{c}$ we
may apply Proposition 5.3.4 to obtain

######  Proposition 5.3.15.

Let $(\theta_{i})_{i\in\mathbb{Z}}\in\widetilde{\mathcal{A}}_{c}$ and
$\tilde{\theta}_{0}\in\mathbb{R}$ a lift of $\theta_{0}$. Then there exists a
unique $(\tilde{\theta}_{i})_{i\in\mathbb{Z}}\in\mathbb{R}^{\mathbb{Z}}$ such
that

$\forall
k<k^{\prime},\quad\sum_{\ell=k}^{k^{\prime}-1}S^{c}(\theta_{\ell},\theta_{\ell+1})=\sum_{\ell=k}^{k^{\prime}-1}\widetilde{S}(\tilde{\theta}_{\ell},\tilde{\theta}_{\ell+1})+c(\tilde{\theta}_{k}-\tilde{\theta}_{k^{\prime}}).$

More precisely, if $k\in\mathbb{Z}$,
$\tilde{\theta}_{k}=\pi_{1}\circ\tilde{f}^{k}(\tilde{\theta}_{0},r_{0})$ where

$r_{0}=-\partial_{1}\widetilde{S}(\tilde{\theta}_{0},\tilde{\theta}_{1})=\partial_{2}\widetilde{S}(\tilde{\theta}_{-1},\tilde{\theta}_{0})=c-\partial_{1}S^{c}(\theta_{0},\theta_{1})=c+\partial_{2}S^{c}(\theta_{-1},\theta_{0}).$

######  Definition 5.3.16.

We denote by $\widetilde{\mathfrak{A}}_{c}\subset\mathbb{R}^{\mathbb{Z}}$ the
set of sequences
$(\tilde{\theta}_{i})_{i\in\mathbb{Z}}\in\mathbb{R}^{\mathbb{Z}}$ given by the
previous proposition.

We denote by $\widehat{\mathfrak{A}}_{c}\subset\mathbb{R}^{2}$ the set of
pairs $(\tilde{\theta}_{0},\tilde{\theta}_{1})$ for
$(\tilde{\theta}_{i})_{i\in\mathbb{Z}}\in\widetilde{\mathfrak{A}}_{c}$.

######  Remark 5.3.17.

  * •

All canonical projections from respectively $\widehat{\mathcal{A}}_{c}$,
$\widetilde{\mathcal{A}}_{c}$ and ${\mathcal{A}}^{*}_{c}$ to
${\mathcal{A}}_{c}$ are bi–Lipschitz homeomorphisms.

  * •

All canonical projections from respectively $\widehat{\mathfrak{A}}_{c}$,
$\widetilde{\mathfrak{A}}_{c}$ and $\mathfrak{A}^{*}_{c}$ to
$\mathfrak{A}_{c}$ are bi–Lipschitz homeomorphisms.

  * •

The sets $\mathfrak{A}_{c}$ and $\mathfrak{A}^{*}_{c}$ are respectively
invariant by horizontal translations $\tilde{\theta}\mapsto\tilde{\theta}+1$
and $(\tilde{\theta},r)\mapsto(\tilde{\theta}+1,r)$.

  * •

The set $(0,c)+{\mathcal{A}}^{*}_{c}=\\{(\theta,c+r),\ \
(\theta,r)\in{\mathcal{A}}^{*}_{c}\\}$ is invariant by $f$ and the set
$(0,c)+\mathfrak{A}^{*}_{c}=\\{(\tilde{\theta},c+r),\ \
(\tilde{\theta},r)\in\mathfrak{A}^{*}_{c}\\}$ by $\tilde{f}$.

This last point is proved using that elements in the projected Aubry sets come
in minimizing sequences that calibrate weak KAM solutions. Hence it is
possible to apply Remark 5.3.2 and Theorem 5.3.8.

We derive the following consequence (that will be improved later in Corollary
5.6.4):

######  Corollary 5.3.18.

Let $c\in\mathbb{R}$. There exists $\rho(c)\in\mathbb{R}$ such that for all
$u_{c}:\mathbb{T}^{1}\to\mathbb{R}$ weak KAM solution at cohomology $c$, if
$(\tilde{\theta}_{k})_{k\leqslant 0}\in\mathbb{R}^{\mathbb{Z}_{-}}$ calibrates
$\tilde{u}_{c}$ then

$\forall k\leqslant
0,\quad|\tilde{\theta}_{k}-\tilde{\theta}_{0}-k\rho(c)|<2.$

###### Proof.

Let $(x_{k})_{k\in\mathbb{Z}}\in\widetilde{\mathcal{A}}_{c}$ that hence
calibrates $u_{c}$. Let $\tilde{x}_{0}\in\mathbb{R}$ such that
$\tilde{x}_{0}\leqslant\tilde{\theta}_{0}<\tilde{x}_{0}+1$. Finally let
$(\tilde{x}_{k})_{k\in\mathbb{Z}}$ be the only sequence that projects on
$(x_{k})_{k\in\mathbb{Z}}$ and calibrates $\tilde{u}_{c}$. Thanks to Theorem
5.1.15, there exists a real number number $\rho$ that is independent on
$u_{c}$ such that

$\forall i\in\mathbb{Z},\quad|\tilde{x}_{i}-\tilde{x}_{0}-i\rho|<1.$

Moreover, by periodicity, the sequences $(\tilde{x}_{k}+1)_{k\in\mathbb{Z}}$
also calibrates $\tilde{u}_{c}$.

If $\tilde{\theta}_{0}=\tilde{x}_{0}$, then $\tilde{\theta}_{k}=\tilde{x}_{k}$
for all $k\leqslant 0$. Indeed recall that
$\tilde{u}^{\prime}_{c}(\tilde{x}_{0})$ exists and then a calibrating sequence
starting at $\tilde{x}_{0}$ is unique (see Theorem 5.3.8).

In the remaining case, by applying Lemma 5.3.12 and a straightforward
induction, one finds that

$\forall k\leqslant 0,\quad\tilde{x}_{k}<\tilde{\theta}_{k}<\tilde{x}_{k}+1,$

and the result follows.

As all sequences
$(x_{k}^{\prime})_{k\in\mathbb{Z}}\in\widetilde{\mathcal{A}}_{c}$ calibrate
$u_{c}$, it follows that the initial $\rho$ does not depend on the initial
choice of $(x_{k})_{k\in\mathbb{Z}}\in\widetilde{\mathcal{A}}_{c}$ (by the
previous argument). Finally, as
$(x_{k})_{k\in\mathbb{Z}}\in\widetilde{\mathcal{A}}_{c}$ calibrates any other
weak KAM solution at cohomology $c$, the real number $\rho$ does only depend
on $c$, independently of the initially chosen weak KAM solution. ∎

###  5.4 Mather measures

Recall that $\widehat{\mathcal{P}}$ is the set of closed measures on
$\mathbb{T}^{1}\times\mathbb{T}^{1}$ (Definition 3.1.1). Then if
$c\in\mathbb{R}$, Theorem 3.1.3 stipulates that

$-\alpha(c)=\min_{\mu\in\widehat{\mathcal{P}}}\int_{\mathbb{T}^{1}\times\mathbb{T}^{1}}S_{c}(\theta,\theta^{\prime})\
{\rm d}\mu(\theta,\theta^{\prime}).$

Moreover, minimizing Mather measures are those $\mu\in\widehat{\mathcal{P}}$
whose support is included in $\widehat{\mathcal{A}}_{c}$. We will denote by
$\widehat{\mathcal{P}}_{c}$ the set of such Mather measures at cohomology $c$.
We aim at obtaining analogous notions involving a cost that does not depend on
$c$.

If $\theta\in\mathbb{T}^{1}$ and $\tilde{\theta}^{\prime}\in\mathbb{R}$ denote
by
$\theta+\tilde{\theta}^{\prime}=\pi(\tilde{\theta}+\tilde{\theta}^{\prime})\in\mathbb{T}^{1}$
where $\tilde{\theta}\in\mathbb{R}$ is any lift of $\theta$. Of course,
$\theta+\tilde{\theta}^{\prime}$ does not depend on the choice of
$\tilde{\theta}$.

######  Definition 5.4.1.

  * •

Let $\tau:{\mathbb{A}}\to\mathbb{T}^{1}$ be defined by
$(\theta,r)\mapsto\theta+r$.

  * •

We say a Borel probability measure $\mu$ on
${\mathbb{A}}=\mathbb{T}^{1}\times\mathbb{R}$ is closed if it has finite first
moment, $\int_{\mathbb{A}}|r|\ {\rm d}\mu(\theta,r)<+\infty$ and if
$\tau_{*}\mu=\pi_{1*}\mu$ meaning that for any continuous function
$g:\mathbb{T}^{1}\to\mathbb{R}$,

$\int_{\mathbb{A}}g(\theta+r)\ {\rm
d}\mu(\theta,r)=\int_{\mathbb{A}}g(\theta)\ {\rm d}\mu(\theta,r).$

The set of closed probability measures on ${\mathbb{A}}$ is denoted by
${\mathcal{P}}^{*}$.

  * •

Given a closed probability measure $\mu\in{\mathcal{P}}^{*}$ we define its
rotation number, $\rho(\mu)=\int_{\mathbb{A}}r\ {\rm d}\mu(\theta,r)$.

  * •

We define $S^{*}:{\mathbb{A}}\to\mathbb{R}$ by
$S^{*}(\theta,r)=\widetilde{S}(\tilde{\theta},\tilde{\theta}+r)$ where
$\tilde{\theta}\in\mathbb{R}$ is any lift of $\theta$. Of course, the result
does not depend on the choice of $\tilde{\theta}$ by Proposition 5.1.4.

######  Proposition 5.4.2.

The following holds

$-\alpha(c)=\min_{\mu^{*}\in{\mathcal{P}}^{*}}\int_{{\mathbb{A}}}\big{[}S^{*}(\theta,r)-cr\big{]}\
{\rm d}\mu^{*}(\theta,r).$ (5.12)

Moreover, a closed measure is minimizing if and only if it is supported on the
set of pairs $(\theta,\delta)\in{\mathbb{A}}$ such that
$\theta\in{\mathcal{A}}_{c}$ and

$\delta=\pi_{1}\circ\tilde{f}\left((\pi_{1|\mathfrak{A}_{c}^{*}})^{-1}(\tilde{\theta})+(0,c)\right)-\tilde{\theta}=\pi_{1}\circ\tilde{f}(\tilde{\theta},c+r_{\theta})-\tilde{\theta},$

where $\tilde{\theta}\in\mathbb{R}$ is any lift of $\theta$ and
$r_{\theta}\in\mathbb{R}$ is the unique real number such that
$(\tilde{\theta},r_{\theta})\in\mathfrak{A}_{c}^{*}$.

###### Proof.

Let $u:\mathbb{T}^{1}\to\mathbb{R}$ be a continuous subsolution for $S^{c}$
that is strict outside of $\widehat{\mathcal{A}}_{c}$ (Theorem 1.4.1), meaning
that

$\forall(\theta,\theta^{\prime})\in\mathbb{T}^{1}\times\mathbb{T}^{1},\quad
u(\theta^{\prime})-u(\theta)\leqslant
S^{c}(\theta,\theta^{\prime})+\alpha(c),$

with strict inequality as soon as
$(\theta,\theta^{\prime})\notin\widehat{\mathcal{A}}_{c}$. By definition of
$S^{c}$ it follows that if $\tilde{u}$ is a lift of $u$,

$\forall(\tilde{\theta},\tilde{\theta}^{\prime})\in\mathbb{R}\times\mathbb{R},\quad\tilde{u}(\tilde{\theta}^{\prime})-\tilde{u}(\tilde{\theta})\leqslant\widetilde{S}(\tilde{\theta},\tilde{\theta}^{\prime})+c(\tilde{\theta}-\tilde{\theta}^{\prime})+\alpha(c).$

This can in turn be written as follows:

$\forall(\theta,\delta)\in\mathbb{T}^{1}\times\mathbb{R},\quad
u(\theta+\delta)-u(\theta)\leqslant S^{*}(\theta,\delta)-c\delta+\alpha(c).$

Integrating the previous inequalities against a closed measure
$\mu^{*}\in{\mathcal{P}}^{*}$ yields

$0=\int_{{\mathbb{A}}}u(\theta+\delta)\ {\rm
d}\mu^{*}(\theta,\delta)-\int_{{\mathbb{A}}}u(\theta)\ {\rm
d}\mu^{*}(\theta,\delta)\leqslant\int_{{\mathbb{A}}}\big{[}S^{*}(\theta,\delta)-c\delta+\alpha(c)\big{]}\
{\rm d}\mu^{*}(\theta,\delta)$

and
$-\alpha(c)\leqslant\int_{{\mathbb{A}}}\big{[}S^{*}(\theta,\delta)-c\delta\big{]}{\rm
d}\mu^{*}(\theta,\delta)$.

Moreover, equality holds if and only if $\mu^{*}$ is supported on pairs
$(\theta,\delta)$ such that
$u(\theta+\delta)-u(\theta)=S^{*}(\theta,\delta)-c\delta+\alpha(c)$. As

$u(\theta+\delta)-u(\theta)\leqslant
S^{c}(\theta,\theta+\delta)+\alpha(c)\leqslant
S^{*}(\theta,\delta)-c\delta+\alpha(c),$

we deduce that for such $(\theta,\delta)\in\mathrm{supp}(\mu^{*})$,

$u(\theta+\delta)-u(\theta)=S^{c}(\theta,\theta+\delta)+\alpha(c).$

In turn, we deduce that $(\theta,\theta+\delta)\in\widehat{\mathcal{A}}_{c}$,
in particular $S^{c}$ is differentiable at $(\theta,\theta+\delta)$. Then

$S^{c}(\theta,\theta+\delta)+\alpha(c)=S^{*}(\theta,\delta)-c\delta+\alpha(c)=\widetilde{S}(\tilde{\theta},\tilde{\theta}+\delta)-c\delta+\alpha(c),$

where $\tilde{\theta}$ is a lift of $\theta$. As weak KAM solutions are
calibrated by points of the $2$-Aubry set and derivable on the Aubry set,
Theorem 5.3.8 (see also Proposition 5.3.15) gives that

$\tilde{\theta}+r=\pi_{1}\circ\tilde{f}(\tilde{\theta},-\partial_{1}\widetilde{S}\big{(}\tilde{\theta},\tilde{\theta}+r)\big{)}$

and

$\big{(}\tilde{\theta},-c-\partial_{1}\widetilde{S}(\tilde{\theta},\tilde{\theta}+r)\big{)}=\big{(}\tilde{\theta},-\partial_{1}S^{c}(\theta,\theta+r)\big{)}\in\mathfrak{A}_{c}^{*}.$

With the notation of the current Proposition, this is rewritten
$r_{\theta}=-\partial_{1}S^{c}(\theta,\theta+r)$.

It remains to prove that such a closed measure realizing equality exists. To
that aim let us start from a minimizing Mather measure $\hat{\mu}_{c}$ such
that
$-\alpha(c)=\int_{\mathbb{T}^{1}\times\mathbb{T}^{1}}S^{c}(\theta,\theta^{\prime})\
{\rm d}\hat{\mu}_{c}(\theta,\theta^{\prime})$, that is henceforth supported on
$\widehat{\mathcal{A}}_{c}$ by Theorem 3.1.3. If
$\tilde{\theta}\in\mathfrak{A}_{c}$ and
$\widetilde{\Theta}\in\mathfrak{A}_{c}$ is the unique element such that
$(\tilde{\theta},\widetilde{\Theta})\in\widehat{\mathfrak{A}}_{c}$, we denote
$R(\tilde{\theta})=\widetilde{\Theta}-\tilde{\theta}$. By periodicity, it is
immediate that $R(\tilde{\theta})$ only depends on
$\theta=\pi(\tilde{\theta})\in{\mathcal{A}}_{c}$ hence we will also refer to
it as $R(\theta)$.

Recall that $\tilde{\theta}\mapsto\widetilde{\Theta}$ is the biLipschitz
homoeomorphism
$\pi_{2}\circ(\pi_{1|\widehat{\mathfrak{A}}_{c}})^{-1}:\mathfrak{A}_{c}\to\mathfrak{A}_{c}$,
as stated in Remark 5.3.17, hence $\theta\mapsto R(\theta)$ is Lipschitz.

We now define a Borel probability measure $\mu^{*}_{c}$ on
$\mathbb{T}^{1}\times\mathbb{R}$ by setting, for any continuous function
$G:\mathbb{T}^{1}\times\mathbb{R}\to\mathbb{R}$,

$\int_{\mathbb{T}^{1}\times\mathbb{R}}G(\theta,r)\ {\rm
d}\mu^{*}_{c}(\theta,r)=\int_{\widehat{\mathcal{A}}_{c}}G\big{(}\theta,R(\theta)\big{)}\
{\rm d}\hat{\mu}_{c}(\theta,\theta^{\prime}).$ (5.13)

As the $R(\theta)$ are uniformly bounded, the measure $\mu^{*}_{c}$ is
compactly supported hence $\int_{\mathbb{A}}|r|\ {\rm
d}\mu^{*}_{c}(\theta,r)<+\infty$. Let us verify it is closed: let
$g:\mathbb{T}^{1}\to\mathbb{R}$ be a continuous function, then

$\int_{\mathbb{T}^{1}\times\mathbb{R}}\big{(}g(\theta+r)-g(\theta)\big{)}\
{\rm d}\mu^{*}_{c}(\theta,r)\\\
=\int_{\widehat{\mathcal{A}}_{c}}\big{(}g\big{(}\theta+R(\theta)\big{)}-g(\theta)\big{)}\
{\rm
d}\hat{\mu}_{c}(\theta,\theta^{\prime})=\int_{\widehat{\mathcal{A}}_{c}}\big{(}g(\theta^{\prime})-g(\theta)\big{)}\
{\rm d}\hat{\mu}_{c}(\theta,\theta^{\prime})\\\
=\int_{\mathbb{T}^{1}\times\mathbb{T}^{1}}\big{(}g(\theta^{\prime})-g(\theta)\big{)}\
{\rm d}\hat{\mu}_{c}(\theta,\theta^{\prime})=0,$

where was used that if $(\theta,\theta^{\prime})\in\widehat{\mathcal{A}}_{c}$
then $\theta^{\prime}=\theta+R(\theta)$ and that
$\hat{\mu}_{c}\in\widehat{\mathcal{P}}$ is closed. Hence
$\mu^{*}_{c}\in{\mathcal{P}}^{*}$ is a closed Borel probability measure on
${\mathbb{A}}$.

To conclude, by definition, $\mu^{*}$ is defined on pairs
$\big{(}\theta,R(\theta)\big{)}$ where $\theta\in{\mathcal{A}}_{c}$ and

$R(\theta)=\pi_{1}\circ\tilde{f}\left((\pi_{1|\mathfrak{A}_{c}^{*}})^{-1}(\tilde{\theta})+(0,c)\right)$

hence it verifies
$-\alpha(c)=\int_{{\mathbb{A}}}\big{[}S^{*}(\theta,r)-cr\big{]}\ {\rm
d}\mu_{c}^{*}(\theta,r)$ by the first part of the proof. ∎

######  Remark 5.4.3.

  * •

We will also call minimizing Mather measures (at cohomology $c$) on
${\mathbb{A}}$ closed measures $\mu^{*}_{c}$ verifying
$-\alpha(c)=\int_{{\mathbb{A}}}\big{[}S^{*}(\theta,r)-cr\big{]}\ {\rm
d}\mu_{c}^{*}(\theta,r)$.

  * •

As seen in the previous proof, if $\hat{\mu}_{c}$ is a minimizing Mather
measure on $\mathbb{T}^{1}\times\mathbb{T}^{1}$ (hence supported on
$\widehat{\mathcal{A}}_{c}$) then there is a closed Mather measure
$\mu^{*}_{c}$ on ${\mathbb{A}}$ that is naturally associated (see (5.13)).

  * •

Reciprocally, if $\mu^{*}_{c}$ is a closed Mather measure on ${\mathbb{A}}$,
then it is verified the same way that
$\hat{\mu}_{c}=\big{(}(\pi_{|\widehat{\mathcal{A}}_{c}})^{-1}\circ\pi_{1}\big{)}^{*}\mu_{c}^{*}$
is a closed Mather measure on $\mathbb{T}^{1}\times\mathbb{T}^{1}$.

  * •

The two mappings $\hat{\mu}_{c}\mapsto\mu^{*}_{c}$ and
$\mu^{*}_{c}\mapsto\hat{\mu}_{c}$ are inverses of one another.

We now relate two notions of rotation number, hence clarifying the
terminology.

######  Proposition 5.4.4.

Let $c\in\mathbb{R}$ be a cohomology class and $\mu^{*}_{c}$ be a minimizing
Mather measure on ${\mathbb{A}}$. Then $\rho(\mu^{*}_{c})=\rho(c)$ where the
first rotation number is provided by Definition 5.4.1 and the second by
Corollary 5.3.18.

###### Proof.

By definition, $\rho(\mu^{*}_{c})=\int_{{\mathbb{A}}}r \ {\rm
d}\mu^{*}_{c}(\theta,r)$. By Proposition 5.4.2 and using notations therein,
$\rho(\mu^{*}_{c})=\int_{{\mathcal{A}}_{c}\times\mathbb{R}}R(\theta) \ {\rm
d}\mu^{*}_{c}(\theta,r)$. As $\mu^{*}_{c}$ is closed and supported on pairs of
the form $\big{(}\theta,R(\theta)\big{)}$, we derive that

$\rho(\mu^{*}_{c})=\int_{{\mathcal{A}}_{c}\times\mathbb{R}}R\big{(}\theta+R(\theta)\big{)}
\ {\rm
d}\mu^{*}_{c}(\theta,r)=\int_{{\mathcal{A}}_{c}\times\mathbb{R}}R_{2}(\theta)
\ {\rm d}\mu^{*}_{c}(\theta,r).$

And by induction it follows that for all positive integer $n>0$,

$\rho(\mu^{*}_{c})=\int_{{\mathcal{A}}_{c}\times\mathbb{R}}R_{n}(\theta) \
{\rm
d}\mu^{*}_{c}(\theta,r)=\int_{{\mathcal{A}}_{c}\times\mathbb{R}}\frac{1}{n}\sum_{k=0}^{n-1}R_{k}(\theta)
\ {\rm d}\mu^{*}_{c}(\theta,r),$ (5.14)

where $R_{n}:{\mathcal{A}}_{c}\to\mathbb{R}$ verifies the induction relation
$R_{n+1}(\theta)=R_{n}\big{(}\theta+R(\theta)\big{)}$ and $R_{1}=R$.

Let $\theta_{0}\in{\mathcal{A}}_{c}$, $\tilde{\theta}_{0}$ a lift and
$(\tilde{\theta}_{k})_{k\in\mathbb{Z}}\in\widetilde{\mathfrak{A}}_{c}$ the
associated sequence given by Definition 5.3.16 and
$(\theta_{k})_{k\in\mathbb{Z}}\in\widetilde{\mathcal{A}}_{c}$ the sequence of
projections. For all $k\in\mathbb{Z}$ it then holds that
$R(\tilde{\theta}_{k})=\tilde{\theta}_{k+1}-\tilde{\theta}_{k}$. One then
readily verifies that
$R_{n}(\theta_{0})=R_{n}(\tilde{\theta}_{0})=\tilde{\theta}_{n+1}-\tilde{\theta}_{n}$
so that
$\sum\limits_{k=0}^{n-1}R_{k}(\theta_{0})=\tilde{\theta}_{n}-\tilde{\theta}_{0}$.
By Theorem 5.1.15 and Corollary 5.3.18 it is infered that
$\frac{1}{n}\sum\limits_{k=0}^{n-1}R_{k}(\theta)$ uniformly converges to
$\rho(c)$ as $n\to+\infty$. It is finally deduced from (3.2.13) that
$\rho(\mu^{*}_{c})=\rho(c)$ as desired. ∎

######  Remark 5.4.5.

The idea behind the previous proof is that all the dynamics of $f$ restricted
to the Aubry set ${\mathcal{A}}^{*}_{c}$ can be translated by projecting to a
dynamics on ${\mathcal{A}}_{c}$. This dynamics is then the restriction of a
circle orientation preserving diffeomorphism (by generalizations of Theorem
5.1.15, see [21]). Then, the push forward of any minimizing closed measures on
$\mathbb{T}^{1}$ is invariant by this circle diffeomorphism. Hence the result
turns out to be an emanation of Birkhoff’s ergodic theorem with this point of
view.

Another fact that is apparent from the previous proofs is that if
$(\theta,r)\in\mathbb{A}$ is in the support of a minimizing measure
$\mu^{*}_{c}$ as above and if $\tilde{\theta}\in\mathbb{R}$ is a lift of
$\theta$, then there exists a unique minimizing sequence
$(\tilde{\theta}_{k})_{k\in\mathbb{Z}}\in\widetilde{\mathfrak{A}}_{c}$ such
that $\tilde{\theta}_{0}=\tilde{\theta}$ and
$\tilde{\theta}+r=\tilde{\theta}_{1}$. Moreover this sequence only depends on
the measure $\mu^{*}_{c}$ (meaning it can be recovered without knowing $c$).

Now introducing Mather’s $\beta$ function:

######  Definition 5.4.6.

Mather’s $\beta$ function is defined by

$\forall\rho_{0}\in\mathbb{R},\quad\beta(\rho_{0})=-\inf_{\begin{subarray}{c}\mu^{*}\in{\mathcal{P}}^{*}\\\
\rho(\mu^{*})=\rho_{0}\end{subarray}}\int_{{\mathbb{A}}}S^{*}(\theta,r)\ {\rm
d}\mu^{*}(\theta,r)=\sup_{\begin{subarray}{c}\mu^{*}\in{\mathcal{P}}^{*}\\\
\rho(\mu^{*})=\rho_{0}\end{subarray}}-\int_{{\mathbb{A}}}S^{*}(\theta,r)\ {\rm
d}\mu^{*}(\theta,r).$

Note that for all $\rho_{0}\in\mathbb{R}$ the set of closed measures on
${\mathbb{A}}$ with rotation number $\rho_{0}$ is not empty. One may for
instance consider the pull back on the circle
$\mathbb{T}^{1}\times\\{\rho_{0}\\}$ of the Lebesgue measure on the circle
$\mathbb{T}^{1}$. Moreover, it is immediate that the rotation number function
$\rho:{\mathcal{P}}^{*}\to\mathbb{R}$ is linear. Arguing as in the first part
of Proposition 5.3.7 immediately yields:

######  Proposition 5.4.7.

The function $\beta$ takes values in $\mathbb{R}\cup\\{+\infty\\}$ and is
convex.

Actually this can be improved by the following Theorem of Mather ([143]):

######  Theorem 5.4.8.

The function $\beta$ is finite–valued, convex and superlinear. Moreover,
$\alpha$ and $\beta$ are convex dual one another meaning that

$\forall\rho_{0}\in\mathbb{R},\quad\beta(\rho_{0})=\max_{c\in\mathbb{R}}\rho_{0}c-\alpha(c),$
$\forall c_{0}\in\mathbb{R},\quad\alpha(c_{0})=\max_{\rho\in\mathbb{R}}\rho
c_{0}-\beta(\rho).$

###### Proof.

We start from the measure characterization of $\alpha$ (given by (5.12) in
Proposition 5.4.2) that we rewrite in separating measures according to their
rotation number:

$\alpha(c)=\max_{\mu^{*}\in\widehat{\mathcal{P}}^{*}}-\int_{{\mathbb{A}}}\big{[}S^{*}(\theta,r)-cr\big{]}\
{\rm d}\mu^{*}(\theta,r)\\\
=\max_{\varrho\in\mathbb{R}}\sup_{\begin{subarray}{c}\mu^{*}\in\widehat{\mathcal{P}}^{*}\\\
\rho(\mu^{*})=\varrho\end{subarray}}c\varrho-\int_{{\mathbb{A}}}S^{*}(\theta,r)\
{\rm d}\mu^{*}(\theta,r)=\max_{\varrho\in\mathbb{R}}c\varrho-\beta(\varrho).$

We recognize here the Fenchel dual of $\beta$: $\alpha=\beta^{*}$ (see [87,
159]). By basic properties of the Fenchel dual, as $\beta^{*}$ is everywhere
finite, it follows that $\beta$ is superlinear. Finally, as $\beta$ is also
convex, then $\beta=\beta^{**}=\alpha^{*}$ as was to be proved. ∎

We deduce from properties of the Fenchel transform ([87, 159]), together with
Corollary 5.3.18 and Proposition 5.4.4:

######  Theorem 5.4.9.

The following relations, given $c_{0}$ and $\rho_{0}$ real numbers, are
equivalent:

  * •

$c_{0}\in\partial^{-}\beta(\rho_{0})$;

  * •

$\beta(\rho_{0})+\alpha(c_{0})=c_{0}\rho_{0}$;

  * •

$\rho_{0}\in\partial^{-}\alpha(c_{0})$;

  * •

there exists a minimizing measure $\mu^{*}_{c_{0}}\in{\mathcal{P}}^{*}$ at
cohomology $c_{0}$ with rotation number $\rho_{0}$,

  * •

all minimizing measures $\mu^{*}_{c_{0}}\in{\mathcal{P}}^{*}$ at cohomology
$c_{0}$ have rotation number $\rho_{0}$,

  * •

there exists a minimizing infinite chain, calibrating a weak KAM solution at
cohomology $c_{0}$ that has rotation number $\rho_{0}$,

  * •

all minimizing infinite chains, calibrating a weak KAM solution at cohomology
$c_{0}$ have rotation number $\rho_{0}$.

As for all $c_{0}\in\mathbb{R}$, the subdifferential $\partial^{-}(c_{0})$ is
a singleton, we deduce that the function $\alpha$ is $C^{1}$ and the function
$\beta$ is strictly convex. The function $c\mapsto\rho(c)=\alpha^{\prime}(c)$
is continuous surjective and non–decreasing.

These last results were first published by Mather in [141] where he attributes
them to Aubry.

###  5.5 Order properties of weak KAM solutions

Results in this section appeared in Arnaud–Zavidovique’s works [14, 16]. Some
similar statements also can be found in Zhang’s work [178] and related results
in the setting of the torus $\mathbb{T}^{2}$ in [66]. Interestingly, non
variational versions of some among these results date back to Katznelson and
Ornstein [127].

Our first result is that weak KAM solutions and their full pseudographs
(Definition 5.3.10) are vertically ordered with respect to the rotation
number.

######  Proposition 5.5.1.

Let $c<c^{\prime}$ be two cohomology classes such that
$\rho(c)<\rho(c^{\prime})$. Let $u_{c}:\mathbb{T}^{1}\to\mathbb{R}$ be a weak
KAM solution at cohomology $c$ and
$u_{c^{\prime}}:\mathbb{T}^{1}\to\mathbb{R}$ be a weak KAM solution at
cohomology $c^{\prime}$. If $\theta\in\mathbb{T}^{1}$ and $r,r^{\prime}$ are
such that $(\theta,r)\in\mathcal{PG}(c+u^{\prime}_{c})$ and
$(\theta,r^{\prime})\in\mathcal{PG}(c^{\prime}+u^{\prime}_{c^{\prime}})$, then
$r<r^{\prime}$.

In particular, if $u_{c}$ and $u_{c^{\prime}}$ are derivable at $\theta$ then
$c+u^{\prime}_{c}(\theta)<c^{\prime}+u^{\prime}_{c^{\prime}}(\theta)$ and the
function
$\tilde{\theta}\mapsto(\tilde{u}_{c^{\prime}}-\tilde{u}_{c})(\tilde{\theta})+(c^{\prime}-c)\tilde{\theta}$
is increasing.

###### Proof.

As $u_{c}$ and $u_{c^{\prime}}$ are semiconcave, it is enough to prove the
following about right and left derivatives:
$c^{\prime}+u^{\prime}_{c^{\prime}+}(\theta)>c+u^{\prime}_{c-}(\theta)$. Set
$r_{0}=c+u^{\prime}_{c-}(\theta)$ and
$r^{\prime}_{0}=c^{\prime}+u^{\prime}_{c^{\prime}+}(\theta)$. Let
$\tilde{\theta}_{0}=\tilde{\theta}_{0}^{\prime}\in\mathbb{R}$ be a lift of
$\theta$. For integers $n<0$, we define
$(\tilde{\theta}_{n},r_{n})=\tilde{f}^{n}(\tilde{\theta}_{0},r_{0})$ and
$(\tilde{\theta}^{\prime}_{n},r^{\prime}_{n})=\tilde{f}^{n}(\tilde{\theta}^{\prime}_{0},r^{\prime}_{0})$.
By Theorem 5.3.8 the sequence $(\tilde{\theta}_{n})_{n\leqslant 0}$ (resp.
$(\tilde{\theta}^{\prime}_{n})_{n\leqslant 0}$) calibrate $\tilde{u}_{c}$ with
cohomology $c$ (resp. $\tilde{u}_{c^{\prime}}$ with cohomology $c^{\prime}$).
Hence both sequences are minimizing and by Corollary 5.3.18 verify
$\lim\limits_{n\to-\infty}\frac{\tilde{\theta}_{n}}{n}=\rho(c)$ and
$\lim\limits_{n\to-\infty}\frac{\tilde{\theta}^{\prime}_{n}}{n}=\rho(c^{\prime})$.
It follows, as $\rho(c)\neq\rho(c^{\prime})$, that $r_{0}\neq r_{0}^{\prime}$.

We now argue by contradiction and assume that $r_{0}^{\prime}<r_{0}$. As
$\tilde{f}$ twists verticals to the right, $\tilde{f}^{-1}$ twists verticals
to the left implying that $\tilde{\theta}_{-1}<\tilde{\theta}^{\prime}_{-1}$.
As $\rho(c^{\prime})>\rho(c)$ it follows that for large $n<0$,
$\frac{\tilde{\theta}^{\prime}_{n}}{n}>\frac{\tilde{\theta}_{n}}{n}$ and then,
for large $n<0$, $\theta^{\prime}_{n}<\theta_{n}$. We deduce that the
sequences $(\tilde{\theta}_{n})_{n\leqslant 0}$ and
$(\tilde{\theta}^{\prime}_{n})_{n\leqslant 0}$ cross at least twice, once at
$0$ and then once at some negative integer or between two consecutive ones.
This contradicts Proposition 5.1.11.

∎

With a similar flavor, here is a result on the actions of Lax-Oleinik
semigroups (Definition 5.3.5):

######  Lemma 5.5.2.

Let $c_{1}<c_{2}$ be two real numbers. Let
$v_{1},v_{2}:\mathbb{T}^{1}\to\mathbb{R}$ be continuous functions.

If the function
$\theta\mapsto(\tilde{v}_{2}-\tilde{v}_{1})(\tilde{\theta})+(c_{2}-c_{1})\tilde{\theta}$
is non–decreasing, then so is the function
$\tilde{\theta}\mapsto(\widetilde{T}^{c_{2}}\tilde{v}_{2}-\widetilde{T}^{c_{1}}\tilde{v}_{1})(\tilde{\theta})+(c_{2}-c_{1})\tilde{\theta}$.

###### Proof.

Let $\tilde{\theta}<\tilde{\theta}^{\prime}$ be two real numbers. By
definition of the operators $T^{c_{i}}$ and $\widetilde{T}^{c_{i}}$ there
exist $\tilde{\theta}_{2}^{\prime}$ and $\tilde{\theta}_{1}$ such that

$\widetilde{T}^{c_{2}}\tilde{v}_{2}(\tilde{\theta}^{\prime})=\tilde{v}_{2}(\tilde{\theta}^{\prime}_{2})+\widetilde{S}(\tilde{\theta}^{\prime}_{2},\tilde{\theta}^{\prime})+c_{2}(\tilde{\theta}^{\prime}_{2}-\tilde{\theta}^{\prime}),$
$\widetilde{T}^{c_{1}}\tilde{v}_{1}(\tilde{\theta})=\tilde{v}_{1}(\tilde{\theta}_{1})+\widetilde{S}(\tilde{\theta}_{1},\tilde{\theta})+c_{1}(\tilde{\theta}_{1}-\tilde{\theta}).$

There are two cases to consider:

  * •

if $\tilde{\theta}_{2}^{\prime}<\tilde{\theta}_{1}$ we use Aubry & Le Daeron’s
fundamental Lemma 5.1.10 to obtain

$\displaystyle\widetilde{T}^{c_{2}}\tilde{v}_{2}(\tilde{\theta}^{\prime})+\widetilde{T}^{c_{1}}\tilde{v}_{1}(\tilde{\theta})$
$\displaystyle=\tilde{v}_{2}(\tilde{\theta}^{\prime}_{2})+\widetilde{S}(\tilde{\theta}^{\prime}_{2},\tilde{\theta}^{\prime})+c_{2}(\tilde{\theta}^{\prime}_{2}-\tilde{\theta}^{\prime})$
$\displaystyle\quad\quad+\tilde{v}_{1}(\tilde{\theta}_{1})+\widetilde{S}(\tilde{\theta}_{1},\tilde{\theta})+c_{1}(\tilde{\theta}_{1}-\tilde{\theta})$
$\displaystyle>\tilde{v}_{2}(\tilde{\theta}^{\prime}_{2})+\widetilde{S}(\tilde{\theta}^{\prime}_{2},\tilde{\theta})+c_{2}(\tilde{\theta}^{\prime}_{2}-\tilde{\theta}^{\prime})$
$\displaystyle\quad\quad+\tilde{v}_{1}(\tilde{\theta}_{1})+\widetilde{S}(\tilde{\theta}_{1},\tilde{\theta}^{\prime})+c_{1}(\tilde{\theta}_{1}-\tilde{\theta})$
$\displaystyle\geqslant\widetilde{T}^{c_{2}}\tilde{v}_{2}(\tilde{\theta})+\widetilde{T}^{c_{1}}\tilde{v}_{1}(\tilde{\theta}^{\prime})+(c_{2}-c_{1})(\tilde{\theta}-\tilde{\theta}^{\prime}).$

After rearranging the terms, this reads

$\widetilde{T}^{c_{2}}\tilde{v}_{2}(\tilde{\theta}^{\prime})-\widetilde{T}^{c_{1}}\tilde{v}_{1}(\tilde{\theta}^{\prime})+(c_{2}-c_{1})\tilde{\theta}^{\prime}>\widetilde{T}^{c_{2}}\tilde{v}_{2}(\tilde{\theta})-\widetilde{T}^{c_{1}}\tilde{v}_{1}(\tilde{\theta})+(c_{2}-c_{1})\tilde{\theta}.$

  * •

if $\tilde{\theta}_{2}^{\prime}\geqslant\tilde{\theta}_{1}$ we use the
hypothesis on
$\tilde{\theta}\mapsto(\tilde{v}_{2}-\tilde{v}_{1})(\tilde{\theta})+(c_{2}-c_{1})\tilde{\theta}$
to show that
$\tilde{v}_{2}(\tilde{\theta}^{\prime}_{2})+\tilde{v}_{1}(\tilde{\theta}_{1})\geqslant\tilde{v}_{2}(\tilde{\theta}_{1})+\tilde{v}_{1}(\tilde{\theta}^{\prime}_{2})+(c_{2}-c_{1})(\tilde{\theta}_{1}-\tilde{\theta}^{\prime}_{2})$
and then

$\displaystyle\widetilde{T}^{c_{2}}\tilde{v}_{2}(\tilde{\theta}^{\prime})+\widetilde{T}^{c_{1}}\tilde{v}_{1}(\tilde{\theta})$
$\displaystyle=\tilde{v}_{2}(\tilde{\theta}^{\prime}_{2})+\widetilde{S}(\tilde{\theta}^{\prime}_{2},\tilde{\theta}^{\prime})+c_{2}(\tilde{\theta}^{\prime}_{2}-\tilde{\theta}^{\prime})$
$\displaystyle\quad\quad+\tilde{v}_{1}(\tilde{\theta}_{1})+\widetilde{S}(\tilde{\theta}_{1},\tilde{\theta})+c_{1}(\tilde{\theta}_{1}-\tilde{\theta})$
$\displaystyle\geqslant\tilde{v}_{2}(\tilde{\theta}_{1})+\widetilde{S}(\tilde{\theta}^{\prime}_{2},\tilde{\theta}^{\prime})+c_{2}(\tilde{\theta}_{1}-\tilde{\theta}^{\prime})$
$\displaystyle\quad\quad+\tilde{v}_{1}(\tilde{\theta}^{\prime}_{2})+\widetilde{S}(\tilde{\theta}_{1},\tilde{\theta})+c_{1}(\tilde{\theta}^{\prime}_{2}-\tilde{\theta})$
$\displaystyle\geqslant\widetilde{T}^{c_{2}}\tilde{v}_{2}(\tilde{\theta})+\widetilde{T}^{c_{1}}\tilde{v}_{1}(\tilde{\theta}^{\prime})+(c_{2}-c_{1})(\tilde{\theta}-\tilde{\theta}^{\prime}).$

As before, this gives the result after rearranging terms.

∎

We now state our main result of the section. The rest will be devoted to
providing elements of its proof. We chose to present the parts which best
illustrate discrete weak KAM theory and the first Chapters of this text.

######  Theorem 5.5.3.

There exists a function $u:\mathbb{T}^{1}\times\mathbb{R}\to\mathbb{R}$ that
is locally Lipschitz and that verifies the following properties:

  1. 1.

For all $c\in\mathbb{R}$, $u(0,c)=0$.

  2. 2.

For all $c\in\mathbb{R}$, the function
$u_{c}=u(\cdot,c):\mathbb{T}^{1}\to\mathbb{R}$ is a weak KAM solution at
cohomology $c$.

  3. 3.

If $c<c^{\prime}$, the function
$\tilde{\theta}\mapsto(\tilde{u}_{c^{\prime}}-\tilde{u}_{c})(\tilde{\theta})+(c^{\prime}-c)\tilde{\theta}$
is non–decreasing.

If $u:\mathbb{T}^{1}\to\mathbb{R}$ is any continuous function verifying
properties 1. and 2. above, then

  1. a.

The map $c\mapsto\mathcal{PG}(c+u^{\prime}_{c})$ is continuous for the
Hausdorff topology.

  2. b.

The entire annulus is filled by pseudographs:
${\mathbb{A}}=\bigcup\limits_{c\in\mathbb{R}}\mathcal{PG}(c+u^{\prime}_{c})$.

The whole proof of this Theorem is quite long and can be found in [14,
16]555In those references a result of Mather is used: the $\beta$ function is
derivable on $\mathbb{R}\setminus\mathbb{Q}$. We will give here a strategy of
proof that avoids using this result that we will recover later.. We give the
main steps and ideas bellow.

The first point in the Theorem, $u(0,c)=0$, is a normalization condition and
is easily enforced as the set of weak KAM solutions (at any cohomology class)
is invariant by addition of constants.

Let us continue by noticing that a function verifying the first three points
of the Theorem is automatically locally Lipschitz. Indeed, weak KAM solutions
are locally uniformly in $c$ equiLipschitz in $\theta$ (as the costs $S^{c}$
are). Moreover, if $c\in\mathbb{R}$, then
$\tilde{u}_{c}(1)=\tilde{u}_{c}(0)=0$. Then if $\theta\in\mathbb{T}^{1}$,
$0\leqslant\tilde{\theta}\leqslant 1$ is a lift, and $c<c^{\prime}$,

$0=(\tilde{u}_{c^{\prime}}-\tilde{u}_{c})(0)+(c^{\prime}-c)\times 0\\\
\leqslant(\tilde{u}_{c^{\prime}}-\tilde{u}_{c})(\tilde{\theta})+(c^{\prime}-c)\tilde{\theta}\\\
\leqslant(\tilde{u}_{c^{\prime}}-\tilde{u}_{c})(1)+(c^{\prime}-c)\times
1=c^{\prime}-c.$

The result follows from

$c-c^{\prime}\leqslant-\tilde{\theta}(c^{\prime}-c)\leqslant(u_{c^{\prime}}-u_{c})(\theta)\leqslant(1-\tilde{\theta})(c^{\prime}-c)\leqslant
c^{\prime}-c.$

Point $a.$ follows from a more general fact, stated in the next Proposition.
As we did not find its proof in the literature we provide it. Note that it is
valid in any dimension $N$. In this setting, superdifferentials are linear
forms on $\mathbb{R}^{N}$:

######  Proposition 5.5.4.

Let $N>0$ be an integer, $(v_{n})_{n\in\mathbb{N}}$ a sequence of (real
valued) equi-semiconcave functions defined on $\mathbb{T}^{N}$. Assume that
the sequence $(v_{n})_{n\in\mathbb{N}}$ uniformly converges to a function
$v:\mathbb{T}^{N}\to\mathbb{R}$ (that is semiconcave). Then
$\big{(}\mathcal{PG}(v_{n})\big{)}_{n\in\mathbb{N}}$ converges to
$\mathcal{PG}(v)$ for the Hausdorff distance.

###### Proof.

By hypothesis there exists a constant $K>0$ such that all
$\tilde{v}_{n}-K\|\cdot\|^{2}:\mathbb{R}^{N}\to\mathbb{R}$ are strictly
concave (where $\|\cdot\|$ denotes the Euclidean norm and $\tilde{v}_{n}$ the
lift of $v_{n}$ to $\mathbb{R}^{N}$). It follows that
$\tilde{v}-K\|\cdot\|^{2}:\mathbb{R}^{N}\to\mathbb{R}$ is also concave and let
us assume it is strictly concave up to taking a slightly larger $K$. Let
$O\subset\mathbb{R}^{N}$ be a relatively compact, convex, open set. We will
show that, restricted to $O$,
$\big{(}\mathcal{PG}(\tilde{v}_{n|O})\big{)}_{n\in\mathbb{N}}$ converges to
$\mathcal{PG}(u_{|O})$ for the Hausdorff distance, which implies the result.

  * •

Let $X\in O$ and let $k_{n}$ be an increasing sequence of integers and
$(x_{k_{n}},p_{k_{n}})\in O\times\mathbb{R}^{N}$ such that
$p_{k_{n}}\in\partial^{+}\tilde{v}_{k_{n}}(x_{k_{n}})$ for all
$n\in\mathbb{N}$ and $x_{k_{n}}\to X$. Assume moreover that $p_{k_{n}}\to p$
as $n\to+\infty$. We will prove that $p\in\partial^{+}\tilde{v}(X)$. By
hypothesis, for all $n\in\mathbb{N}$, $0\in\partial^{+}w_{k_{n}}$ where
$w_{k_{n}}:x\mapsto\tilde{v}_{k_{n}}(x)-K\|x-x_{k_{n}}\|^{2}-p_{k_{n}}(x)$ is
defined on $O$. It follows that $x_{k_{n}}$ is a (strict) maximum point of the
(strictly) concave function $w_{k_{n}}$:

$\forall y\in O,\quad w_{k_{n}}(y)\leqslant w_{k_{n}}(x_{k_{n}}).$

As $(w_{k_{n}})_{n\in\mathbb{N}}$ uniformly converges to the function
$w:x\mapsto\tilde{v}(x)-K\|x-X\|^{2}-p(x)$ we can fix $y\in O$ and pass to the
limit in the previous inequality to find that $w(y)\leqslant w(X)$. As this is
true for all $y\in O$, $X$ is a maximum point of $w$ which proves, going back
to $\tilde{v}$ that $p\in\partial^{+}\tilde{v}(X)$.

  * •

Let now $X\in O$ and $p\in\partial^{+}\tilde{v}(X)$. We now define
$\varpi:x\mapsto\tilde{v}(x)-K\|x-X\|^{2}-p(x)$ that has a strict maximum at
$X$. Let $\varpi_{n}:x\mapsto\tilde{v}_{n}(x)-K\|x-X\|^{2}-p(x)$ for all
integer $n$, then $\varpi_{n}$ converges to $\varpi$ uniformly on $O$. Let
$\varepsilon>0$ small enough such that $B(X,2\varepsilon)\subset O$. Let
$\eta>0$ such that if $\|x-X\|=\varepsilon$ then $\varpi(x)<\varpi(X)-\eta$.
Let $n_{0}$ such that for all $n>n_{0}$,
$\|\varpi_{n}-\varpi\|_{\infty,O}<\eta/3$. If $n>n_{0}$ and $x\in O$ such that
$\|x-X\|=\varepsilon$ it follows that

$\varpi_{n}(x)<\varpi(x)+\frac{\eta}{3}<\varpi(X)-\frac{2\eta}{3}<\varpi_{n}(X)-\frac{\eta}{3}.$

It follows that $\varpi_{n}$ admits a local (hence global) maximum in the ball
$B(X,\varepsilon)$ denoted $x_{n}$ and at which we infer that
$0\in\partial^{+}\varpi_{n}(x_{n})$.

Using this argument applied to a decreasing sequence $\varepsilon_{k}\to 0$ it
is easy to construct a sequence $x^{\prime}_{n}$ that converges to $X$ defined
for $n>n_{1}$ large enough, and such that
$0\in\partial^{+}\varpi_{n}(x^{\prime}_{n})$ for all integer $n>n_{1}$. Going
back to the initial functions, we conclude that

$\forall n>n_{1},\quad
p+2K(x_{n}^{\prime}-X)=p_{n}\in\partial^{+}\tilde{v}_{n}(x_{n}^{\prime}).$

Clearly, $p_{n}\to p$.

The result follows from the two previous points. ∎

We now give elements of the construction of a continuous choice of weak KAM
solutions with respect to the cohomology class:

###### Elements of Proof.

The first fundamental step is:

######  Lemma 5.5.5.

Let $\rho_{0}\in\mathbb{R}$, we denote by
$[a,b]=[\beta^{\prime}_{-}(\rho_{0}),\beta^{\prime}_{+}(\rho_{0})]=\rho^{-1}(\\{\rho_{0}\\})$
(by Theorem 5.4.9). Then there exists a unique weak KAM solution at cohomology
$a$ (resp. $b$), denoted $u_{a}$ (resp. $u_{b}$) such that $u_{a}(0)=0$ (resp.
$u_{b}(0)=0$).

Let $(c_{n})_{n\in\mathbb{N}}$ be a sequence converging to $a$ (resp.
converging to $b$). Let $(v_{n})_{n\in\mathbb{N}}$ be a sequence of functions
on $\mathbb{T}^{1}$ such that $v_{n}$ is a weak KAM solution at cohomology
$c_{n}$ for all $n$ verifying $v_{n}(0)=0$. Then the sequence
$(v_{n})_{n\in\mathbb{N}}$ uniformly converges towards $u_{a}$ (resp.
$u_{b}$).

###### Proof.

We will prove the result for $a$, the rest being similar. Let us first
consider an increasing sequence $(c_{n})_{n\in\mathbb{N}}$ that converges to
$a$. It follows from the hypothesis that $\rho(c_{n})<\rho_{0}$ for all
$n\in\mathbb{N}$.

Let $(v_{n})_{n\in\mathbb{N}}$ be a sequence of functions on $\mathbb{T}^{1}$
such that $v_{n}$ is a weak KAM solution at cohomology $c_{n}$ for all $n$
verifying $v_{n}(0)=0$. The functions $v_{n}$ are equiLipschitz (by
Proposition 1.1.3 and its proof, since the $S^{c_{n}}$ are) hence by Ascoli’s
Theorem, the sequence $(v_{n})_{n\in\mathbb{N}}$ is relatively compact. As all
limit points of $(v_{n})_{n\in\mathbb{N}}$ are weak KAM solutions at
cohomology $a$ 666It can be verified with the use of Remark 5.3.2 that
$c\mapsto S^{c}$ is continuous. that vanish at $0$, it is enough to prove that
there is only one such function. To this end, let
$u_{a}:\mathbb{T}^{1}\to\mathbb{R}$ be a weak KAM solution at cohomology $a$
such that $u_{a}(0)=0$. Let $(k_{n})_{n\in\mathbb{N}}$ be an increasing
sequence such that $(v_{k_{n}})_{n\in\mathbb{N}}$ uniformly converges to a
function $w:\mathbb{T}^{1}\to\mathbb{R}$ that is hence a weak KAM solution at
cohomology $a$ such that $w(0)=0$. Finally, let
$\mathcal{D}\subset\mathbb{T}^{1}$ be a full measure set such that all
$(v_{n})_{n\in\mathbb{N}}$, $u_{a}$, $w$ are derivable on $\mathcal{D}$ and
$\widetilde{\mathcal{D}}\subset\mathbb{R}$ its lift. By Proposition 5.5.1, for
all $n\in\mathbb{N}$, $s\in\mathcal{D}$,
$c_{n}+v^{\prime}_{n}(s)<a+u^{\prime}_{a}(s)$. By Proposition 5.5.4, for all
$s\in\mathcal{D}$,
$w^{\prime}(s)=\lim\limits_{n\to+\infty}v^{\prime}_{k_{n}}(s)$ hence
$a+w^{\prime}(s)\leqslant a+u_{a}^{\prime}(s)$. Integrating, it follows that

$\forall
x\in[0,1],\quad\tilde{u}_{a}(x)=\int_{\widetilde{\mathcal{D}}\cap[0,x]}\tilde{u}^{\prime}_{a}(s)\
{\rm d}s\geqslant\int_{\widetilde{\mathcal{D}}\cap[0,x]}\tilde{w}^{\prime}(s)\
{\rm d}s=\tilde{w}(x).$

For $x=1$, $\tilde{u}_{a}(1)=\tilde{w}(1)=0$. It implies that equality
$\tilde{u}_{a}^{\prime}(s)=\tilde{w}^{\prime}(s)$ holds for almost every
$s\in[0,1]$ and then, integrating as above, that
$\tilde{u}_{a}(x)=\tilde{w}(x)$ for all $x\in[0,1]$. We have thus proved that
$u_{a}=w$ and the uniqueness of $u_{a}$ follows.

If now $(c_{n})_{n\in\mathbb{N}}$ is any sequence converging to $a$ and
$(v_{n})_{n\in\mathbb{N}}$ is a sequence of weak KAM solutions at cohomology
$c_{n}$ for all $n$ verifying $v_{n}(0)=0$ then again, the sequence
$(v_{n})_{n\in\mathbb{N}}$ is relatively compact and as all its limit points
are weak KAM solutions at cohomology $a$ vanishing at $0$, by what has been
proved, the sequence $(v_{n})_{n\in\mathbb{N}}$ converges to $u_{a}$.

∎

Keeping the notations of the Lemma, a straightforward corollary of the
preceding Lemma and Proposition 5.5.1 is,

######  Corollary 5.5.6.

Let $c\in\mathbb{R}$ and $v_{c}:\mathbb{T}^{1}\to\mathbb{R}$ be a weak KAM
solution at cohomology $c$. Then

  * •

$t\mapsto(\tilde{u}_{a}-\tilde{v}_{c})(t)+(a-c)t$ is

    * –

non–increasing if $c>a$,

    * –

increasing if $c<a$.

  * •

$t\mapsto(\tilde{u}_{b}-\tilde{v}_{c})(t)+(b-c)t$ is

    * –

non–decreasing if $c<b$,

    * –

decreasing if $c>b$.

We denote by
$\mathcal{I}=\\{\beta^{\prime}_{-}(\rho_{0}),\beta^{\prime}_{+}(\rho_{0}),\ \
\rho_{0}\in\mathbb{R}\\}\subset\mathbb{R}$ then the previous Lemma tells that
any choice of weak KAM solutions $u_{c}$ at cohomology $c$ for
$c\in\mathbb{R}$ such that $u_{c}(0)=0$ for all $c\in\mathbb{R}$ is
automatically continuous on $\mathcal{I}$. Note that $\mathcal{I}$ is closed
as it can alternatively be defined by
$\mathcal{I}=\overline{\\{\beta^{\prime}(r),\ \ r\in\mathbb{R}\ \mathrm{such\
that}\ \beta^{\prime}(r)\ \mathrm{exists}\\}}$.

The rest of the construction consists in focusing on its complement,
$\mathbb{R}\setminus\mathcal{I}$, that is on intervals of the form
$\big{(}\beta^{\prime}_{-}(\rho_{0}),\beta^{\prime}_{+}(\rho_{0})\big{)}=(a,b)$
that are not empty (that is where $\beta$ is not derivable at $\rho_{0}$777We
will prove later that such $\rho_{0}$ are necessarily rational.). If this is
the case, we use in a crucial way that the Aubry set ${\mathcal{A}}_{c}$ does
not depend on
$c\in\big{(}\beta^{\prime}_{-}(\rho_{0}),\beta^{\prime}_{+}(\rho_{0})\big{)}$
by a result of Massart ([135, Proposition 6]). We hence denote it
${\mathcal{A}}$ until the end of the construction. Let us then interpolate
linearly between $u_{a}$ and $u_{b}$ restricted to ${\mathcal{A}}$ and use
Proposition 2.2.5 and Theorem 2.2.4. If $t\in(0,1)$ let $c_{t}=at+(1-t)b$ and
$v_{t}=tu_{a}+(1-t)u_{b}$. As $\alpha(c_{t})=t\alpha(a)+(1-t)\alpha(b)$ by
duality between $\alpha$ and $\beta$, it follows that for all $t\in(0,1)$,
$v_{t}$ is a critical subsolution for $S^{c_{t}}$. Hence by Proposition 2.2.5
and Theorem 2.2.4 there exists a unique weak KAM solution at cohomology
$c_{t}$, that we note $u_{c_{t}}=u_{t}$ for short, such that
$u_{t|{\mathcal{A}}}=v_{t|{\mathcal{A}}}$. The continuity of $t\mapsto u_{t}$
is a consequence of uniqueness and Ascoli’s Theorem.

Finally, item $3$ of the Theorem 5.5.3 follows from successive applications of
Lemma 5.5.2 and the alternative characterization:
$u_{t}=\lim\limits_{n\to+\infty}(T^{c_{t}})^{n}v_{t}$.

Again, point $a.$ follows from Proposition 5.5.4. The last thing to prove in
the Theorem is $b.$ It uses in a crucial way 2 dimensional topology and
Jordan’s curve Theorem. It is quite technical and we refer the interested
reader to [14, 16]. ∎

###  5.6 Structure of infinite minimizing chains

We continue exploring the results of [14, 16] by studying infinite minimizing
chains (indexed by non–positive numbers) of the twist map
$\tilde{f}:\mathbb{R}^{2}\to\mathbb{R}^{2}$. As we will see, all such
minimizing chains calibrate a weak KAM solution, hence fall within the scope
of the previous sections. The present results generalize classical Aubry-
Mather theory as the latter studies full orbits that are minimizing. After
obtaining the results, the authors discovered that many had already appeared
in Bangert’s [22]. In the latter, Bangert studies Busemann functions that are
weak KAM solutions on the universal cover $\mathbb{R}$. We adopt the
alternative approach to study and use weak KAM solutions on $\mathbb{T}^{1}$,
as previously defined. Consequently most proofs differ slightly from [22].

We start by a consequence of Theorem 5.5.3 that is a generalization of
Corollary 5.3.18.

######  Proposition 5.6.1.

Let $(\tilde{\theta}_{k})_{k\leqslant 0}\in\mathbb{R}^{\mathbb{Z}_{-}}$ be a
minimizing chain. Then there exists $\rho\in\mathbb{R}$ such that

$\forall k\leqslant 0,\quad|\tilde{\theta}_{k}-\tilde{\theta}_{0}-k\rho|<2.$

###### Proof.

We denote as usual for all $k\leqslant 0$,
$r_{k}=\partial_{2}\widetilde{S}(\tilde{\theta}_{k-1},\tilde{\theta}_{k})$ in
such a way that $(\tilde{\theta}_{k},r_{k})_{k\leqslant 0}$ is a piece of
orbit of $\tilde{f}$. By Theorem 5.5.3 there exists $c\in\mathbb{R}$ and a
weak KAM solution at cohomology $c$, $u_{c}:\mathbb{T}^{1}\to\mathbb{R}$ such
that $(\tilde{\theta}_{0},r_{0})\in\mathcal{PG}(c+\tilde{u}^{\prime}_{c})$.
The result is proved with $\rho=\rho(c)$. Let
$r_{0}^{\pm}=c+\tilde{u}_{c\pm}^{\prime}(\tilde{\theta}_{0})$ (so that
$r_{0}^{+}\leqslant r_{0}^{-}$). Finally, for $k\geqslant 0$, we set
$(\tilde{\theta}_{k}^{\pm},r_{k}^{\pm})=\tilde{f}^{k}(\tilde{\theta}_{0},r_{0}^{\pm})$.
It follows from Theorem 5.3.8, and the following Remark 5.3.9, that both
chains $(\tilde{\theta}_{k}^{+})_{k\leqslant 0}$ and
$(\tilde{\theta}_{k}^{-})_{k\leqslant 0}$ calibrate $\tilde{u}_{c}$ hence are
minimizing.

If $r_{0}$ coincides with either $r_{0}^{+}$ or $r_{0}^{-}$ then the result is
a particular case of Corollary 5.3.18. We now assume otherwise which
translates to $r_{0}^{+}<r_{0}<r_{0}^{-}$. As the chains
$(\tilde{\theta}_{k})_{k\leqslant 0}$ and
$(\tilde{\theta}_{k}^{+})_{k\leqslant 0}$ (resp.
$(\tilde{\theta}_{k})_{k\leqslant 0}$ and
$(\tilde{\theta}_{k}^{-})_{k\leqslant 0}$) cross at $k=0$, they cannot cross
anywhere else. We deduce from the twist hypothesis that
$\tilde{\theta}_{-1}^{-}<\tilde{\theta}_{-1}<\tilde{\theta}_{-1}^{+}$. Hence
we conclude from the non–crossing that
$\tilde{\theta}_{k}^{-}<\tilde{\theta}_{k}<\tilde{\theta}_{k}^{+}$ for all
$k<0$. Finally, from Corollary 5.3.18 we conclude

$\forall k<0,\quad
k\rho(c)-2<\tilde{\theta}_{k}^{-}-\tilde{\theta}_{0}<\tilde{\theta}_{k}-\tilde{\theta}_{0}<\tilde{\theta}_{k}^{+}-\tilde{\theta}_{0}<k\rho(c)+2.$

∎

######  Remark 5.6.2.

Note that the previous proposition clearly implies that the hypothesis in
Proposition 5.1.13, that $|\tilde{\theta}_{i+1}-\tilde{\theta}_{i}|$ is
bounded, is automatically verified by any minimizing chain, hence can be
dropped.

We anticipate on the two next sections by stating a result that will be proved
later in the text:

######  Theorem 5.6.3.

Let $(\tilde{\theta}_{k})_{k\leqslant 0}\in\mathbb{R}^{\mathbb{Z}_{-}}$ be a
minimizing chain. Then there exists a cohomology class $c\in\mathbb{R}$ and a
weak KAM solution $u_{c}:\mathbb{T}^{1}\to\mathbb{R}$ at cohomology $c$ such
that $(\tilde{\theta}_{k})_{k\leqslant 0}$ calibrates $\tilde{u}_{c}$.

As a consequence, we already state an improvement of Proposition 5.6.1. The
proof of the next result uses ideas from Aubry–Mather theory ([21]) that play
a central role in the study of minimizing chains with rational rotation
numbers.

######  Corollary 5.6.4.

Let $(\tilde{\theta}_{k})_{k\leqslant 0}\in\mathbb{R}^{\mathbb{Z}_{-}}$ be a
minimizing chain. Then there exists $\rho\in\mathbb{R}$ such that

  * •

for all pairs of integers $(p,q)$ with $q<0$ such that $p/q<\rho$, then
$\tilde{\theta}_{q}-\tilde{\theta}_{0}<p$,

  * •

for all pairs of integers $(p,q)$ with $q<0$ such that $p/q>\rho$, then
$\tilde{\theta}_{q}-\tilde{\theta}_{0}>p.$

In particular,

$\forall k\leqslant 0,\quad|\tilde{\theta}_{k}-\tilde{\theta}_{0}-k\rho|<1.$

###### Proof.

Of course, $\rho$ is the same as the one given by Proposition 5.6.1 and is
also $\rho(c)$ where $c$ is given by the previous Theorem.

Let us prove the first point, the second is similar. By contradiction, assume
the existence of an integer $p$ and $q<0$ such that
$\tilde{\theta}_{q}-\tilde{\theta}_{0}\geqslant p>q\rho$. By the previous
Theorem 5.6.3, there exists $c\in\mathbb{R}$ and
$u_{c}:\mathbb{T}^{1}\to\mathbb{R}$ such that
$(\tilde{\theta}_{k})_{k\leqslant 0}$ calibrates $\tilde{u}_{c}$.

  * •

We first exclude the equality $\tilde{\theta}_{q}=\tilde{\theta}_{0}+p$.
Indeed, $\tilde{u}_{c}$ is derivable at $\tilde{\theta}_{q}$ by Theorem 5.3.8.
Hence assuming by contradiction that
$\tilde{\theta}_{q}=\tilde{\theta}_{0}+p$, by periodicity, $\tilde{u}_{c}$ is
also derivable at $\tilde{\theta}_{0}$ with
$\tilde{u}_{c}^{\prime}(\tilde{\theta}_{0})=\tilde{u}_{c}^{\prime}(\tilde{\theta}_{q})$.
With the usual notations,
$r_{k}=c+\tilde{u}^{\prime}_{c}(\tilde{\theta}_{k})$, recalling that
$(\tilde{\theta}_{k},r_{k})_{k\leqslant 0}$ is a piece of orbit of
$\tilde{f}$, it follows that

$\forall k\leqslant 0,\quad\tilde{\theta}_{k+q}=\tilde{\theta}_{k}+p.$

By induction, we deduce that $\tilde{\theta}_{nq}=\tilde{\theta}_{0}+np$ for
all $n>0$ and finally, dividing by $nq$ and letting $n\to+\infty$ we conclude
that $\rho=\frac{p}{q}$. This contradicts the right inequality $p>q\rho$.

  * •

We are left with the hypothesis that
$\tilde{\theta}_{q}-\tilde{\theta}_{0}>p>q\rho$. By periodicity, the chain
$(\tilde{\theta}_{q+k}-p)_{k\leqslant 0}$ also calibrates $\tilde{u}_{c}$. By
Lemma 5.3.12 we deduce that $\tilde{\theta}_{k+q}-p>\tilde{\theta}_{k}$ for
all $k\leqslant 0$. By induction we readily obtain that for all $k\leqslant
0$, the sequence $(\tilde{\theta}_{k+nq}-np)_{n\geqslant 0}$ is increasing.
Applying for $k=0$ and dividing by $nq<0$ yields
$\dfrac{\tilde{\theta}_{nq}}{nq}-\dfrac{p}{q}<\dfrac{\tilde{\theta}_{0}}{nq}$.
Finally, letting $n\to+\infty$ entails $\rho\leqslant\dfrac{p}{q}$ that is
again a contradiction.

To prove the final statement, let us argue by contradiction and assume that
there exists $q<0$ such that
$|\tilde{\theta}_{q}-\tilde{\theta}_{0}-q\rho|\geqslant 1$. Then one of the
following holds:

$\exists p\in\mathbb{Z},\quad\tilde{\theta}_{q}-\tilde{\theta}_{0}\leqslant
p<q\rho,$ $\exists
p\in\mathbb{Z},\quad\tilde{\theta}_{q}-\tilde{\theta}_{0}\geqslant p>q\rho,$

both impossible by what was proved above. ∎

We now turn to making Proposition 5.1.13 more precise.

######  Proposition 5.6.5.

Let $c\in\mathbb{R}$ be a cohomology class. Let
$(\tilde{\theta}_{i})_{i\leqslant 0}$ and
$(\tilde{\theta}_{i}^{c})_{i\leqslant 0}$ be two minimizing chains such that

  * •

$\tilde{\theta}_{0}=\tilde{\theta}_{0}^{c}$,

  * •

$(\tilde{\theta}_{i})_{i\leqslant 0}$ and
$(\tilde{\theta}_{i}^{c})_{i\leqslant 0}$ are $\alpha$-asymptotic,

  * •

there exists a weak KAM solution $u_{c}:\mathbb{T}^{1}\to\mathbb{R}$ at
cohomology $c$, such that $(\tilde{\theta}_{i}^{c})_{i\leqslant 0}$ calibrates
$\tilde{u}_{c}$.

Then $(\tilde{\theta}_{i})_{i\leqslant 0}$ also calibrates $\tilde{u}_{c}$.

###### Proof.

We argue by contradiction assuming that there exist $n_{0}<0$ and
$\varepsilon>0$ such that

$\tilde{u}_{c}(\tilde{\theta}_{0})<\tilde{u}_{c}(\tilde{\theta}_{n_{0}})+\sum_{i=n_{0}}^{-1}\widetilde{S}(\tilde{\theta}_{i},\tilde{\theta}_{i+1})+c(\tilde{\theta}_{n_{0}}-\tilde{\theta}_{0})+|n_{0}|\alpha(c)-\varepsilon.$