# Realization of Anosov Diffeomorphisms on the Torus

Tamara Kucherenko  Department of Mathematics, The City College of New York,
New York, NY, 10031, USA<EMAIL_ADDRESS>and  Anthony Quas
Department of Mathematics and Statistics, University of Victoria, Victoria, BC
Canada<EMAIL_ADDRESS>
###### Abstract.

We study area preserving Anosov maps on the two-dimensional torus within a
fixed homotopy class. We show that the set of pressure functions for Anosov
diffeomorphisms with respect to the geometric potential is equal to the set of
pressure functions for the linear Anosov automorphism with respect to Hölder
potentials. We use this result to provide a negative answer to the
$C^{1+\alpha}$ version of the question posed by Rodriguez Hertz on whether two
homotopic area preserving $C^{\infty}$ Anosov difeomorphisms whose geometric
potentials have identical pressure functions must be $C^{\infty}$ conjugate.

###### Key words and phrases:

Anosov diffeomorphisms, smooth conjugacy problem, thermodynamic formalism,
pressure function, equilibrium states, Hölder potentials

###### 2020 Mathematics Subject Classification:

37D35, 37B10, 37A60, 37C15, 37D20

T.K. is supported by grants from the Simons Foundation #430032.

A.Q. is supported by a grant from NSERC

##  1\. Introduction

We consider an Anosov diffeomorphism $T$ of the two-dimensional torus
$\mathbb{T}^{2}$. That is, there is a continuous splitting of the tangent
bundle of $\mathbb{T}^{2}$ into a direct sum $E^{u}\oplus E^{s}$ which is
preserved by the derivative $DT$ and such that the unstable subbundle $E^{u}$
is uniformly expanded by $DT$ and the stable subbundle $E^{s}$ is uniformly
contracted by $DT$. Any such Anosov diffeomorphism $T$ is homotopic and
topologically conjugate to a hyperbolic toral automorphism $L$ given by an
integer matrix with determinant one and no eigenvalues of absolute value one.
This was first proven by Franks in 1969 [6] under the assumption that all
points on the torus are non-wandering (in fact, his result was for an
$n$-dimensional torus). A year later Newhouse [22] pointed out that this
assumption is satisfied when either $\dim E^{s}=1$ or $\dim E^{u}=1$, which
provided the classification of Anosov diffeomorphisms up to topological
conjugacy in dimensions 2 and 3. The case of dimension $n\geq 4$ was settled
by Manning [18] in 1974.

Suppose $T_{1}$ and $T_{2}$ are two $C^{r}\,(r>1)$ Anosov diffeomorphisms in
the homotopy class of a fixed hyperbolic automorphism $L$. It follows from the
above that there is a homeomorphism $h$ such that $h\circ T_{1}=T_{2}\circ h$.
The problem of determining when $h$ has the same regularity as the maps
$T_{1}$ and $T_{2}$ is known as the smooth conjugacy problem and has been
studied extensively, see e.g. [14, 12, 7, 8]. Already in 1967 Anosov [1]
constructed examples which showed that $h$ may be merely Hölder even for
highly regular $T_{1}$ and $T_{2}$, which initially discouraged further study
of the problem (see comments in [25]). However, a series of papers [19, 15,
20, 16], authored, in various combinations, by de la Llave, Marco, and
Moriyón, appeared in the 1980s focusing on the study of the conjugacy of
$C^{\infty}$ diffeomorphisms on $\mathbb{T}^{2}$. The culmination of their
work is the following theorem.

######  Theorem.

[16] Let $T_{1}$ and $T_{2}$ be $C^{\infty}$ Anosov diffeomorphisms of
$\mathbb{T}^{2}$. If they are topologically conjugate and the Lyapunov
exponents at corresponding periodic orbits are the same, then the conjugating
homeomorphism is $C^{\infty}$.

Later it was shown that the equality of the corresponding Lyapunov exponents
for $C^{r}$ Anosov diffeomorphisms on $\mathbb{T}^{2}$ implies that the
conjugacy is $C^{r-\epsilon}$, however it is no longer true on
$\mathbb{T}^{4}$ even for $C^{\infty}$ maps [17]. The case of $\mathbb{T}^{3}$
is still open, with a positive result recently obtained when one of the
diffeomorphisms is an automorphism [5].

Note that if $h$ is differentiable, then for any point $x$ of period $n$ for
$T_{1}$, $h(x)$ is of period $n$ for $T_{2}$ and

$DT_{1}^{n}(x)=Dh^{-1}(h(x))DT_{2}^{n}(h(x))Dh(x).$

We see that the Lyapunov exponents of $x$ under $T_{1}$ and $h(x)$ under
$T_{2}$ coincide. The result of [16] is quite remarkable since a condition,
which is a priori weaker than $h$ being $C^{1}$, is shown to imply that $h$ is
$C^{\infty}$. F. Rodriguez Hertz asked whether we can get away with even less.
He proposed to replace the assumption of equality of the Lyapunov exponents by
the equality of the pressure functions of the geometric potentials.

To introduce the pressure function we first define the topological pressure
using the variational principle. The topological pressure of a continuous
potential $\phi:\mathbb{T}^{2}\to\mathbb{R}$ with respect to a dynamical
system $T:\mathbb{T}^{2}\to\mathbb{T}^{2}$ is given by

$P_{\rm top}(T,\phi)=\sup_{\mu}\left\\{h_{\mu}(T)+\int\phi\,d\mu\right\\},$

where $\mu$ runs over the set of all $T$-invariant probability measures on
$\mathbb{T}^{2}$ and $h_{\mu}(T)$ is the measure-theoretic entropy of $\mu$. A
measure $\mu$ which realizes the supremum is called an equilibrium state of
$\phi$. By a celebrated result of Bowen [2], for an Anosov diffeomorphism $T$
any Hölder potential $\phi:\mathbb{T}^{2}\to\mathbb{R}$ has a unique
equlibrium state $\mu_{\phi}$. Equilibrium states are mathematical
generalizations of Gibbs distributions in statistical physics. The most
important ones are the measure of maximal entropy, which is the equilibrium
state of a constant potential, and the SRB measure, which is the equilibrium
state of the _geometric potential_. The geometric potential is the negative
logarithm of the Jacobian of $T$ along the unstable bundle $E^{u}$,

$\phi_{T}^{u}(x)=-\log\big{|}D_{u}T(x)\big{|}.$

The _pressure function_ of a potential $\phi$ is the map $t\mapsto P_{\rm
top}(T,t\phi)$, where $t$ is a real valued parameter. Information about
various dynamical properties of an Anosov system is encoded into the pressure
function of the geometric potential. For example, when $T$ is area preserving,
the positive Lyapunov exponent of $T$ with respect to the normalized Lebesgue
measure (which is the equilibrium state of $\phi_{T}^{u}$) is given by the
negative derivative of the pressure function of $\phi_{T}^{u}$ at $t=1$, while
the derivative at $t=0$ gives the Lyapunov exponent with respect to the
measure of maximal entropy of $T$. F. Rodriguez Hertz asked whether
information on the regularity of the conjugating homeomorphism can also be
extracted from the pressure functions of the geometric potentials of the
corresponding maps. More precisely,

######  Question 1.

[11, attr. F. Rodriguez Hertz] Let $T_{1}$ and $T_{2}$ be $C^{\infty}$ area-
preserving Anosov diffeomorphisms on $\mathbb{T}^{2}$ that are homotopic.
Assume $P_{\rm top}(T_{1},t\phi^{u}_{T_{1}})=P_{\rm
top}(T_{2},t\phi^{u}_{T_{2}})$ for all $t$. Does this imply that $T_{1}$ and
$T_{2}$ are $C^{\infty}$ conjugate?

We point out that the answer to the above question is positive when one of the
diffeomorphisms is an automorphism. Indeed if $T_{1}$ is an automorphism, then
$\phi^{u}_{T_{1}}$ is constant, so that $P_{\rm
top}(T_{1},t\phi^{u}_{T_{1}})$, and hence $P_{\rm
top}(T_{2},t\phi^{u}_{T_{2}})$ is affine. However, pressure functions of
Hölder continuous functions are known to be strictly convex unless the
underlying potential is cohomologous to a constant. Hence $\phi^{u}_{T_{2}}$
is cohomologous to the constant $\phi^{u}_{T_{1}}$. This guarantees that the
Lyapunov exponents of periodic points of $T_{2}$ match those of periodic
orbits of $T_{1}$, so that $T_{1}$ and $T_{2}$ are $C^{\infty}$ conjugate by
the above result.

One reason that Anosov diffeomorphisms on $\mathbb{T}^{2}$ are well-understood
is that they admit symbolic codings. Using a Markov partition of
$\mathbb{T}^{2}$ one can find a finite set $\mathcal{A}$ (indexing the set of
rectangles of the Markov partition) and a mixing subshift of finite type
$\Omega\subset\mathcal{A}^{\mathbb{Z}}$ such that there exists a finite-to-one
factor map $\pi:\Omega\to\mathbb{T}^{2}$ which is Hölder. Then
$\phi^{u}_{T}\circ\pi$ is a Hölder potential on $\Omega$. It turns out that in
the symbolic setting, a related question to Question 1 has been studied by
Pollicott and Weiss in [23].

Suppose $(\Omega,\sigma)$ is a subshift of finite type and
$\psi:\Omega\to\mathbb{R}$ is a Hölder potential. Denote the Birkhoff sum of
$\psi$ by $S_{n}\psi(x)=\sum_{k=0}^{n-1}\psi(\sigma^{k}x)$. The multi-set
$\\{(S_{n}\psi(x),n):\sigma^{n}x=x\\}$ is called the _unmarked orbit spectrum
of $\psi$_. In [23] the extent to which a potential is determined by its
periodic orbit invariants such as its orbit spectrum and its pressure function
was investigated. Note that for subshifts of finite type the pressure function
can be defined topologically as

$P_{\rm
top}(\sigma,t\psi)=\lim_{n\to\infty}\frac{1}{n}\log\left(\sum_{\sigma^{n}x=x}e^{tS_{n}\psi(x)}\right),$

and therefore any two potentials with the same unmarked orbit spectrum must
have identical pressure functions. The converse is not true. It was shown by
Pollicott and Weiss that there exists an uncountable family of Hölder
continuous functions on a full shift with different unmarked orbit spectra,
but all sharing the same pressure function.

Since for Anosov $T:\mathbb{T}^{2}\to\mathbb{T}^{2}$ we have
$-\log\big{|}D_{u}T^{n}\big{|}=S_{n}\phi^{u}_{T}$, the equality of the
Lyapunov exponents at periodic orbits for torus diffeomorphisms $T_{1}$ and
$T_{2}$ corresponds to the equality of the unmarked orbit spectra of their
geometric potentials. Hence Question 1 may be seen as asking whether Hölder
functions arising from geometric potentials of Anosov diffeomorphisms on the
torus are special enough that the equality of their pressure functions implies
the equality of their unmarked orbit spectrum. That turns out not to be the
case.

We show that the set of pressure functions for Anosov diffeomorphisms with
respect to their geometric potentials is equal to the set of pressure
functions for the hyperbolic automorphism with respect to Hölder potentials.

######  Theorem 1.

Let $L$ be a hyperbolic automorphism of $\mathbb{T}^{2}$ and let $\mu$ be the
equilibrium state for a Hölder continuous potential $\phi$ with $P_{\rm
top}(L,\phi)=0$. Then there exists a $C^{1+H}$ area-preserving Anosov
diffeomorphism $T$ of $\mathbb{T}^{2}$ such that

  * •

the system
$T\colon(\mathbb{T}^{2},\mathsf{Leb})\to(\mathbb{T}^{2},\mathsf{Leb})$ is
conjugate to $L\colon(\mathbb{T}^{2},\mu)\to(\mathbb{T}^{2},\mu)$ by a map
$h$;

  * •

the potential $-\log|D_{u}T|\circ h$ is cohomologous to $\phi$.

In this theorem, and throughout the paper, we write $T$ is $C^{1+H}$ to mean
that there exists $0<\alpha<1$ where $T$ is $C^{1+\alpha}$.

A statement similar to the above theorem could be deduced from the work by
Cawley [4] which establishes a bijection between Teichmüller space of an
Anosov diffeomorphism and the quotient of Hölder functions by the subspace of
coboundaries plus constants. However the proofs in [4] appear to be rather
opaque. Our approach is constructive where the main step – the change of
coordinates – is given by an explicit formula in terms of the equilibrium
state of $\phi$.

In view of Theorem 1, to solve Question 1 we need to find Hölder potentials
having identical pressure functions with respect to an automorphism $L$, but
different unmarked orbit spectra. From the work of Pollicott and Weiss one
might expect uncountably many such potentials on the corresponding subshift of
finite type. However, there is no reason to expect that any of these
potentials will be Hölder continuous on the torus. Hence we have to employ
another construction to produce torus continuous examples. We obtain

######  Theorem 2.

There exist homotopic $C^{1+H}$ area-preserving Anosov diffeomorphisms $T_{1}$
and $T_{2}$ on $\mathbb{T}^{2}$ such that $P_{\rm
top}(T_{1},t\phi^{u}_{T_{1}})=P_{\rm top}(T_{2},t\phi^{u}_{T_{2}})$ for all
$t$, but $T_{1}$ and $T_{2}$ fail to be $C^{1}$ conjugate.

In fact our results give countably many homotopic Hölder differentiable area-
preserving Anosov diffeomorphisms, none of which are $C^{1}$ conjugate, but
all having the same pressure function. We do not know whether one can find
uncountably many such maps, as would be suggested by the result in [23].

We remark that our examples, which are in the $C^{1+H}$ category, do not
directly respond to the $C^{\infty}$ question of Rodriguez Hertz; however they
strongly suggest a negative answer to that question also.

Acknowledgement. Part of this work was completed during our one-week stay at
the Centre International de Rencontres Mathématiques in Luminy, France through
the Research in Residence program. We thank CIRM for the support and
hospitality.

##  2\. Preliminary Results

###  2.1. Gibbs Measures and Radon-Nikodym Derivative

In recent works an invariant measure is termed Gibbs if the weight of the
Bowen balls of order $n$ satisfies the growth estimate given in [2, Theorem
1.2]. We recall the original definition of a Gibbs state introduced by Ruelle
[24] and Capocaccia [3], which is equivalent to Bowen’s property from [2] in
our situation. Let $T:M\to M$ be an expansive homeomorphism on a compact
metric space $M$. A map $\chi$ from some open set $U\subset M$ into $M$ is
called _conjugating_ for the system $(M,T)$ if
$d(T^{n}\circ\chi(x),T^{n}(x))\to 0$ for $|n|\to\infty$ uniformly in $x\in U$.
In the case of an Anosov automorphism $L$, the conjugating homeomorphisms are
locally given by $x\mapsto x+v$ where $v$ is homoclinic to 0. For this
article, we only need the global conjugating homeomorphisms $x\mapsto x+v$.

Suppose $\phi$ is a continuous function on $M$. A probability measure $\mu$ on
$M$ is a _Gibbs state_ for $\phi$ if for every conjugating homeomorphism
$\chi:U\to\chi(U)$ where $U=U_{\chi}$ is an open set in $M$ the measure
$\chi_{*}(\mu|_{U})$ is absolutely continuous with respect to
$\mu|_{\chi(U)}$, with Radon-Nikodym derivative

(1) $\frac{d\chi_{*}\mu}{d\mu}=\exp\sum_{n\in\mathbb{Z}}\big{[}\phi\circ
T^{n}\circ\chi^{-1}-\phi\circ T^{n}\big{]}.$

For an axiom A diffeomorphism the equilibrium state of a Hölder potential
$\phi$ is also a Gibbs state for $\phi$, which is proven in Ruelle’s book [24,
Theorem 7.18]. A result of Haydn [9] is that the converse holds as well. In
fact, Haydn and Ruelle show in [10] that equilibrium states and Gibbs states
are equivalent for expansive homeomorphisms with specification and Bowen
potentials.

We need the regularity properties of the Radon-Nikodym derivative (1).
Although the question of regularity seems to be very natural, we were not able
to locate a corresponding result in the literature. We provide a proof in the
case of Anosov automorphisms, however the same argument can be
straightforwardly generalized to Anosov diffeomorphisms, Axiom A
diffeomorphisms or more general Smale spaces.

######  Lemma 3.

Let $L:\mathbb{T}^{2}\to\mathbb{T}^{2}$ be an Anosov automorphism, let $v$ be
homoclinic to 0 and let $\tau(x)=x-v$. Let $\phi$ be a Hölder continuous
function and let $\mu$ be the corresponding equilibrium state. Then the Radon-
Nikodym derivative $\frac{d\tau_{*}\mu}{d\mu}$ in (1) above is Hölder
continuous.

###### Proof.

Let $\lambda$ be the expanding eigenvalue of $L$. Then there exist $C_{1}$ and
$C_{2}$ such that $d(L^{n}v,0)\leq C_{1}\lambda^{-|n|}$ and $d(L^{n}x,0)\leq
C_{2}\lambda^{|n|}d(x,0)$ for all $n\in\mathbb{Z}$. We let $C_{3}>0$ and
$\alpha\in(0,1)$ be such that $|\phi(x)-\phi(y)|\leq C_{3}d(x,y)^{\alpha}$ for
all $x,y\in\mathbb{T}^{2}$.

We define

$\theta(x)=\sum_{n\in\mathbb{Z}}\left[\phi(L^{n}(x+v))-\phi(L^{n}x)\right].$

Suppose $x,y\in\mathbb{T}^{2}$ satisfy $d(x,y)<\lambda^{-2k}$ for some $k$.
Then we calculate

$\displaystyle|\theta(y)$
$\displaystyle-\theta(x)|\leq\sum_{n\in\mathbb{Z}}\big{|}\phi(L^{n}(y+v))-\phi(L^{n}y)-\phi(L^{n}(x+v))+\phi(L^{n}x)\big{|}$
$\displaystyle\leq\sum_{|n|\leq
k}\big{[}|\phi(L^{n}(y+v))-\phi(L^{n}(x+v))|+|\phi(L^{n}(y))-\phi(L^{n}(x))|\big{]}$
$\displaystyle+\sum_{|n|>k}\big{[}|\phi(L^{n}(y+v))-\phi(L^{n}(y))|+|\phi(L^{n}(x+v))-\phi(L^{n}(x))|\big{]}.$

We bound the sums by geometric series and obtain

$|\phi(L^{n}y)-\phi(L^{n}x)|\leq C_{3}(C_{2}\lambda^{|n|}d(x,y))^{\alpha}\leq
C_{3}C_{2}^{\alpha}\lambda^{-(2k-|n|)\alpha},$

with the same bound for $|\phi(L^{n}(y+v))-\phi(L^{n}(x+v))|$. Likewise,
$|\phi(L^{n}(x+v))-\phi(L^{n}(x))|\leq
C_{3}C_{1}^{\alpha}\lambda^{-|n|\alpha}$ with the same bound for
$|\phi(L^{n}(y+v))-\phi(L^{n}(y))|$. Summing the geometric series, we obtain
$|\theta(y)-\theta(x)|\leq K\lambda^{-k\alpha}$, where
$K=4C_{3}(C_{1}^{\alpha}+C_{2}^{\alpha})/(1-\lambda^{-\alpha})$ showing that
$\theta$ is Hölder as required. ∎

###  2.2. Coding for Toral Automorphisms

Let $L$ be a mixing toral automorphism of $\mathbb{T}^{2}$ and we let
$\mathcal{P}$ be a generating Markov partition, which we assume to consist of
(closed) rectangles whose boundaries are pieces of the unstable and stable
manifolds through the origin. We make the further assumption that if $A$ and
$B$ are elements of the partition, then $(A+v)\cap B$ is connected (either a
rectangle or an empty set). This condition is automatically satisfied if
$\operatorname{diam}(\mathcal{P})<\frac{1}{2}$, and so may be assumed without
loss of generality by replacing $\mathcal{P}$ with a Markov partition of the
form $\bigvee_{j=0}^{m-1}L^{-j}\mathcal{P}$ if necessary.

For $\mathcal{A}=\\{0,...,\\#(\mathcal{P})-1\\}$ let
$\Omega\subset\mathcal{A}^{\mathbb{Z}}$ be the corresponding shift of finite
type and let $\pi\colon\Omega\to\mathbb{T}^{2}$ be the corresponding finite-
to-one factor map from $(\Omega,\sigma)$ to $(\mathbb{T}^{2},L)$. The map
$\pi$ is one-to-one on a set of measure 1 with respect to any invariant
measure on $\Omega$. We equip $\Omega$ with the standard metric on $\Omega$
where $d(\omega,\omega^{\prime})=2^{-n}$ if $\omega_{j}=\omega^{\prime}_{j}$
whenever $|j|<n$, but $\omega_{\pm n}\neq\omega^{\prime}_{\pm n}$.

If $\phi$ is a Hölder continuous function on $\mathbb{T}^{2}$, we let $\mu$ be
its equilibrium measure. We also set $\psi=\phi\circ\pi$ to be the
corresponding potential on $\Omega$ and let $\nu$ be the equilibrium measure
of $\psi$. Since $\pi$ is one-to-one $\nu$-almost everywhere,
$\pi_{*}\nu=\mu$. Let $\Omega^{+}\subset\mathcal{A}^{\mathbb{N}_{0}}$ be the
one-sided version of $\Omega$ that is the image of $\Omega$ under the map
$p_{+}\colon\mathcal{A}^{\mathbb{Z}}\to\mathcal{A}^{\mathbb{N}_{0}}$ defined
by $p_{+}(\omega)_{n}=\omega_{n}$ for $n\geq 0$. Similarly, let
$\Omega^{-}\subset\mathcal{A}^{-\mathbb{N}}$ be the image of $\Omega$ under
the restriction map
$p_{-}\colon\mathcal{A}^{\mathbb{Z}}\to\mathcal{A}^{-\mathbb{N}}$. Then
$\nu^{+}=(p_{+})_{\ast}\nu$ and $\nu^{-}=(p_{-})_{\ast}\nu$ are the measures
corresponding to $\nu$ on $\Omega^{+}$ and $\omega^{-}$ respectively.

The main symbolic result we are using is the local product structure of $\nu$.
Ruelle proves in [24, Lemma 5.9] that $\nu$ has _local product structure_ ,
i.e.
$d\nu(\omega)=\hat{\varrho}(\omega)\,d\hat{\nu}^{+}(p_{+}(\omega))\,d\hat{\nu}^{-}(p_{-}(\omega))$
where $\hat{\nu}^{+}$ is a probability measure on $\Omega^{+}$,
$\hat{\nu}^{-}$ is a probability measure on $\Omega^{-}$, and $\hat{\varrho}$
is a positive continuous function on $\Omega$. Furthermore, it is shown in
[24, Lemma 5.23] that $\hat{\varrho}$ is Hölder on $\Omega$, and the functions
$\hat{\varrho}^{+}(\omega^{+})=\int\hat{\varrho}(\omega)\,d\hat{\nu}^{-}(\omega^{-})$,
$1/\hat{\varrho}^{+}(\omega^{+})$ are Hölder on $\Omega^{+}$. Analogous
statements hold for $\hat{\varrho}^{-}(\omega^{-})$. Note that for each
$\omega^{+}\in\Omega^{+}$ the integral is taken over the set
$\\{\omega^{-}\in\Omega^{-}\colon\omega^{-}_{-1}\omega^{+}_{0}\text{ is legal
in $\Omega$}\\}$. In this case the measure $\nu^{+}$ on $\Omega^{+}$ is given
by $d\nu^{+}=\varrho^{+}(\omega^{+})\,d\hat{\nu}^{+}$; similarly for
$\nu^{-}$.

We are mostly concerned with the structure of $\nu$ on the cylinder
$[0]=\\{\omega\in\Omega:\omega_{0}=0\\}$. We let
$A^{-}=\\{\omega^{-}\in\Omega^{-}\colon\omega^{-}_{-1}0\text{ is legal in
$\Omega$}\\}$. For $\omega^{-}\in A^{-}$ and $\omega^{+}\in p_{+}([0])$ we
write
$\varrho^{+}(\omega^{+})=\int_{A^{-}}\hat{\varrho}(\omega^{-}\omega^{+})\,d\hat{\nu}^{-}(\omega^{-})$,
$\varrho^{-}(\omega^{-})=\int_{[0]}\hat{\varrho}(\omega^{-}\omega^{+})\,d\hat{\nu}^{+}(\omega^{+})$,
and
$\varrho(\omega^{-}\omega^{+})=\frac{\hat{\varrho}(\omega^{-}\omega^{+})}{\varrho^{-}(\omega^{-})\varrho^{+}(\omega^{+})}$,
so that
$d\nu(\omega)=\rho(\omega)\,d\nu^{+}(\omega^{+})\,d\nu^{-}(\omega^{-})$. In
particular,

(2)
$\begin{split}\int_{A^{-}}\varrho(\omega^{-}\omega^{+})\,d\nu^{-}(\omega^{-})&=\int_{A^{-}}\frac{\hat{\varrho}(\omega^{-}\omega^{+})}{\varrho^{-}(\omega^{-})\varrho^{+}(\omega^{+})}\,d\nu^{-}(\omega^{-})\\\
&=\frac{1}{\varrho^{+}(\omega^{+})}\int_{A^{-}}\hat{\varrho}(\omega^{-}\omega^{+})\,d\hat{\nu}^{-}(\omega^{-})\\\
&=1\end{split}$

We summarize the above in the following lemma which is frequently used
throughout this article.

######  Lemma 4 (Ruelle [24]).

Let $\psi$ be a Hölder continuous function on a mixing shift of finite type
$\Omega$ and let $\nu$ be its equilibrium state. Then $\nu$ has _local product
structure_. That is, on the cylinder set $[0]$ there exist a positive Hölder
continuous function $\varrho(\omega)$ such that
$d\nu(\omega)=\varrho(\omega)\,d\nu^{+}(\omega^{+})\,d\nu^{-}(\omega^{-})$
where $\nu^{-}$, $\nu^{+}$ are the restrictions of $\nu$ to $\Omega^{+}$,
$\Omega^{-}$ respectively, and $\omega$ denotes the concatenation of
$\omega^{-}$ and $\omega^{+}$.

It is shown by Walters in [26] that under the assumptions of the above lemma
there is a Hölder function $g:\Omega^{+}\to(0,1)$ such that $\log g$ is
cohomologous to $\phi$ and $\nu^{+}$ is the unique $g$-measure for $g$, i.e.
for $\omega^{+}\in\Omega^{+}$

(3) $g(\omega^{+})=\lim_{\begin{subarray}{c}{\rm diam}(S)\to 0\\\
\nu^{+}(S)\neq 0,\,\omega^{+}\in
S\end{subarray}}\frac{\nu^{+}(S)}{\nu^{+}(\sigma_{+}(S))}.$

Since the map $\pi\colon\Omega\to\mathbb{T}^{2}$ is Hölder continuous, given a
Hölder continuous function $\phi$ on the torus, we see that $\phi\circ\pi$ is
Hölder; however many Hölder continuous functions on the shift cannot be
written in the form $\phi\circ\pi$. We call a function $f$ defined on $\Omega$
_torus-Hölder_ if it can be written in the form $\phi\circ\pi$ where $\phi$ is
a Hölder continuous function of the torus. A subset $R$ of $\Omega$ is called
a _rectangle_ if it satisfies the following conditions

  * •

$\omega,\omega^{\prime}\in R$ implies the concatenation
$p_{-}(\omega)p_{+}(\omega^{\prime})$ belongs to $R$;

  * •

$\pi(R)$ is connected;

  * •

$\operatorname{diam}(\pi(R))<\frac{1}{2}$;

  * •

$R=\pi^{-1}(\pi(R))$.

######  Lemma 5.

Let $L$ be an Anosov automorphism of $\mathbb{T}^{2}$ and let $\mathcal{P}$ be
a Markov partition as described above. Let $\Omega$ be the corresponding shift
of finite type and let $\pi\colon\Omega\to\mathbb{T}^{2}$ be the natural
factor map. Let $R$ be a rectangular subset of a cylinder set $[i]$ in
$\Omega$ and suppose that $f\colon R\to\mathbb{R}$ is a Hölder continuous
function. If $f$ has the property that $f(\omega)=f(\omega^{\prime})$ whenever
$\pi(\omega)=\pi(\omega^{\prime})$, then $f$ may be expressed as $h\circ\pi$
where $h$ is a Hölder continuous function defined on
$\pi(R)\subset\mathbb{T}^{2}$.

###### Proof.

Since $f(\omega)=f(\omega^{\prime})$ when $\pi(\omega)=\pi(\omega^{\prime})$,
we see that $f$ takes the same value on each element of $\pi^{-1}(x)$ for any
$x\in\pi(R)$. Hence $h(x):=f(\pi^{-1}x)$ is well-defined on the rectangle
$A:=\pi(R)$ which has sides parallel to the stable and unstable directions.
Since $f$ is Hölder continuous, let $c$ and $\alpha$ be such that
$|f(\omega)-f(\omega^{\prime})|\leq c\alpha^{n}$ whenever
$d(\omega,\omega^{\prime})\leq 2^{-n}$.

Since $A$ is a rectangle in $\mathbb{T}^{2}$, we define for $x,y\in A$,
$\llbracket x,y\rrbracket_{A}$ to be the unique point $z$ in $A$ such that the
line segments $[x,z]$ and $[z,y]$ lie in $A$ with $[x,z]$ in the stable
direction and the $[z,y]$ in the unstable direction. We now estimate
$|h(x)-h(z)|$. An exactly similar estimate applies to $|h(z)-h(y)|$. Let $C$
be the constant (depending only on the angle between the stable and unstable
directions) so that if $x,y$ lie in $A$ then $d(x,\llbracket
x,y\rrbracket_{A}),d(y,\llbracket x,y\rrbracket_{A})\leq Cd(x,y)$. Let
$\lambda$ be the expanding eigenvalue and let $n$ be the smallest natural
number such that $C^{-1}\operatorname{diam}(\mathcal{P})\lambda^{-n}\leq
d(x,y)$.

Let $x=\pi(\xi)$ and $\llbracket x,y\rrbracket_{A}=\pi(\zeta)$. Then either
$x$ and $\llbracket x,y\rrbracket_{A}$ lie in the same element of
$L^{j}\mathcal{P}$ for each $0\leq j<n$, in which case $|h(x)-h(\llbracket
x,y\rrbracket_{A})|=|f(\xi)-f(\zeta)|\leq c\alpha^{n}$ or there exists a point
$w$ in $\partial L^{-(n-1)}\mathcal{P}\cap[x,\llbracket x,y\rrbracket_{A}]$.
Since $d(x,w)$ and $d(\llbracket x,y\rrbracket_{A},w)$ are less than
$\operatorname{diam}(\mathcal{P})\lambda^{-(n-1)}$ and $w$ is on the boundary,
$x$ and $w$ must belong to a common element of $L^{-(n-1)}\mathcal{P}$ and
similarly for $w$ and $\llbracket x,y\rrbracket_{A}$, see Fig 1.

$\mathbb{T}^{2}$$A_{0}$$x$$\llbracket x,y\rrbracket_{A}$$w$ Figure 1. On
$\mathbb{T}^{2}$ the unstable and stable directions are shown as north-east
and north-west respectively.

Now write $w=\pi(\eta)=\pi(\eta^{\prime})$ where
$\eta_{-\infty}^{n-1}=\xi_{-\infty}^{n-1}$ and
${\eta^{\prime}}_{-\infty}^{n-1}=\zeta_{-\infty}^{n-1}$. We then have
$|h(x)-h(z)|=|f(\xi)-f(\zeta)|\leq|f(\xi)-f(\eta)|+|f(\eta^{\prime})-f(\zeta)|\leq
2c\alpha^{n},$ where we made use of the fact that $f(\eta)=f(\eta^{\prime})$.
Combining this with the analogous estimate for $|h(\llbracket
x,y\rrbracket_{A})-h(y)|$, we see $|h(x)-h(y)|\leq 4c\alpha^{n}\leq
4c\big{(}Cd(x,y)/(\lambda\operatorname{diam}(\mathcal{P})\big{)}^{-\log\alpha/\log\lambda}$,
so that $h$ is Hölder as required. ∎

##  3\. Anosov realization

In this section we show that given a hyperbolic automorphism $L$ for any
positive Hölder continuous potential $\phi$ with zero topological pressure
there exists a conjugate Anosov diffeomorphism $T$ for which the geometric
potential is cohomologous to $\phi$.

######  Theorem 6.

Let $L$ be an Anosov automorphism of $\mathbb{T}^{2}$ and let $\mu$ be the
equilibrium state for a Hölder continuous potential $\phi$ with $P_{\rm
top}(L,\phi)=0$. Then there exists a $C^{1+H}$-atlas on $\mathbb{T}^{2}$ with
respect to which $L$ is an Anosov diffeomorphism with Hölder derivative and
its geometric potential is cohomologous to $\phi$.

We prove the theorem in a number of steps.

###  3.1. Definition of new $C^{1+H}$ atlas

We let $\mathcal{H}$ denote the collection of points of $\mathbb{T}^{2}$ that
are homoclinic to 0 under the action of $L$. Since $L$ is an automorphism, it
follows that if $v\in\mathcal{H}$ and $x\in\mathbb{T}^{2}$ then
$d(L^{n}(x+v),L^{n}(x))=d(L^{n}v,0)\to 0$ as $|n|\to\infty$. Recall that the
points homoclinic to 0 are dense in $\mathbb{T}^{2}$ (see e.g. [21]).

For the remainder of this section $A_{0}$ denotes the element of the partition
$\mathcal{P}$ which corresponds to the cylinder $[0]$ in $\Omega$, i.e.
$\pi([0])=A_{0}$.

######  Lemma 7.

Let $w\in\mathcal{H}$ and suppose that $A_{0}\cap(A_{0}-w)$ has non-empty
interior. Then there exist vectors $u,v\in\mathcal{H}$ such that:

  * •

$u+v=w$;

  * •

if $x\in\text{Int}(A_{0}\cap(A_{0}-u))$ then the line segment $[x,x+u]$ lies
in $\text{Int}(A_{0})$ and is parallel to the stable direction;

  * •

if $x\in\text{Int}(A_{0}\cap(A_{0}-v))$ then the line segment $[x,x+v]$ lies
in $\text{Int}(A_{0})$ and is parallel to the unstable direction;

  * •

$\text{Int}(A_{0}\cap(A_{0}-w))=\text{Int}\big{(}A_{0}\cap(A_{0}-u)\cap(A_{0}-v)\big{)}$.

###### Proof.

For any $x\in\text{Int}(A_{0}\cap(A_{0}-w))$, since $A_{0}$ is a parallelogram
with edges parallel to the stable and unstable directions, the vector $w$ may
be expressed as a sum of pieces $u$ and $v$ parallel to the stable and
unstable directions, where $[x,x+u]$ and $[x,x+v]$ lie in $A_{0}$. Note that
$x+u$ is the point of intersection of the stable manifold of $x$ and the
unstable manifold of $x+w$. Linearity of $L$ implies that $u$ belongs to the
stable manifold of 0 and unstable manifold of $w$. Since $w\in\mathcal{H}$, we
conclude that $u\in\mathcal{H}$ as well. Similarly, $x+v$ is the point of
intersection of the unstable manifold of $x$ and the stable manifold of $x+w$,
so $v\in\mathcal{H}$. ∎

We define two functions $\xi_{1}$ and $\xi_{2}$ on $A_{0}$. Let $\xi_{1}(x)$
be the $\mu$-measure of the rectangle contained in $A_{0}$ lying to the left
of the connected portion of the stable manifold of $x$ within $A_{0}$ as
illustrated in Figure 2. Similarly, let $\xi_{2}(x)$ be the $\mu$-measure of
the rectangle contained in $A_{0}$ lying below the connected portion of the
unstable manifold of $x$ within $A_{0}$. We denote
$\xi(x)=(\xi_{1}(x),\xi_{2}(x))$.

$\mathbb{T}^{2}$$A_{0}$$x$ Figure 2.  $\xi_{1}(x)$ is the measure of the
region shaded with horizontal lines; $\xi_{2}(x)$ is the measure of the region
shaded with vertical lines.

We introduce a new family of charts on $\mathbb{T}^{2}$. For
$v\in\mathcal{H}$, let $\tau_{v}$ denote the translation $\tau_{v}(x)=x+v$. We
then define a chart $\alpha_{v}$ with domain $\text{Int}(A_{0})-v$ by
$\alpha_{v}=\xi\circ\tau_{v}$. Since $\mathcal{H}$ is dense in
$\mathbb{T}^{2}$, the collection of charts covers all of $\mathbb{T}^{2}$. Our
goal for the reminder of this subsection is to show that the family of charts
$\\{(\alpha_{v},\text{Int}(A_{0})-v)\\}_{v\in\mathcal{H}}$ forms a
$C^{1+H}$-differentiable atlas on $\mathbb{T}^{2}$. We first prove a key
lemma.

Let $v\in\mathcal{H}$ be such that $A_{0}\cap(A_{0}-v)$ has non-empty interior
and such that for any $x\in\text{Int}(A_{0}\cap(A_{0}-v))$, the line segment
joining $x$ and $x+v$ lies in $\text{Int}(A_{0})$ and is parallel to the
unstable direction. Using the notation from Section 2.2 we consider the
function $\xi_{1}(\pi(\omega))$ defined on
$\pi^{-1}\big{(}\text{Int}(A_{0}\cap(A_{0}-v))\big{)}\subset[0]$ in $\Omega$
and study the limit

(4)
$\ell(\omega):=\lim_{\omega^{\prime}\to\omega}\frac{\xi_{1}(\pi(\omega^{\prime})+v)-\xi_{1}(\pi(\omega)+v)}{\xi_{1}(\pi(\omega^{\prime}))-\xi_{1}(\pi(\omega))}.$

Here the limit is taken over those $\omega^{\prime}$ such that
$\xi_{1}(\pi(\omega^{\prime}))\neq\xi_{1}(\pi(\omega))$, that is those
$\omega^{\prime}$ such that $\pi(\omega^{\prime})$ does not lie in the same
local stable manifold as $\pi(\omega)$. This is illustrated in Figure 3.

$\mathbb{T}^{2}$$A_{0}$$\pi(\omega)$$\pi(\omega^{\prime})$$v$ Figure 3. The
numerator and denominator in the limit are respectively the measures of the
right and left shaded rectangles.

######  Lemma 8.

Let $v\in\mathcal{H}$ be as described above. Then the limit $\ell(\omega)$,
defined above, exists for all $\omega$ in
$\pi^{-1}\big{(}\text{Int}(A_{0}\cap(A_{0}-v))\big{)}$ and the function
$\ell(\omega)$ is torus-Hölder on its domain.

###### Proof.

Letting $R[\omega,\omega^{\prime}]$ be the rectangle bounded on the top and
bottom by the boundary of $A_{0}$ and the left and right by the stable
manifolds through $\pi(\omega)$ and $\pi(\omega^{\prime})$, we see that

(5)
$\frac{\xi_{1}(\pi(\omega^{\prime})+v)-\xi_{1}(\pi(\omega)+v)}{\xi_{1}(\pi(\omega^{\prime}))-\xi_{1}(\pi(\omega))}=\frac{\mu(R[\omega,\omega^{\prime}]+v)}{\mu(R[\omega,\omega^{\prime}])}$

We now apply the discussion of Section 2.1 to the case when $T$ is the toral
automorphism $L$. For any $v\in\mathbb{T}^{2}$ homoclinic to 0 under $L$ the
map $x\mapsto x+v$ is a (global) conjugating homeomorphism of
$\mathbb{T}^{2}$. It follows from Lemma 3 that for an equilibrium state $\mu$
of a Hölder potential $\phi$ we have

$\frac{d\mu(x+v)}{d\mu(x)}=\theta_{v}(x),$

where

(6)
$\theta_{v}(x)=\exp\left(\sum_{n\in\mathbb{Z}}\big{[}\phi(L^{n}(x+v))-\phi(L^{n}(x))\big{]}\right).$

Recall that by Lemma 3 the function
$\theta_{v}\colon\mathbb{T}^{2}\to\mathbb{R}$ is Hölder continuous.

We can now rewrite (4) as

$\displaystyle\ell(\omega)$
$\displaystyle=\lim_{\omega^{\prime}\to\omega}\frac{\mu(R[\omega,\omega^{\prime}]+v)}{\mu(R[\omega,\omega^{\prime}])}$
$\displaystyle=\lim_{\omega^{\prime}\to\omega}\frac{\int_{R[\omega,\omega^{\prime}]}\theta_{v}(x)\,d\mu(x)}{\int_{R[\omega,\omega^{\prime}]}1\,d\mu(x)}.$

We observe that $\pi^{-1}R[\omega,\omega^{\prime}]$ is a subset of $\Omega$
consisting of points $\zeta$ such that $\zeta_{0}^{\infty}$ are the non-
negative coordinates of points lying between $\pi(\omega)$ and
$\pi(\omega^{\prime})$. There is no restriction on the negative coordinates
other than that $\zeta\in\Omega$ and $\zeta_{0}=0$. Write
$A^{+}[\omega,\omega^{\prime}]$ for
$\\{\zeta^{+}\in\Omega^{+}\colon\zeta^{+}\text{ are the non-negative
coordinates of a point in $R[\omega,\omega^{\prime}]$}\\}$ and $A^{-}$ for
$\\{\zeta^{-}\in\Omega^{-}\colon\text{ $\zeta^{-}_{-1}0$ is legal in
$\Omega$}\\}$. We now apply Lemma 4, giving

(7)
$\ell(\omega)=\lim_{\omega^{\prime}\to\omega}\frac{\int_{A^{-}}\int_{A^{+}[\omega,\omega^{\prime}]}\varrho(\zeta)\theta_{v}(\pi(\zeta))\,d\nu^{+}(\zeta^{+})\,d\nu^{-}(\zeta^{-})}{\int_{A^{-}}\int_{A^{+}[\omega,\omega^{\prime}]}\varrho(\zeta)\,d\nu^{+}(\zeta^{+})\,d\nu^{-}(\zeta^{-})}.$

Since $\varrho$ and $\theta_{v}\circ\pi$ are continuous, the integrands in the
numerator and denominator may be approximated for $\omega^{\prime}$ close to
$\omega$ by $\varrho(\zeta^{-}\omega^{+})\theta_{v}(\pi(\zeta^{-}\omega^{+}))$
and $\varrho(\zeta^{-}\omega^{+})$ respectively. Since these new integrands
don’t depend on $\zeta^{+}$, the inner integrals of the approximation to (7)
are just the product of the integrand and
$\nu^{+}(A^{+}[\omega,\omega^{\prime}])$. Since $\rho$ is strictly positive,
cancelling the common factor, we now see that the limit exists, and

$\displaystyle\ell(\omega)$
$\displaystyle=\frac{\int_{A^{-}}\varrho(\zeta^{-}\omega^{+})\theta_{v}(\pi(\zeta^{-}\omega^{+}))\,d\nu^{-}(\zeta^{-})}{\int_{A^{-}}\varrho(\zeta^{-}\omega^{+})\,d\nu^{-}(\zeta^{-})}$
$\displaystyle=\int_{A^{-}}\varrho(\zeta^{-}\omega^{+})\theta_{v}(\pi(\zeta^{-}\omega^{+}))\,d\nu^{-}(\zeta^{-}),$

where the second equality follows from (2). Further, since $\varrho$ and
$\theta_{v}\circ\pi$ are Hölder continuous functions on $\Omega$, we can see
that $\ell(\omega)$ is a Hölder continuous function of $\omega$ on $[0]$,
depending only on the non-negative coordinates of $\omega$.

In order to show that $\ell(\omega)$ is also torus-continuous, we consider
$\omega$ belonging to the stable manifold of 0 (so that $\pi(\omega)$, which
we assumed to lie in $\text{Int}(A_{0})$, lies on the boundary of two elements
of $L^{-j}\mathcal{P}$ for some $j>0$: one on the left and one on the right).
In this case, $p_{+}^{-1}(\pi(\omega))$ consists of two elements, say
$\omega^{+}$ and $\eta^{+}$. We will show that
$\ell({\omega^{-}\omega^{+}})=\ell({\omega^{-}\eta^{+}})$.

It will be convenient to find another expression for $\ell(\omega)$ in which
$\pi(\omega)$ is translated by another homoclinic vector $\tilde{v}$ (which by
Lemma 7 we can assume to be parallel to the unstable direction and to satisfy
$[\pi(\omega),\pi(\omega)+\tilde{v}]\subset\text{Int}(A_{0})$). Let
$\tilde{R}[\omega,\omega^{\prime}]=R[\omega,\omega^{\prime}]+\tilde{v}$ and
denote by $\tilde{A}^{+}[\omega,\omega^{\prime}]$ the set of future codes of
points in the rectangle $\tilde{R}[\omega,\omega^{\prime}]$.

By similar arguments to those above and using the fact that
$\theta_{v-\tilde{v}}(x)=\theta_{-\tilde{v}}(x)\theta_{v}(x-\tilde{v})$ which
is immediate from the expression of the Radon-Nikodym derivative (6), we
obtain

$\displaystyle\ell(\omega)$
$\displaystyle=\lim_{\omega^{\prime}\to\omega}\frac{\mu(\tilde{R}[\omega,\omega^{\prime}]+v-\tilde{v})}{\mu(\tilde{R}[\omega,\omega^{\prime}]-\tilde{v})}$
$\displaystyle=\lim_{\omega^{\prime}\to\omega}\frac{\int_{A^{-}}\int_{\tilde{A}^{+}[\omega,\omega^{\prime}]}\varrho(\zeta)\theta_{-\tilde{v}}(\pi(\zeta))\theta_{v}(\pi(\zeta)-\tilde{v})\,d\nu^{+}(\zeta^{+})\,d\nu^{-}(\zeta^{-})}{\int_{A^{-}}\int_{\tilde{A}^{+}[\omega,\omega^{\prime}]}\varrho(\zeta)\theta_{-\tilde{v}}(\pi(\zeta))\,d\nu^{+}(\zeta^{+})\,d\nu^{-}(\zeta^{-})},$

As before, taking a limit as $\omega^{\prime}$ approaches $\omega$, we see
that

(8)
$\ell(\omega^{-}\omega^{+})=\frac{\int_{A^{-}}\varrho(\zeta^{-}\tilde{\omega}^{+})\theta_{-\tilde{v}}(\pi(\zeta^{-}\tilde{\omega}^{+}))\theta_{v}(\pi(\zeta^{-}\omega^{+}))\,d\nu^{-}(\zeta^{-})}{\int_{A^{-}}\varrho(\zeta^{-}\tilde{\omega}^{+})\theta_{-\tilde{v}}(\pi(\zeta^{-}\tilde{\omega}^{+}))\,d\nu^{-}(\zeta^{-})},$

where $\tilde{\omega}^{+}$ is the future coding of $\pi(\omega)+\tilde{v}$
corresponding to $\omega^{+}$.

Letting $\tilde{\eta}^{+}$ be the future coding of $\pi(\omega)+\tilde{v}$
corresponding to $\eta^{+}$ we get

(9)
$\begin{split}\ell({\omega^{-}\eta^{+}})&=\frac{\int_{A^{-}}\varrho(\zeta^{-}\tilde{\eta}^{+})\theta_{-\tilde{v}}(\pi(\zeta^{-}\tilde{\eta}^{+}))\theta_{v}(\pi(\zeta^{-}\eta^{+}))\,d\nu^{-}(\zeta^{-})}{\int_{A^{-}}\varrho(\zeta^{-}\tilde{\eta}^{+})\theta_{-v^{\prime}}(\pi(\zeta^{-}\tilde{\eta}^{+}))\,d\nu^{-}(\zeta^{-})}\\\
&=\frac{\int_{A^{-}}\varrho(\zeta^{-}\tilde{\eta}^{+})\theta_{-\tilde{v}}(\pi(\zeta^{-}\tilde{\omega}^{+}))\theta_{v}(\pi(\zeta^{-}\omega^{+}))\,d\nu^{-}(\zeta^{-})}{\int_{A^{-}}\varrho(\zeta^{-}\tilde{\eta}^{+})\theta_{-\tilde{v}}(\pi(\zeta^{-}\tilde{\omega}^{+}))\,d\nu^{-}(\zeta^{-})},\end{split}$

where we used the facts
$\pi(\zeta^{-}\tilde{\eta}^{+})=\pi(\zeta^{-}\tilde{\omega}^{+})$ and
$\pi(\zeta^{-}\eta^{+})=\pi(\zeta^{-}\omega^{+})$.

Comparing (8) and (9), we see that the only place where they differ is that in
the numerator and denominator, $\varrho(\zeta^{-}\tilde{\omega}^{+})$ is
replaced by $\varrho(\zeta^{-}\tilde{\eta}^{+})$. However if $\tilde{v}$ is
chosen so that $\pi(\omega)+\tilde{v}$ does not lie on the stable boundary of
any element of $\bigvee_{0\leq j<n}L^{-j}\mathcal{P}$, then $\tilde{\eta}^{+}$
and $\tilde{\omega}^{+}$ agree for at least $n$ symbols. Since $\varrho$ is
Hölder continuous,
$\varrho(\zeta^{-}\tilde{\eta}^{+})/\varrho(\zeta^{-}\tilde{\omega}^{+})$ is
uniformly exponentially close to 1 as $\zeta^{-}$ runs over $A^{-}$. It
follows that $\ell(\omega)=\ell({\omega^{-}\zeta^{+}})$, so that $\ell$ is
torus-continuous. ∎

We are now ready to establish that the atlas
$\\{(\alpha_{v},\text{Int}(A_{0}-v))\colon v\in\mathcal{H}\\}$ is $C^{1+H}$.
We need to prove that for $v_{0},v_{1}\in\mathcal{H}$ with the property that
$\text{Int}(A_{0}-v_{0})\cap\text{Int}(A_{0}-v_{1})\neq\emptyset$, the map
$\alpha_{v_{1}}\circ\alpha_{v_{0}}^{-1}$ is differentiable with Hölder
continuous derivative. In this case, observe
$\alpha_{v_{1}}\circ\alpha_{v_{0}}^{-1}=(\xi\circ\tau_{v_{1}})\circ(\xi\circ\tau_{v_{0}})^{-1}=\xi\circ\tau_{w}\circ\xi^{-1}$,
where $w=v_{1}-v_{0}\in\mathcal{H}$.

Using Lemma 7, we write $w=v+u$, where $v$ is in the unstable direction and
$u$ is in the stable direction. Moreover, if both $x$ and $x+w$ are in
$\text{Int}(A_{0})$, then the line segment joining $x$ and $x+v$ lies in
$\text{Int}(A_{0})$, so that $v$ satisfies the conditions of Lemma 8. Let
$h_{1}$ be the Hölder continuous function on
$\text{Int}(A_{0})\cap\text{Int}(A_{0}-w)$ such that $\ell=h_{1}\circ\pi$ on
their domain.

We now evaluate the derivative of $\xi\circ\tau_{w}\circ\xi^{-1}$ using the
function $\ell$. If $(a,b)$ and $(a,b^{\prime})$ have the same first
coordinate and are in the range of $\xi\circ\tau_{w}\circ\xi^{-1}$, then we
see from the definition of $\xi$ that $\tau_{w}\circ\xi^{-1}(a,b)$ and
$\tau_{w}\circ\xi^{-1}(a,b^{\prime})$ lie on the same stable manifold, so that
the first coordinates of $\xi\circ\tau_{w}\circ\xi^{-1}(a,b)$ and
$\xi\circ\tau_{w}\circ\xi^{-1}(a,b^{\prime})$ agree. Similarly the second
coordinates of $\xi\circ\tau_{w}\circ\xi^{-1}(a,b)$ and
$\xi\circ\tau_{w}\circ\xi^{-1}(a^{\prime},b)$ agree, so that
$\xi\circ\tau_{w}\circ\xi^{-1}(a,b)$ is of the form $(f_{1}(a),f_{2}(b))$. We
see from the definition of $\ell$ that for $(a,b)$ in the domain of
$\xi\circ\tau_{w}\circ\xi^{-1}$,
$f_{1}^{\prime}(a)=h_{1}(\xi^{-1}(a,b))=h_{1}(\xi_{1}^{-1}(a)\cap\xi_{2}^{-1}(b))$.
Since $h_{1}$ is constant on local stable manifolds, this can also be written
as $h_{1}(\xi_{1}^{-1}(a))$.

We verify that $f_{1}^{\prime}$ is Hölder; an almost identical argument will
show that $f_{2}^{\prime}$ is Hölder. Let $e_{u}$ be the unit unstable
direction and $z$ be the bottom left corner of $A_{0}$. Using
$\iota(t)=\xi_{1}(z+te_{u})$ we can write
$f_{1}^{\prime}(a)=h_{1}(z+\kappa^{-1}(a)e_{u})$. To show that
$f_{1}^{\prime}$ is Hölder, it therefore suffices to show that $\kappa^{-1}$
is Hölder, which follows from an estimate of the form
$|\kappa(t^{\prime})-\kappa(t)|\geq c|t-t^{\prime}|^{\beta}$. We conclude the
proof by establishing an estimate of this form. Let $t^{\prime}>t$ and let $n$
be such that $|t-t^{\prime}|\geq
2\operatorname{diam}(\mathcal{P})\lambda^{-n}$ (as before, $\lambda$ denotes
the expanding eigenvalue of the matrix defining $L$). Then between the local
stable manifolds through $z+t\,e_{u}$ and $z+t^{\prime}\,e_{u}$, there is at
least one full element of $\bigvee_{j=0}^{n-1}L^{-j}\mathcal{P}$. By the Gibbs
inequality, these elements have measure at least $c^{\prime}e^{-\delta n}$ for
some $c^{\prime}$ and $\delta$ that are independent of $t$ and $t^{\prime}$,
so that $|\kappa(t^{\prime})-\kappa(t)|\geq c^{\prime}e^{-\delta n}$. But from
the bound on $|t-t^{\prime}|$, we deduce $|\kappa(t^{\prime})-\kappa(t)|\geq
c|t-t^{\prime}|^{\beta}$ for some $c$ and $\beta$ as required.

###  3.2. Differentiability of $L$ with respect to the new atlas

We proved in Section 3.1 that the family of charts
$\Xi=\\{(\alpha_{v},\text{Int}(A_{0})-v)\\}_{v\in\mathcal{H}}$ form a
$C^{1+H}$-differentiable atlas on $\mathbb{T}^{2}$. In this section we show
that $L:(\mathbb{T}^{2},\Xi)\to(\mathbb{T}^{2},\Xi)$ is $C^{1+H}$.

We first consider the case when $A_{0}\cap L^{-1}A_{0}$ has non-empty
interior. We claim that it suffices to establish that $\xi\circ
L\circ\xi^{-1}$ is $C^{1+H}$ on $\xi(A_{0}\cap L^{-1}A_{0})$. To see this, let
$v_{0},v_{1}\in\mathcal{H}$ be such that the domain of $\alpha_{v_{1}}\circ
L\circ\alpha_{v_{0}}^{-1}$, i.e. $U:=(A_{0}-v_{0})\cap L^{-1}(A_{0}-v_{1})$,
has non-empty interior. Let $(a,b)\in\alpha_{v_{0}}(U)$ and write
$(a,b)=\alpha_{v_{0}}(x)=\xi(v_{0}+x)$. Let $w\in\mathcal{H}$ be such that
$x+v_{0}+w\in\text{Int}(A_{0}\cap L^{-1}A_{0})$. We now see that on a
neighbourhood of $(a,b)$

$\displaystyle\alpha_{v_{1}}\circ L\circ\alpha_{v_{0}}^{-1}$
$\displaystyle=\xi\circ\tau_{v_{1}}\circ L\circ\tau_{-v_{0}}\circ\xi^{-1}$
$\displaystyle=(\xi\circ\tau_{v_{1}-Lv_{0}-Lw}\circ\xi^{-1})\circ(\xi\circ
L\circ\xi^{-1})\circ(\xi\circ\tau_{w}\circ\xi^{-1}):$

$\xi\circ\tau_{w}\circ\xi^{-1}(a,b)=\xi(x+v_{0}+w)$; $\xi\circ
L\circ\xi^{-1}(\xi(x+v_{0}+w))=\xi(Lx+Lv_{0}+Lw)\in\xi(A_{0})$;
$\xi\circ\tau_{v_{1}-Lv_{0}-Lw}\circ\xi^{-1}(\xi(Lx+Lv_{0}+Lw))=\xi(Lx+v_{1})=\alpha_{v_{1}}\circ
L\circ\alpha_{v_{0}}^{-1}(a,b)$. Once we establish that $\xi\circ
L\circ\xi^{-1}$ is $C^{1+H}$ on $\xi(A_{0}\cap L^{-1}A_{0})$, it will follow
from the results of the previous section that $\alpha_{v_{1}}\circ
L\circ\alpha_{v_{0}}^{-1}$ is $C^{1+H}$ on a neighbourhood of $(a,b)$.

A similar argument to that in Section 3.1 shows that $\xi\circ
L\circ\xi^{-1}(c,d)$ is of the form $(f_{1}(c),f_{2}(d))$ on its domain. We
establish Hölder differentiability of $\xi\circ L\circ\xi^{-1}$ following the
strategy of the previous section: first we show that $f_{1}^{\prime}$ is
shift-Hölder and then we verify that $f_{1}^{\prime}$ is torus-continuous. We
compute

(10) $f^{\prime}_{1}(a)=\lim_{h\to 0}\frac{\xi\circ
L\circ\xi^{-1}(a+h,b)-\xi\circ L\circ\xi^{-1}(a,b)}{h}.$

From the definition of $\xi$ we see that $h$ is the $\mu$-measure of the
rectangle in $A_{0}$ lying between the stable manifolds through $x$ and
$x^{\prime}=\xi^{-1}(a+h,b)$. Assuming that $h$ is small enough that
$x^{\prime}$ is also in $A_{0}\cap L^{-1}A_{0}$, we can write the numerator in
the limit (10) as the $\mu$-measure of the rectangle in $A_{0}$ lying between
the stable manifolds through $L(x)$ and $L(x^{\prime})$. We provide an
illustration in Figure 4 below.

$\mathbb{T}^{2}$$A_{0}$$x$$x^{\prime}$$L$$\mathbb{T}^{2}$$A_{0}$$L(x)$$L(x^{\prime})$$\mathbb{R}^{2}$$\xi^{-1}$$b$$a$$h$$\\}$$\mathbb{R}^{2}$$\xi$$\\}$$\xi\circ
L(x^{\prime})-\xi\circ L(x)$ Figure 4. The $\mu$-measures of the shaded
rectangles on the right and left are the numerator and the denominator in the
limit (10) respectively.

The derivative of $f_{1}$ can be represented symbolically on
$[00]\subset\Omega$ as

$\ell(\omega)=\lim_{\omega^{\prime}\to\omega}\frac{\mu(R[\sigma(\omega),\sigma(\omega^{\prime})])}{\mu(R[\omega,\omega^{\prime}])},$

where, as before, $R[\omega,\omega^{\prime}]$ and
$R[\sigma(\omega),\sigma(\omega^{\prime})]$ are the rectangles bounded on the
top and bottom by the boundary of $A_{0}$ and on the sides by the stable
manifolds through $\pi(\omega)$, $\pi(\omega^{\prime})$ and $L(\pi(\omega))$,
$L(\pi(\omega^{\prime}))$ respectively. Again, we observe that
$\pi^{-1}(R[\omega,\omega^{\prime}])=A^{-}\times
A^{+}[\omega,\omega^{\prime}]$ and
$\pi^{-1}(R[\sigma(\omega),\sigma(\omega^{\prime})])=A^{-}\times
A^{+}[\omega,\omega^{\prime}]$ where $A^{-}$, $A^{+}[\omega,\omega^{\prime}]$
are defined as in Section 3.1. On the other hand,
$\pi^{-1}R[\sigma(\omega),\sigma(\omega^{\prime})]$ is a subset of $\Omega$
consisting of points $\zeta$ such that $\zeta_{0}^{\infty}$ are the non-
negative coordinates of points in $L(R[\omega,\omega^{\prime}])$ and there are
no additional restrictions on the negative coordinates. Using Lemma 4 we
obtain

$\ell(\omega)=\lim_{\omega^{\prime}\to\omega}\frac{\nu(A^{-}\times\sigma_{+}(A^{+}[\omega,\omega^{\prime}]))}{\nu(A^{-}\times
A^{+}[\omega,\omega^{\prime}])}=\lim_{\omega^{\prime}\to\omega}\frac{{\nu}^{+}(\sigma_{+}(A^{+}[\omega,\omega^{\prime}]))}{{\nu}^{+}(A^{+}[\omega,\omega^{\prime}])}.$

Since $\operatorname{diam}(A^{+}[\omega,\omega^{\prime}])\to 0$ as
$\omega^{\prime}\to\omega$ and $\omega\in A^{+}[\omega,\omega^{\prime}]$, we
conclude that $\ell(\omega)=\frac{1}{g(p_{+}(\omega))}$, where $g$ is the
$g$-function for measure $\nu^{+}$. Since $g$ is strictly positive and Hölder
on $\Omega^{+}$, $\ell$ is Hölder on $\Omega$.

To prove that $\ell$ is torus continuous suppose that $x=\pi(\omega)$ lies on
the stable manifold boundary of two elements of the partition
$L^{-j}\mathcal{P}$ for some $j\in\mathbb{N}$. Let $\omega^{-}\omega^{+}$ and
$\omega^{-}\eta^{+}$ be two different symbolic representations of $x$. To show
that $\ell(\omega^{-}\omega^{+})=\ell(\omega^{-}\eta^{+})$ we apply the same
steps as in Section 3.1. For any $N\in\mathbb{N}$, let $v\in\mathcal{H}$ be
parallel to the unstable direction satisfying $x+v\in\text{Int}(A_{0}\cap
L^{-1}A_{0})$ and $x+v\notin\bigcup_{|k|<N}\partial L^{k}\mathcal{P}$. Let
$\tilde{R}[\omega,\omega^{\prime}]=R[\omega,\omega^{\prime}]+v$ and denote by
$\tilde{A}^{+}[\omega,\omega^{\prime}]$ the set of future coordinates of
points in $\tilde{R}[\omega,\omega^{\prime}]$. Using the expression for the
Radon-Nikodym derivative (6) we obtain

$\mu(R[\omega,\omega^{\prime}])=\int_{A^{-}}\int_{\tilde{A}^{+}[\omega,\omega^{\prime}]}\varrho(\zeta^{-}\zeta^{+})\theta_{-v}(\pi(\zeta^{-}\zeta^{+}))\,d\nu^{+}(\zeta^{+})\,d\nu^{-}(\zeta^{-}).$

Similarly, let
$\tilde{R}[\sigma(\omega),\sigma(\omega^{\prime})]=R[\sigma(\omega),\sigma(\omega^{\prime})]+L(v)$
and obtain

$\mu(R[\sigma(\omega),\sigma(\omega^{\prime})])=\int_{A^{-}}\int_{\sigma^{+}(\tilde{A}^{+}[\omega,\omega^{\prime}])}\varrho(\zeta^{-}\zeta^{+})\theta_{-L(v)}(\pi(\zeta^{-}\zeta^{+}))\,d\nu^{+}\,d\nu^{-}.$

Consider $\omega=\omega^{-}\omega^{+}$ and denote be $\tilde{\omega}^{+}$ the
corresponding future coding of $\pi(\omega)+v$. By continuity of $\varrho$ and
$\theta$ for each $\zeta^{-}$ the inner integral of
$\mu(R[\omega,\omega^{\prime}])$ is approximately
$\varrho(\zeta^{-}\tilde{\omega}^{+})\theta_{-v}(\pi(\zeta^{-}\sigma^{+}(\tilde{\omega}^{+})))\nu^{+}(\tilde{A}^{+}[\omega,\omega^{\prime}])$
and similarly the inner integral of
$\mu(R[\sigma(\omega),\sigma(\omega^{\prime})])$ is approximately
$\varrho(\zeta^{-}\tilde{\omega}^{+})\theta_{-L(v)}(\pi(\zeta^{-}\sigma^{+}(\tilde{\omega}^{+})))\nu^{+}(\tilde{A}^{+}[\sigma(\omega),\sigma(\omega^{\prime})])$
whenever $\zeta^{+}$ is close enough to $\tilde{\omega}^{+}$. As
$\omega^{\prime}\to\omega$ the diameter of
$\tilde{A}^{+}[\omega,\omega^{\prime}]$ tends to zero while
$\tilde{\omega}^{+}\in\tilde{A}^{+}[\omega,\omega^{\prime}]$, so that
$\frac{{\nu}^{+}(\sigma_{+}(\tilde{A}^{+}[\omega,\omega^{\prime}]))}{{\nu}^{+}(\tilde{A}^{+}[\omega,\omega^{\prime}])}\to\frac{1}{g(\tilde{\omega}^{+})}$.
Therefore,

$\ell(\omega^{-}\omega^{+})=\frac{\int_{A^{-}}\varrho(\zeta^{-}\sigma^{+}(\tilde{\omega}^{+}))\theta_{-L(v)}(\pi(\zeta^{-}\sigma^{+}(\tilde{\omega}^{+})))\,d\nu^{-}(\zeta^{-})}{\int_{A^{-}}\varrho(\zeta^{-}\tilde{\omega}^{+})\theta_{-v}(\pi(\zeta^{-}\tilde{\omega}^{+}))\,d\nu^{-}(\zeta^{-})}\cdot\frac{1}{g(\tilde{\omega}^{+})}$

Letting $\tilde{\eta}^{+}$ be the future coding of $\pi(\omega)+v$
corresponding to $\eta^{+}$ we get

$\ell(\omega^{-}\eta^{+})=\frac{\int_{A^{-}}\varrho(\zeta^{-}\sigma^{+}(\tilde{\eta}^{+}))\theta_{-L(v)}(\pi(\zeta^{-}\sigma^{+}(\tilde{\eta}^{+})))\,d\nu^{-}(\zeta^{-})}{\int_{A^{-}}\varrho(\zeta^{-}\tilde{\eta}^{+})\theta_{-v}(\pi(\zeta^{-}\tilde{\eta}^{+}))\,d\nu^{-}(\zeta^{-})}\cdot\frac{1}{g(\tilde{\eta}^{+})}.$

Note that since $\omega^{-}\tilde{\omega}^{+}$ and
$\omega^{-}\tilde{\eta}^{+}$ are two symbolic codings of a single point $x+v$
in $\text{Int}(\pi([00]))$, $\sigma(\omega^{-}\tilde{\omega}^{+})$ and
$\sigma(\omega^{-}\tilde{\eta}^{+})$ are two symbolic codings of the point
$L(x+v)$ in $\text{Int}(\pi([0])$. Hence,
$\pi(p_{+}^{-1}(\sigma_{+}(\tilde{\omega}^{+})))$ and
$\pi(p_{+}^{-1}(\sigma_{+}(\tilde{\eta}^{+})))$ are the same local stable
manifold inside $\pi([0])$. Both points
$\pi(\omega^{-}\sigma_{+}\tilde{\omega}^{+})$ and
$\pi(\omega^{-}\sigma_{+}\tilde{\eta}^{+})$ lie on the intersection of this
local stable manifold and the local unstable manifold
$\pi(p_{-}^{-1}(\omega^{-}))$ inside $\pi([0])$, so they must coincide.

Repeating the argument at the end of Section 3.1 completes the proof. Since
$x+v$ is not on the boundary of the partition $\bigvee_{0\leq
k<N}L^{-k}\mathcal{P}$, $\tilde{\omega}^{+}$ and $\tilde{\eta}^{+}$ agree on
at least $N$ symbols. Now Hölder continuity of $\varrho$ and $g$ implies that
the ratio $\ell(\omega^{-}\omega^{+})/\ell(\omega^{-}\eta^{+})$ can be made
arbitrarily close to one when by choosing $N$ sufficiently large, so that
$\ell$ is torus continuous.

So far, we have completed the proof that $L$ is $C^{1+H}$ in the new charts in
the case that $A_{0}\cap L^{-1}A_{0}$ has non-empty interior. An essentially
identical argument shows that if $A_{0}\cap L^{-n}A_{0}$ has non-empty
interior, then $L^{n}$ is $C^{1+H}$ in the new charts. (The only modification
is that the $g$-function has to be replaced by $g^{(n)}$ defined by
$g^{(n)}(x)=g(x)g(\sigma(x))\cdots g(\sigma^{n-1}x)$). Since Anosov
automorphisms are topologically mixing, $A_{0}\cap L^{-n}A_{0}$ has non-empty
interior for all sufficiently large $n$. In particular there is $n$ such that
$L^{n}$ and $L^{n+1}$ are both $C^{1+H}$ diffeomorphisms. It follows that
$L=(L^{n})^{-1}\circ L^{n+1}$ is $C^{1+H}$ as required.

###  3.3. Cohomology of $\phi$ and the geometric potential of $L$ in the new
atlas.

######  Lemma 9.

Let $L$ and $\mathcal{P}$ be as above. There exist $\gamma>0$ and $k>0$ such
that if $R\subset A_{0}$ is of the form $R=\pi(C_{-}\times S)$ where $C_{-}$
is an $n$-cylinder in $\Omega_{-}$ and $S\subset[0]\subset\Omega_{+}$, then
$\mu(R)\leq ke^{-\gamma n}\mu(\pi\circ p_{+}^{-1}S)$.

The proof is an application of the product structure outlined in Section 2.2
together with the fact that $\nu^{-}$ is a $g$-measure with $g_{-}$ bounded
away from 1.

######  Lemma 10.

The map $L$ is expanding in the unstable direction in the new coordinate
system: for any finite sub-atlas there exists $n\in\mathbb{N}$ such that for
any $x\in\mathbb{T}^{2}$ and for any charts in the sub-atlas containing $x$
and $L^{n}x$ respectively, $D_{u}L^{n}>2$ when computed in the respective
charts.

###### Proof.

Let the finite sub-atlas be $\\{\alpha_{u_{1}},\ldots,\alpha_{u_{N}}\\}$. Let
$M$ and $M^{\prime}$ be positive constants such that
$\tfrac{1}{M}\leq\theta_{u_{i}}(x)\leq M$ for $1\leq i\leq N$ and all of the
maps $\alpha_{u_{i}}\circ\alpha_{u_{j}}^{-1}$ have derivatives between
$M^{\prime}$ and ${M^{\prime}}^{-1}$ when $1\leq i,j\leq N$. By compactness,
there exists $\delta>0$ such that for each $x\in\mathbb{T}^{2}$, there exists
an $i$ with $x+u_{i}\in A_{0}$ and $d(x+u_{i},\partial A_{0})>\delta$. Let $n$
be a fixed integer sufficiently large that $\lambda^{-n}<\delta$ and also
satisfying $e^{\gamma n}>2kM^{2}{M^{\prime}}^{2}$, where $\lambda$ is the
expanding eigenvalue of $L$ and $k$, $\gamma$ are as in Lemma 9.

Let $u,v\in\\{u_{1},...,u_{N}\\}$ be such that $x+u$ and $L^{n}x+v$ both lie
in $\text{int}_{\delta}(A_{0})$. Let $B_{1}+u$ be a rectangle in $A_{0}$ whose
projection in $A_{0}$ onto the stable direction is all of the stable manifold
segment defining $A_{0}$ and whose unstable projection in $A_{0}$ is
sufficiently narrow that $L^{n}B_{1}+v\subset A_{0}$. Let $B_{2}$ be the
rectangle in $A_{0}$ whose projection onto the stable direction is the stable
manifold segment defining $A_{0}$ and whose unstable projection is the same as
that of $L^{n}B_{1}+v$. Then by Lemma 9,

(11) $\mu(L^{n}B_{1}+v)\leq ke^{-\gamma n}\mu(B_{2}).$

We then have

(12) $\begin{split}\mu(L^{n}B_{1}+v)&\geq\tfrac{1}{M}\mu(L^{n}B_{1})\\\
\mu(L^{n}B_{1})&=\mu(B_{1})\\\
\mu(B_{1})&\geq\tfrac{1}{M}\mu(B_{1}+u).\end{split}$

Combining equations (11) and (12), by the choice of $n$ we see

$\mu(B_{2})\geq\frac{e^{\gamma n}}{kM^{2}}\mu(B_{1}+u)\geq 2M^{\prime
2}\mu(B_{1}+u).$

Shrinking $B_{1}$ so that $B_{1}+u$ shrinks to the segment of the stable
manifold of $x$ lying in $A_{0}$, we deduce the unstable derivative of $L^{n}$
in the $(\alpha_{u},\alpha_{v})$ charts is at least $2{M^{\prime}}^{2}$. Now
if $u^{\prime}$ and $v^{\prime}$ are such that $\alpha_{u^{\prime}}$ and
$\alpha_{v^{\prime}}$ are arbitrary charts in the sub-atlas containing $x$ and
$L^{n}x$ in their domain then the unstable derivative of $L^{n}$ in the
$(\alpha_{u^{\prime}},\alpha_{v^{\prime}})$ charts is at least 2. This
completes the proof. ∎

######  Lemma 11.

There exists $M>0$ such that for any $n\in\mathbb{N}$, any cylinder set $C$ in
$\Omega$ of the form $[0a_{1}\ldots a_{n-1}0]$, and any $\omega\in C$ we have

$\frac{1}{M}\leq|D_{u}L^{n}(\pi(\omega))|\cdot\exp(S_{n}\phi(\pi(\omega)))\leq
M,$

where the unstable derivative of $L$ is computed using the new charts.

###### Proof.

The proof is based on a standard argument that the fibre maps of uniformly
expanding maps have bounded distortion (see e.g. [21, Chapter III]). Suppose
that $x,y$ are points in $A_{0}$ which lie on the same local unstable manifold
and are such that $x=\pi(\omega),y=\pi(\eta)$ with
$\omega,\eta\in[0a_{1}\ldots a_{n-1}0]\subset\Omega$. Recall from Section 3.2
that in terms of charts of the new atlas, the map $L$ has the form
$\alpha_{v_{1}}\circ L\circ\alpha_{v_{0}}^{-1}(a,b)=(f_{1}(a),f_{2}(b))$ where
the functions $f_{1}$ and $f_{2}$, which depend on the choice of $v_{0}$ and
$v_{1}$, are differentiable with Hölder continuous derivatives. Since
$\sigma^{n}(\omega),\sigma^{n}(\eta)\in[0]$ we have that both $L^{n}(x)$ and
$L^{n}(y)$ are in $A_{0}$ and hence $d(L^{j}(x),L^{j}(y))\leq\lambda^{-(n-j)}$
for $0\leq j<n$, where $\lambda$ is the expanding eigenvalue of $L$ (and here
the distance is computed using the original metric). Denote by $f_{1,j}$ the
first component of $L$ computed in the charts corresponding to $L^{j}x$ and
$L^{j+1}x$. Applying the chain rule we see that

$\left|\frac{D_{u}L^{n}(x)}{D_{u}L^{n}(y)}\right|=\prod_{j=0}^{n-1}\left|\frac{f^{\prime}_{1,j}(L^{j}x)}{f^{\prime}_{1,j}(L^{j}y)}\right|.$

It follows from Hölder continuity of the derivatives and Lemma 10 that there
are $K>0$ and $\gamma\in(0,1)$ such that for $0\leq j<n$

$\left|\frac{f_{1,j}(L^{j}(x))}{f_{1,j}(L^{j}(y))}\right|\leq
1+Kd(L^{j}(x),L^{j}(y))^{\gamma}\leq 1+K\lambda^{-(n-j)}.$

Setting $M=\prod_{j=1}^{\infty}(1+K\lambda^{-j})$ we obtain that
$|D_{u}L^{n}(x)|/|D_{u}L^{n}(y)|\leq M$ for all $x,y$ lying in a segment of
the local unstable manifold contained in a single partition element.

Now suppose $C$ is a cylinder set $[a_{0}a_{1}\ldots a_{n}]$ in $\Omega$ with
$a_{0}=a_{n}=0$. Let $\omega^{-}$ be a compatible past and set
$U=\pi(\omega^{-}C)$, a piece of unstable manifold that is mapped bijectively
by $L^{n}$ onto a fibre of the unstable manifold crossing the partition
element $A_{0}$. By the mean value theorem, the length (in the new charts) of
$L^{n}U$ (which is the same as the width of the 0 partition element) is the
product of the length of $U$ and the unstable derivative at some point $u\in
U$. Since coordinates (and hence lengths) in the unstable direction are
computed using by $\mu$-measures the measure $\mu=\pi_{\ast}\nu$, this gives,
for any $\omega\in U$,

$\frac{1}{M}\leq|(D_{u}L^{n})(\pi(\omega))|\cdot\nu(C)\leq M.$

Now applying the Bowen definition [2] for the Gibbs state $\nu$ of the
potential $\phi\circ\pi$ together with the fact that $P_{\rm
top}(\sigma,\phi\circ\pi)=0$,

$\frac{1}{M^{\prime}}\exp(S_{n}\phi(\pi(\omega)))\leq\mu(C)\leq
M^{\prime}\exp(S_{n}\phi(\pi(\omega)))$

Substituting in the previous inequality gives the required statement. ∎

######  Lemma 12.

Let $\phi$ be as in the statement of Theorem 1, and let the charts be
constructed as above. Then potential $\phi(x)$ is cohomologous to
$-\log|D_{u}L(x)|$, where the unstable derivative is computed using the new
charts.

###### Proof.

We rely on Livšic’s theorem [13]: if $T$ is a hyperbolic dynamical system and
$\psi$ is a Hölder continuous function such that $S_{n}\psi(p)=0$ whenever
$T^{n}p=p$, then $\psi$ is a coboundary (with Hölder continuous transfer
function).

As a corollary, if $\Omega$ is a mixing subshift of finite type and there
exists an $M$ such that $|S_{n+1}\psi(\omega)|\leq M$ whenever $\omega\in[0]$
and $\sigma^{n}\omega\in[0]$, then $\psi$ is a Hölder coboundary.

Lemma 11 shows that the function $\psi(\omega)=\chi\circ\pi(\omega)$ where
$\chi(x)=\log|D_{u}L(x)|+\phi(x)$ satisfies the hypothesis of this corollary
of Livšic’s theorem, so that $\psi$ is a Hölder coboundary. It follows that
$\chi$ sums to zero around any periodic orbit in $\mathbb{T}^{2}$, so that
$\chi$ is also a Hölder coboundary, using Livšic’s theorem again. ∎

##  4\. Application to the smooth conjugacy problem.

In this section we explicitly construct a countable family of Hölder
potentials in the homotopy class of the toral automorphism $L$ whose geometric
potentials have identical pressure functions, yet they are not $C^{1}$
conjugate.

######  Lemma 13.

Let $L$ be an automorphism of $\mathbb{T}^{2}$, let $k\in\mathbb{N}$ and let
$M_{k}(x)=kx\bmod 1$. Then for any continuous function $\phi$ on
$\mathbb{T}^{2}$

$P_{\rm top}(L,\phi)=P_{\rm top}(L,\phi\circ M_{k}).$

###### Proof.

We use the topological definition of pressure:

$P_{\rm top}(L,\phi)=\lim_{\epsilon\to
0}\limsup_{n\to\infty}\frac{1}{n}\log\sup\left\\{\sum_{x\in
E}e^{S_{n}\phi(x)}:E\text{ is $(n,\epsilon)$-separated}\right\\},$

where a subset $E$ of $\mathbb{T}^{2}$ is _$(n,\epsilon)$ -separated_ (with
respect to $L$) if for any distinct elements $x,y\in E$, there exists $0\leq
j<n$ such that $d(L^{j}x,L^{j}y)\geq\epsilon$.

Denote $\phi_{k}=\phi\circ M_{k}$. We first show that $P_{\rm
top}(L,\phi_{k})\geq P_{\rm top}(L,\phi)$. Let $E$ be an
$(n,\epsilon)$-separated subset of $\mathbb{T}^{2}$. We define a subset
$E^{\prime}$ of $\mathbb{T}^{2}$ by

$E^{\prime}=M_{k}^{-1}(E)=\\{(x+\mathbf{n})/k\colon x\in
E,\mathbf{n}\in\\{0,\ldots,k-1\\}^{2}\\}$

and claim $E^{\prime}$ is $(n,\frac{\epsilon}{k})$-separated. In the case when
$x\in E$ and $\mathbf{m},\mathbf{n}$ are distinct elements of
$\\{0,\ldots,k-1\\}^{2}$, we claim
$d(L^{i}(\frac{x+\mathbf{m}}{k}),L^{i}(\frac{x+\mathbf{n}}{k}))\geq\frac{1}{k}$
for each $i$. Since $L$ is an automorphism, it suffices to show that
$d(L^{i}(\frac{\mathbf{p}}{k}),0)\geq\frac{1}{k}$ for each
$\mathbf{p}\in\\{0,\frac{1}{k},\ldots,\frac{k-1}{k}\\}^{2}\setminus\\{(0,0)\\}$
and $i\in\mathbb{N}$. Since the matrix $A$ defining $L$ has an inverse with
integer entries, it is not hard to see that $L$ is a permutation of the points
$\\{0,\ldots,\frac{k-1}{k}\\}^{2}$. Since $L$ is injective, it follows that
$d(L^{i}(\frac{\mathbf{p}}{k}),0)\geq\frac{1}{k}$ for each $i$. In the case
when $x,y$ are distinct elements of $E$ and $\mathbf{m},\mathbf{n}$ are
elements of $\\{0,\ldots,k-1\\}^{2}$ (not necessarily distinct), letting
$u=\frac{x+\mathbf{m}}{k}$ and $v=\frac{y+\mathbf{n}}{k}$, we have

$d(L^{i}u,L^{i}v)\geq\tfrac{1}{k}d(M_{k}(L^{i}u),M_{k}(L^{i}v))=\tfrac{1}{k}d(L^{i}x,L^{i}y).$

Since $\max_{i<n}d(L^{i}x,L^{i}y)\geq\epsilon$, it follows that
$\max_{i<n}d(L^{i}u,L^{i}v)\geq\frac{\epsilon}{k}$. Hence we have established
that $E^{\prime}$ is $(n,\frac{\epsilon}{k})$-separated as required.

Note $S_{n}\phi_{k}(\frac{x+\mathbf{m}}{k})=S_{n}\phi(x)$ for each $x\in E$
and $\mathbf{m}\in\\{0,\ldots,k-1\\}^{2}$. Therefore

$\displaystyle\sup$ $\displaystyle\left\\{\sum_{x\in
E}e^{S_{n}\phi_{k}(x)}:E\text{ is
$\textstyle(n,\frac{\epsilon}{k})$-separated}\right\\}$
$\displaystyle\qquad\geq k^{2}\sup\left\\{\sum_{x\in
E}e^{S_{n}\phi(x)}:E\text{ is }(n,\epsilon)\text{-separated}\right\\},$

which gives $P_{\rm top}(L,\phi_{k})\geq P_{\rm top}(L,\phi)$.

For the converse inequality, we first claim that any $u,v\in\mathbb{T}^{2}$
and for any positive $\epsilon<1/(2k\|A\|)$, the following implication holds:

(13) $d(u,v)<\epsilon\text{ and }d(M_{k}(Lu),M_{k}(Lv))<k\epsilon\text{
implies }d(Lu,Lv)<\epsilon.$

Again, by the linearity of $L$, it suffices to show that if $d(u,0)<\epsilon$
and $d(M_{k}(Lu),0)<k\epsilon$ then $d(Lu,0)<\epsilon$. To verify this claim,
suppose $d(u,0)<\epsilon$. By the choice of $\epsilon$, $d(Lu,0)<\frac{1}{2k}$
so that $0$ is the closest element of $M_{k}^{-1}\\{0\\}$ to $Lu$. Since
$d(M_{k}(Lu),0)<k\epsilon$, the fact that $M_{k}$ locally expands distances by
a factor of $k$ implies that $d(Lu,0)<\epsilon$ as required.

Let $\epsilon<\frac{1}{2k\|A\|}$ and let $E^{\prime}$ be an $(n,\epsilon)$
separated set in $\mathbb{T}^{2}$. We define a relation $R$ on $E^{\prime}$ by

$uRv\quad\Leftrightarrow\quad\max_{0\leq
i<n}d(L^{i}M_{k}(u),L^{i}M_{k}(v))<\frac{\epsilon}{2k}.$

Equivalently $uRv$ iff $\max_{0\leq
i<n}d(M_{k}(L^{i}u),M_{k}(L^{i}v))<\frac{\epsilon}{2k}$, since $L\circ
M_{k}=M_{k}\circ L$. We then take the transitive closure of $R$ to form an
equivalence relation $\sim$ on $E^{\prime}$. That is, $u\sim v$ if there exist
$u_{0},u_{1},\ldots,u_{l}$ with $u_{0}=u$, $u_{l}=v$ and $u_{i-1}Ru_{i}$ for
$i=1,\ldots,l$. We claim that each $\sim$-equivalence class has at most
$k^{2}$ elements. We prove this by contradiction. Suppose $C$ is a
$\sim$-equivalence class containing at least $k^{2}+1$ elements. We construct
a subset $D$ of cardinality exactly $k^{2}+1$ such that there is a path
between any two elements of $D$ using steps in $R$. To see this, fix an
initial element of $u_{0}$ of $C$, enumerate the other elements of $C$ and for
each such element $u$, find an $R$-path, from the definition of $\sim$
connecting $u_{0}$ to $u$. We now build $D$ by adding the elements of the
paths one at a time until the cardinality is exactly $k^{2}+1$. (At each step
when a vertex is to be included, $D$ may either increase by one element if the
vertex is new; or remain the same if the vertex has already been added.) By
the construction, each element of $D$ is connected by $R$ to a previous
element of $D$.

Let $D=\\{u_{0},\ldots,u_{k^{2}}\\}$. By the triangle inequality and the
definition of $R$,
$d(M_{k}(L^{i}(u_{0})),M_{k}(L^{i}(u_{j})))<\frac{k\epsilon}{2}$ for each $j$
(since we can get from $u_{0}$ to $u_{j}$ along an $R$ path of length at most
$k^{2}$). In particular, $d(M_{k}(u_{0}),M_{k}(u_{j}))<\frac{k\epsilon}{2}$
for each $j$. Using the fact that $M_{k}$ locally expands distances by a
factor of $k$, for each $0\leq j\leq k^{2}$, $u_{j}$ differs from $u_{0}$ by
an element of $M_{k}^{-1}\\{0\\}=\\{0,\frac{1}{k},\ldots,\frac{k-1}{k}\\}^{2}$
plus a term of size at most $\frac{\epsilon}{2}$. By the pigeonhole principle,
there exist $0\leq j<j^{\prime}\leq k^{2}$ such that $u_{j}$ and
$u_{j^{\prime}}$ differ by at most $\epsilon$. Since $u_{j}\sim
u_{j^{\prime}}$, we see that
$d(L^{i}M_{k}(u_{j}),L^{i}M_{k}(u_{j^{\prime}}))<k\epsilon$ for
$i=0,\ldots,n$. Applying (13) inductively we see
$d(L^{i}u_{j},L^{i}u_{j^{\prime}})<\epsilon$ for $i=0,\ldots,n$. This
contradicts the initial assumption that $E^{\prime}$ was
$(n,\epsilon)$-separated.

Hence we have shown that each $\sim$-equivalence class in $E^{\prime}$ has at
most $k^{2}$ elements. Let the equivalence classes be $C_{1},\ldots,C_{M}$;
and for each equivalence class, pick $u_{i}\in C_{i}$ for which
$S_{n}\phi_{k}(u_{i})$ is maximal in the equivalence class. We now have

$\sum_{u\in C_{i}}\exp(S_{n}\phi_{k}(u))\leq k^{2}\exp(S_{n}\phi_{k}(u_{i})).$

Summing over the equivalence classes, we obtain

$\sum_{u\in E^{\prime}}\exp(S_{n}\phi_{k}(u))\leq
k^{2}\sum_{i=1}^{M}\exp(S_{n}\phi_{k}(u_{i})).$

Let $x_{i}=M_{k}(u_{i})$ for each $i$. Since
$S_{n}\phi_{k}(u_{i})=S_{n}\phi(x_{i})$, rearranging the above inequality
gives

$\sum_{i=1}^{M}\exp(S_{n}\phi(x_{i}))\geq\frac{1}{k^{2}}\sum_{u\in
E^{\prime}}\exp(S_{n}\phi_{k}(u)).$

Finally, we claim that $\\{x_{1},\ldots,x_{M}\\}$ is
$(n,\frac{\epsilon}{2k})$-separated. If not then, there there exist $j,l$ such
that $d(L^{i}x_{j},L^{i}x_{l})<\frac{\epsilon}{2k}$ for $i=0,\ldots,n-1$.
Then, since $x_{j}=M_{k}(u_{j})$ and $x_{l}=M_{k}(u_{l})$, we see from the
definition of $R$ that $u_{j}Ru_{l}$. This contradicts the assumption that the
$u_{i}$’s belong to distinct equivalence classes.

Hence we have shown

$\displaystyle\sup$ $\displaystyle\left\\{\sum_{x\in
E}e^{S_{n}\phi(x)}:E\text{ is
$\textstyle(n,\frac{\epsilon}{2k})$-separated}\right\\}$
$\displaystyle\qquad\geq\frac{1}{k^{2}}\sup\left\\{\sum_{u\in
E^{\prime}}e^{S_{n}\phi_{k}(u)}:E\text{ is
}(n,\epsilon)\text{-separated}\right\\},$

It follows that $P_{\rm top}(L,\phi)\geq P_{\rm top}(L,\phi_{k})$ as required.
∎

###### Proof of Theorem 2.

Let $L$ be the Anosov automorphism of the torus given by the matrix
$\begin{pmatrix}1&1\\\ 1&0\end{pmatrix}$. Note that
$(\frac{1}{2},0),(\frac{1}{2},\frac{1}{2}),(0,\frac{1}{2})$ is the unique
period 3 orbit of $L$. Let $\phi$ be a Hölder continuous function of the torus
of pressure 0 such that
$\phi(0,0)\neq\frac{1}{3}(\phi(\frac{1}{2},0)+\phi(\frac{1}{2},\frac{1}{2})+\phi(0,\frac{1}{2}))$
and let $\phi_{2}(x)=\phi(2x)$ as above. Then

$\textstyle\frac{1}{3}(\phi_{2}(\frac{1}{2},0)+\phi_{2}(\frac{1}{2},\frac{1}{2})+\phi_{2}(0,\frac{1}{2}))=\phi(0,0)\neq\frac{1}{3}(\phi(\frac{1}{2},0)+\phi(\frac{1}{2},\frac{1}{2})+\phi(0,\frac{1}{2})).$

We conclude the proof by showing that if $T$ and $T_{2}$ are the area-
preserving Anosov diffeomorphisms obtained from $\phi$ and $\phi_{2}$
respectively as in Theorem 1, then $T$ and $T_{2}$ are not conjugate, but they
satisfy $P_{\rm top}(T,-sD_{u}T)=P_{\rm top}(T_{2},-sD_{u}T_{2})$ for all
$s\in\mathbb{R}$.

Let $h$ be the conjugacy between $T$ and $L$ obtained in the proof of Theorem
1. Similarly, let $h_{2}$ be the conjugacy between $T_{2}$ and $L$. The
theorem guarantees that $-\log|D_{u}T|\circ h$ is cohomologous to $\phi$ and
$-\log|D_{u}T_{2}|\circ h_{2}$ is cohomologous to $\phi_{2}$. Let
$p=h(\frac{1}{2},0)$ and notice that $\\{p,Tp,T^{2}p\\}$ is the unique period
3 orbit of $T$. Similarly let $p_{2}=h_{2}(\frac{1}{2},0)$ so that
$\\{p_{2},T_{2}p_{2},T_{2}^{2}p_{2}\\}$ is the unique period 3 orbit of
$T_{2}$. Since $-\log|D_{u}T|\circ h$ is cohomologous to $\phi$, we see that

$|D_{u}T^{3}(p)|=|D_{u}T^{3}(Tp)|=|D_{u}T^{3}(T^{2}p)|=e^{\phi(\frac{1}{2},0)+\phi(\frac{1}{2},\frac{1}{2})+\phi(0,\frac{1}{2})},$

while

$|D_{u}T_{2}^{3}(p_{2})|=e^{\phi_{2}(\frac{1}{2},0)+\phi_{2}(\frac{1}{2},\frac{1}{2})+\phi_{2}(0,\frac{1}{2})}=e^{3\phi(0,0)}.$

Since differentiable conjugacies preserve unstable multipliers, we see that
$T$ and $T_{2}$ are not differentiably conjugate.

However,

$\displaystyle P_{\rm top}(T,-s\log|D_{u}T|)$ $\displaystyle=P_{\rm
top}(hLh^{-1},-s\log|D_{u}T|)$ $\displaystyle=P_{\rm
top}(L,-s\log|D_{u}T|\circ h)$ $\displaystyle=P_{\rm top}(L,-s\phi)$

and similarly $P_{\rm top}(T_{2},-s\log|D_{u}T_{2}|)=P_{\rm
top}(L,-s\phi_{2})$. By Lemma 13, $P_{\rm top}(L,-s\phi)=P_{\rm
top}(L,-s\phi\circ M_{2})=P_{\rm top}(L,-s\phi_{2})$ for all $s\in\mathbb{R}$
so that $P_{\rm top}(T,-s\log|D_{u}T|)=P_{\rm top}(T_{2},-s\log|D_{u}T_{2}|)$
for all $s$. ∎

## References

  * [1] D. V. Anosov, _Geodesic flows on closed Riemannian manifolds_ , Proc. Stek. Inst. 90 (1967)
  * [2] R. Bowen, _Equilibrium states and the ergodic theory of Anosov diffeomorphisms_ , Lecture Notes in Mathematics, Vol. 470. corr. 2nd rev. ed. (1st ed. 1975) Springer (2017).
  * [3] D. Capocaccia, _A definition of Gibbs states for a compact set with $\mathbb{Z}^{\nu}$-action_, Comm. Math. Phys. 48, (1976), 85-88.
  * [4] E. Cawley, _The Teichmüller space of an Anosov diffeomorphism of $\mathbb{T}^{2}$_, Invent. Math. 112 (1993), no. 2, 351–376.
  * [5] J. DeWitt and A. Gogolev, _Dominated splitting from constant periodic data and global rigidity of Anosov automorphisms_ , preprint arXiv:2309.07323
  * [6] J. Franks, _Anosov diffeomorphisms on tori_ , Trans. AMS 145, 117-125 (1969)
  * [7] A. Gogolev, _Smooth conjugacy of Anosov diffeomorphisms on higher dimensional tori_ , Journal of Modern Dynamics, 2, no. 4, 645-700 (2008)
  * [8] A. Gogolev and F. Rodriguez Hertz, _Smooth rigidity for very non-algebraic Anosov diffeomorphisms of codimension one_ , Israel Journal of Mathematics, to appear.
  * [9] N. Haydn, _On Gibbs and equilibrium states_ , Ergodic Theory Dyn. Syst. 7, 119-132 (1987).
  * [10] N. Haydn and D. Ruelle, _Equivalence of Gibbs and Equilibrium States for Homeomorphisms Satisfying Expansiveness and Specification_ , Commun. Math. Phys. 148, 155-167 (1992).
  * [11] A. Erchenko, _Flexibility of Lyapunov exponents with respect to two classes of measures on the torus_ , Ergodic Theory and Dynamical Systems, 42, Issue 2, 737 – 776 (2022).
  * [12] B. Kalinin, V. Sadovskaya, _On Anosov diffeomorphisms with asymptotically conformal periodic data_ , Ergodic Theory and Dynamical Systems, 29 Issue 1, 117 - 136 (2009).
  * [13] A. N. Livšic, _Cohomology of dynamical systems_ , Izv. Akad. Nauk SSSR Ser. Mat. 36, 1296–1320 (1972).
  * [14] R. de la Llave, J. M. Marco, and R. Moriyón, _Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation_ , Ann. of Math. (2) 123 no.3, 537–611 (1986).
  * [15] R. de la Llave, _Invariants for smooth conjugacy of hyperbolic dynamical systems II_ , Commun. Math. Phys. 109, 369-378 (1987)
  * [16] R. de la Llave and R. Moriyón, _Invariants for smooth conjugacy of hyperbolic dynamical systems IV_ , Commun. Math. Phys. 116, 185-192 (1988)
  * [17] R. de la Llave, _Smooth Conjugacy and S-R-B Measures for Uniformly and Non-Uniformly Hyperbolic Systems_ , Commun. Math. Phys. 150, 289-320 (1992)
  * [18] A. Manning, _There are no new Anosov diffeomorphisms on tori_ , Amer. J. Math. 96, 422-429 (1974).
  * [19] J. M. Marco and R. Moriyón, _Invariants for smooth conjugacy of hyperbolic dynamical systems I_ , Commun. Math. Phys. 109, 681-689 (1987)
  * [20] J. M. Marco and R. Moriyón, _Invariants for smooth conjugacy of hyperbolic dynamical systems IV_ , Commun. Math. Phys. 112, 317-333 (1987)
  * [21] R. Mañé, _Ergodic Theory and Differentiable Dynamics_ , Springer (1987)
  * [22] S. Newhouse, _On codimension one Anosov diffeomorphisms_ , Amer. J. Math. 92, 761-770 (1970).
  * [23] M. Pollicott and H. Weiss, _Free Energy as a Dynamical Invariant (or Can You Hear the Shape of a Potential?)_ , Commun. Math. Phys. 240, 457–482 (2003)
  * [24] D. Ruelle, _Thermodynamic formalism. The mathematical structures of classical equilibrium statistical mechanics_ , Addison-Wesley, 1978.
  * [25] S. Smale, _Differentiate dynamical systems_ , Bull A.M.S. 73, 747-817 (1967).
  * [26] P. Walters, _Ruelle’s operator theorem and g-measures_ , Trans. Amer. Math. Soc. 214 (1975), 375-387.


# The Quantum Rabi model: Towards Braak’s conjecture

Zeév Rudnick  School of Mathematical Sciences, Tel Aviv University, Tel Aviv
69978, Israel<EMAIL_ADDRESS>
###### Abstract.

We establish a density one version of Braak’s conjecture on the fine structure
of the spectrum of the quantum Rabi model, as well as a recent conjecture of
Braak, Nguyen, Reyes-Bustos and Wakayama on the nearest neighbor spacings of
the spectrum. The proof uses a three-term asymptotics for large eigenvalues
due to Boutet de Monvel and Zielinski, and a number theoretic argument from
uniform distribution theory.

##  1\. Introduction

In this note we address a conjecture of Braak [1] about the fine structure of
the spectrum of the quantum Rabi model (QRM), a fundamental model of light-
matter interaction, which describes the interaction between a two-level atom
(qubit) coupled to a quantized, single-mode harmonic oscillator, see the
survey [7]. The Hamiltonian of the system is

$H=\mathbf{a}^{\dagger}\mathbf{a}+\Delta\sigma_{z}+g\sigma_{x}(\mathbf{a}+\mathbf{a}^{\dagger})$

where $\sigma_{x}=\left(\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}\right)$, $\sigma_{z}=\left(\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}\right)$ are the Pauli matrices of the two-level system,
assumed to have level splitting $2\Delta$; $\mathbf{a}^{\dagger}$ and
$\mathbf{a}$ are the creation and annihilation operators of the harmonic
oscillator with frequency set to be unity; and $g>0$ measures the strength of
the coupling between the systems.

The Rabi Hamiltonian commutes with a parity operator
$P=(-1)^{\mathbf{a}^{\dagger}\mathbf{a}}\sigma_{z}$, and hence the Hilbert
space of states decomposes into the $\pm 1$-eigenspaces of $P$ which are
preserved by $H$, and and the spectrum of $H$ breaks up into a union of two
parity classes $\\{E_{n}^{\pm}\\}$.

The eigenvalues in each parity class satisfy $E_{n}^{\pm}=n-g^{2}+o(1)$ as
$n\to\infty$ [6, 8], so that for $n$ sufficiently large, each interval
$[n,n+1]$ contains at most $4$ shifted eigenvalues $E_{n}^{\pm}+g^{2}$. Braak
[1] conjectured that

######  Conjecture 1.1 (Braak’s G-conjecture).

For a given parity class, all intervals $[n,n+1]$ contains at most two shifted
eigenvalues, two intervals containing no shifted eigenvalues are not adjacent,
and two intervals containing two shifted eigenvalues are also not adjacent.

In this note, we show that Braak’s conjecture holds for “almost all” $n$.

######  Theorem 1.2.

Fix $\Delta>0$ and $g>0$. For all but at most $O(N^{1/2+o(1)})$ values of
$n\leq N$, the interval $(n,n+1)$ contains exactly two shifted eigenvalues of
one of the parity classes, and none for the other parity class, while the
adjacent intervals $(n-1,n)$ and $(n+1,n+2)$ contain exactly two eigenvalues
of the other parity class and none of the first parity class. Moreover,
neither $n$ nor $n\pm 1$ are shifted eigenvalues.

In particular, almost all intervals $[n,n+1]$ contain exactly two elements of
the shifted spectrum.

Concerning the last assertion, there are special choices of the parameters $g$
and $\Delta$ for which there are “exceptional” eigenvalues $E$ such that
$E+g^{2}$ is an integer, see [7, §3.2] and the references therein, and our
theorem excludes $n-g^{2}$ being one of these eigenvalues for almost all $n$.

An application of Theorem 1.2 is to prove a recent conjecture of Braak,
Nguyen, Reyes-Bustos and Wakayama [2] on the nearest neighbor spacings of the
full spectrum. Denote by $\\{E_{n}\\}$ the ordered eigenvalues of $H$ of both
parity classes:

$E_{1}\leq E_{2}\leq\dots$

In [2], the nearest neighbor spacings $\delta_{n}:=E_{n+1}-E_{n}$ were
classified into three types: positive if both $E_{n},E_{n+1}$ fell into the
positive parity class, negative if both fell into the negative parity class,
and mixed if one of the pair was positive and one negative. Based on numerical
observation, it was conjectured [2, eq 14] that

######  Conjecture 1.3 (Spacings conjecture for the QRM).

The frequencies of the three different types of nearest neighbor spacings are
$1/4$,$1/4$,$1/2$, respectively.

This clearly follows from the full conjecture of Braak, but since we establish
that Braak’s conjecture holds for $100\%$ of $n^{\prime}s$, we have also
established Conjecture 1.3.

Finally, we examine the value distribution of the normalized deviations

$\delta_{n}^{\pm}:=n^{1/4}\left(E_{n}^{\pm}-\left(n-g^{2}\right)\right).$

As an application of the method of proof of Theorem 1.2, we show that the
deviations in each parity class satisfy an arcsine law:

######  Theorem 1.4.

For any subinterval $[\alpha,\beta]\subset[-\frac{\Delta}{\sqrt{2\pi
g}},\frac{\Delta}{\sqrt{2\pi g}}]$, we have

$\lim_{N\to\infty}\frac{1}{N}\\#\Big{\\{}n\leq
N:\delta_{n}^{\pm}\in[\alpha,\beta]\Big{\\}}=\int_{\alpha}^{\beta}\frac{dy}{\pi\sqrt{\frac{2\pi
g}{\Delta^{2}}-y^{2}}}.$

The proof of Theorem 1.2 starts with an approximation to the eigenvalues due
to Boutet de Monvel and Zielinski [3] and concludes with a number-theoretic
argument.

Acknowledgement: I thank Masato Wakayama for introducing me to the QRM and for
helpful discussions.

This research was supported by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 786758).

##  2\. The case of good $n$’s

Boutet de Monvel and Zielinski [3] proved a three term expansion for the
eigenvalues in each parity class:

(1) $E_{n}^{\pm}=n-g^{2}\mp\frac{\Delta}{\sqrt{2\pi
g}}\frac{(-1)^{n}\cos(\theta_{n})}{n^{1/4}}+O(n^{-1/2+o(1)}),$

where

$\theta_{n}=4g\sqrt{n}-\frac{\pi}{4}.$

(this approximation was apparently proposed in [4], see also [8]).

Fix $\delta\in(0,1/2)$ small and let $N\gg 1$. We say that $n\in[N/2,N]$ is
“good” if

$|\cos(\theta_{n})|>N^{-1/2+\delta}.$

Otherwise we say that $n$ is “bad”.

Let $x_{n}^{\pm}=E_{n}^{\pm}+g^{2}$ be the shifted eigenvalues, and denote by
$\mathcal{X}^{\pm}=\\{x_{n}^{\pm}\\}$ the shifted spectra in each parity
class.

######  Proposition 2.1.

Let $N\gg 1$. If $n\in[N/2,N]$ is “good” then $n$, $n\pm 1$ are not shifted
eigenvalues and either

i) the interval $(n,n+1)$ contains both $x_{n}^{-}$ and $x_{n+1}^{-}$:
$(n,n+1)\cap\mathcal{X}^{-}=\\{x_{n}^{-},x_{n+1}^{-}\\}$, and no elements of
$\mathcal{X}^{+}$ while the intervals $(n-1,n)$ and $(n+1,n+2)$ contain no
elements of $\mathcal{X}^{-}$, $(n-1,n)$ contains both $x_{n-1}^{+}$ and
$x_{n}^{+}$, while $(n+1,n+2)$ contains both $x_{n+1}^{+}$ and $x_{n+2}^{+}$.

or

ii) Otherwise, the same holds with the roles of $\mathcal{X}^{-}$ and
$\mathcal{X}^{+}$ reversed.

###### Proof.

Let $N\gg 1$ be large, and take $n\in[N/2,N]$. Then

$\theta_{n+1}=\theta_{n}+O\left(\frac{1}{\sqrt{N}}\right)$

since

$\theta_{n+1}-\theta_{n}=4g\sqrt{n+1}-4g\sqrt{n}=\frac{4g}{\sqrt{n+1}+\sqrt{n}}\sim\frac{2g}{\sqrt{N}}.$

Hence

$\cos(\theta_{n+1})=\cos(\theta_{n})+O\left(\frac{1}{\sqrt{N}}\right),$

and likewise

$\cos(\theta_{n-1})=\cos(\theta_{n})+O\left(\frac{1}{\sqrt{N}}\right).$

and the same holds for $\cos(\theta_{n\pm 2})$.

Therefore, for “good” $n$, if $\cos(\theta_{n})>N^{-1/2+\delta}$ then
$\cos(\theta_{n\pm 1}),\cos(\theta_{n\pm 2})>\frac{1}{2}N^{-1/2+\delta}$ and
in particular have the same sign as $\cos(\theta_{n})$, and and analogous
statement holds true if $\cos(\theta_{n})<-N^{-1/2+\delta}$.

Let’s assume that $(-1)^{n}\cos(\theta_{n})>N^{-1/2+\delta}$. Then

$x_{n}^{-}-n=\frac{\Delta}{\sqrt{2\pi
g}}\frac{(-1)^{n}\cos(\theta_{n})}{n^{1/4}}+O(n^{-1/2+o(1)})>\frac{1}{2}N^{-1/2+\delta}>0$

so that $x_{n}^{-}\in(n,n+1)$. Moreover,

$(-1)^{n+1}\cos(\theta_{n\pm 1})=-(-1)^{n}\cos(\theta_{n\pm
1})<-\frac{1}{2}N^{-1/2+\delta}<0$

because $\cos(\theta_{n\pm 1})$ has the same sign and roughly the same size as
$\cos(\theta_{n})$. Hence

$\begin{split}x_{n+1}^{-}-(n+1)&=\frac{\Delta}{\sqrt{2\pi
g}}\frac{(-1)^{n+1}\cos(\theta_{n+1})}{n^{1/4}}+O(n^{-1/2+o(1)})\\\
&<-\;\frac{\Delta}{\sqrt{2\pi g}}\frac{1}{4}N^{-1/2+\delta}<0\end{split}$

so that $x_{n+1}^{-}\in(n,n+1)$. Likewise $x_{n-1}^{-}<n-1$ so that
$x_{n-1}^{-}\in(n-2,n-1)$, and $x_{n+2}^{-}\in(n+2,n+3)$,
$x_{n-2}^{-}\in(n-2,n-1)$. Thus

$\mathcal{X}^{-}\cap(n,n+1)=\\{x_{n}^{-},x_{n+1}^{-}\\},$
$\mathcal{X}^{-}\cap(n-1,n)=\emptyset=\mathcal{X}^{-}\cap(n+1,n+2)$

in this case. Furthermore, for the other parity class, we have

$x_{n}^{+}-n=-\;\frac{\Delta}{\sqrt{2\pi
g}}\frac{(-1)^{n}\cos(\theta_{n})}{n^{1/4}}+O(n^{-1/2+o(1)})<-\frac{1}{2}N^{-1/2+\delta}<0$

so that $x_{n}^{+}\in(n-1,n)$, and arguing as above we see that
$x_{n+1}^{+},x_{n+2}^{+}\in(n+1,n+2)$ and $x_{n-1}^{+},x_{n-2}^{+}\in(n-1,n)$,
so that

$\mathcal{X}^{+}\cap(n-1,n)=\\{x_{n-2}^{+},x_{n-1}^{+}\\},\quad\mathcal{X}^{+}\cap(n+1,n+2)=\\{x_{n+1}^{+},x_{n+2}^{+}\\}$

and $\mathcal{X}^{+}\cap(n,n+1)=\emptyset$.

If $(-1)^{n}\cos(\theta_{n})<-N^{-1/2+\delta}$ then we reverse the roles of
the parity classes. ∎

##  3\. Bounding the exceptional set

To conclude the proof of Theorem 1.2, we need to bound the number of “bad”
$n\in[N/2,N]$, that is $|\cos(\theta_{n})|<N^{-1/2+\delta}$, which follows
from

$\theta_{n}\bmod\pi\in[\frac{\pi}{2}-N^{-1/2+\delta},\frac{\pi}{2}+N^{-1/2+\delta}]$

or from

$((\frac{4g}{\pi}\sqrt{n}+\frac{1}{4}))\in[\frac{1}{2}-N^{-1/2+\delta},\frac{1}{2}+N^{-1/2+\delta}]$

where $((x))=x-\lfloor x\rfloor\in[0,1)$ denotes the fractional part.

An elementary argument due to Fejér (1920) (see e.g. [5, Chapter 1 §2]) shows
that for any $a>0$, and any shift $\gamma\in{\mathbb{R}}$, for suitable
$c=c(a,\gamma)>0$, for $N\gg 1$, for any interval $[\alpha,\beta]\in[0,1]$,

$\left|\\#\left\\{n\in[N/2,N]:((a\sqrt{n}+\gamma))\in[\alpha,\beta]\right\\}-(\beta-\alpha)\frac{N}{2}\right|\leq
c\sqrt{N}.$

and in particular, for an interval of length $N^{1/2+o(1)}$ the number of
fractional parts which fall into that interval is asymptotically $N/2$ times
the length of that interval. In our case, the length of the interval is
$2N^{-1/2+\delta}$ and we obtain that the number of “bad” $n\in[N/2,N]$ is
about $N^{1/2+\delta}$, as claimed.

For the readers’ benefit, we recall Fejér’s argument for the case of the
fractional parts of $\sqrt{n}$. If $k^{2}\leq n<(k+1)^{2}$ then
$((\sqrt{n}))=\sqrt{n}-k$, and then $((\sqrt{n}))\in[\alpha,\beta]$ means
$\alpha\leq\sqrt{n}-k\leq\beta$ or $(k+\alpha)^{2}\leq n\leq(k+\beta)^{2}$, so
that $n$ lies in an interval of length $2k(\beta-\alpha)+O(1)$. Summing over
$k$ we see that the number of $n\in[N/2,N]$ with
$((\sqrt{n}))\in[\alpha,\beta]$ is

$\sum_{\sqrt{N/2}\leq
k<\sqrt{N}}\left\\{2k\left(\beta-\alpha\right)+O\left(1\right)\right\\}=(\beta-\alpha)\frac{N}{2}+O(\sqrt{N}).$

##  4\. Proof of Theorem 1.4

###### Proof.

We want to count $\\#\Big{\\{}n\leq
N:\delta_{n}^{-}\in[\alpha,\beta]\Big{\\}}$. According to (1), we have

$\delta_{n}^{-}=C(-1)^{n}\cos\left(4g\sqrt{n}-\frac{\pi}{4}\right)+O(n^{-1/4+o(1)})$

with

$C=\frac{\Delta}{\sqrt{2\pi g}}.$

For the purpose of understanding the distribution of $\delta_{n}^{-}$, we may
ignore the remainder term.

Writing $\varphi_{n}=\frac{2g}{\pi}\sqrt{n}-\frac{1}{8}$, we observe that
$\cos\left(4g\sqrt{n}-\frac{\pi}{4}\right)=\cos(2\pi\varphi_{n})$ depends only
on the fractional part $((\varphi_{n}))\in[0,1)$ of $\varphi_{n}$.

We split the range $n\in[1,N]$ into even and odd $n$’s. First take $n=2m$
even. Then $\delta_{2m}^{-}\in[\alpha,\beta]$ is equivalent to

$\widehat{\alpha}\leq\cos(2\pi\varphi_{2m})\leq\widehat{\beta}$

where

$\widehat{\alpha}:=\frac{\alpha}{C},\quad\widehat{\beta}:=\frac{\beta}{C}.$

Writing $\mathbf{1}_{[\widehat{\alpha},\widehat{\beta}]}$ for the indicator
function of the interval $[\widehat{\alpha},\widehat{\beta}]$, we have

$\\#\Big{\\{}2m\leq
N:\delta_{2m}^{-}\in[\alpha,\beta]\Big{\\}}=\sum_{m=1}^{N/2}\mathbf{1}_{[\widehat{\alpha},\widehat{\beta}]}\left(\cos\left(2\pi\varphi_{2m}\right)\right).$

Now the function
$\mapsto\mathbf{1}_{[\widehat{\alpha},\widehat{\beta}]}\left(\cos\left(2\pi
x\right)\right)$ is Riemann integrable, and from uniform distribution of the
fractional parts of $\varphi_{2m}$ (Fejér’s theorem) it follows that (see e.g.
[5, Chapter 1, §1])

$\lim_{N\to\infty}\frac{1}{N/2}\sum_{m=1}^{N/2}\mathbf{1}_{[\widehat{\alpha},\widehat{\beta}]}\left(\cos\left(2\pi\varphi_{2m}\right)\right)=\int_{-1/2}^{1/2}\mathbf{1}_{[\widehat{\alpha},\widehat{\beta}]}\left(\cos\left(2\pi
x\right)\right)dx.$

Now

$\begin{split}\int_{-1/2}^{1/2}\mathbf{1}_{[\widehat{\alpha},\widehat{\beta}]}\left(\cos\left(2\pi
x\right)\right)dx&=\frac{1}{\pi}\int_{0}^{\pi}\mathbf{1}_{[\widehat{\alpha},\widehat{\beta}]}\left(\cos\left(y\right)\right)dy\\\
&=\frac{1}{\pi}\int_{\widehat{\alpha}}^{\widehat{\beta}}\frac{dt}{\sqrt{1-t^{2}}}\\\
&=\frac{1}{\pi}\int_{\alpha}^{\beta}\frac{Cdy}{\sqrt{1-C^{2}y^{2}}}=\int_{\alpha}^{\beta}\frac{dy}{\pi\sqrt{\frac{2\pi
g}{\Delta^{2}}-y^{2}}}.\end{split}$

Therefore we obtain

$\lim_{N\to\infty}\frac{1}{N}\\#\Big{\\{}2m\leq
N:\delta_{2m}^{-}\in[\alpha,\beta]\Big{\\}}=\frac{1}{2}\int_{\alpha}^{\beta}\frac{dy}{\pi\sqrt{\frac{2\pi
g}{\Delta^{2}}-y^{2}}}.$

The same considerations are valued in the case that $n=2m+1$ is odd, except
that we require

$\widehat{\alpha}\leq-\cos(2\pi\varphi_{2m})\leq\widehat{\beta},$

leading to the integral

$\int_{-1/2}^{1/2}\mathbf{1}_{[-\widehat{\beta},-\widehat{\alpha}]}\left(\cos\left(2\pi
x\right)\right)dx$

which gives the same result.

Altogether we obtain

$\operatorname{Im}_{N\to\infty}\frac{1}{N}\\#\Big{\\{}\delta_{n}^{-}\in[\alpha,\beta]\Big{\\}}=\int_{\alpha}^{\beta}\frac{dy}{\pi\sqrt{\frac{2\pi
g}{\Delta^{2}}-y^{2}}}.$

The argument for $\delta_{n}^{+}$ is identical. ∎

## References

  * [1] Braak, D. Integrability of the Rabi Model. Phys. Rev. Lett. 107, 100401 (2011).
  * [2] Braak, D.; Nguyen, T.H.L.; Reyes-Bustos, C. and Wakayama, M. Spacing distribution for quantum Rabi models. arXiv:2310.09811 [math-ph]
  * [3] Boutet de Monvel, A. and Zielinski, L. Oscillatory behavior of large eigenvalues in quantum Rabi models. Int. Math. Res. Not. IMRN (2021), no.7, 5155–5213.
  * [4] Feranchuk I. D.; Komarov L. I. and Ulyanenkov, A. P. Two-level system in a one-mode quantum field: numerical solution on the basis of the operator method, J. Phys. A: Math. Gen. 29 (1996) no. 14, 4035–4047.
  * [5] Kuipers, L. and Niederreiter, H. Uniform distribution of sequences. Pure Appl. Math. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.
  * [6] Tur, É. A. Jaynes–Cummings model: solution without rotating wave approximation. Opt. Spectrosc. 89, no. 4 (2000): 574–8.
  * [7] Xie, Qiongtao; Zhong, Honghua; Batchelor, Murray T.; Lee, Chaohong. The quantum Rabi model: solution and dynamics. J. Phys. A 50 (2017), no. 11, 113001, 40 pp.
  * [8] Yanovich, E. A. Asymptotics of eigenvalues of an energy operator in a problem of quantum physics. In Operator Methods in Mathematical Physics (OTAMP 2010, Bedlewo), 165–77. Operator Theory: Advances and Applications, 227. Basel: Birkhauser/Springer Basel AG, 2013.