# A finite dimensional proof of a result of Hutchings about irrational pseudo-
rotations

Patrice Le Calvez  Sorbonne Université, Université Paris-Cité, CNRS, IMJ-PRG,
F-75005, Paris, France & Institut Universitaire de France  patrice.le-
<EMAIL_ADDRESS>
###### Abstract.

We prove that the Calabi invariant of a $C^{1}$ pseudo-rotation of the unit
disk, that coincides with a rotation on the unit circle, is equal to its
rotation number. This result has been shown some years ago by Michael
Hutchings (under very slightly stronger hypothesis). While the original proof
used Embedded Contact Homology techniques, the proof of this article uses
generating functions and the dynamics of the induced gradient flow.

Keywords: Irrational pseudo-rotation, Calabi invariant, generating function,
rotation number, linking number

MSC 2020: 37E30, 37E45, 37J11

##  1\. Introduction

###  1.1. Statement of the main theorem

We denote by $\mathbb{D}$ the closed unit disk of the Euclidean plane and by
$\mathbb{S}$ the unit circle. We will furnish $\mathbb{D}$ with the standard
area form $\omega=dx\wedge dy$ and will denote by
$\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$ the group of diffeomorphisms of class
$C^{1}$ that preserve $\omega$ (we will say that $f$ is symplectic). We will
also denote by $\mathrm{Homeo}_{*}(\mathbb{D})$ the group of orientation
preserving homeomorphisms of $\mathbb{D}$.

If $f\in\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$ fixes every point in a
neighborhood of $\mathbb{S}$, its Calabi invariant
$\mathrm{Cal}(f)\in\mathbb{R}$ is a well-studied object that has several
interpretations (see [Ca], [Fa], [GaGh] for instance). It admits a natural
extension to $\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$. We will explain this
extension, as described by Benoit Joly in his thesis [J]. If $\kappa$ is a
primitive of $\omega$, and $A:\mathbb{D}\to\mathbb{R}$ a primitive of
$f^{*}(\kappa)-\kappa$, then $\int_{\mathbb{S}}A\,d\mu$ does not depend on
$\mu$, where $\mu$ is a Borel probability measure invariant by
$f_{|\mathbb{S}}$111Such a measure is unique when the rotation number of
$f_{|\mathbb{S}}$ is irrational and supported in the periodic point set
otherwise.. Consequently, there exists a unique primitive $A_{f,\kappa}$ of
$f^{*}(\kappa)-\kappa$ satisfying $\int_{\mathbb{S}}A_{f,\kappa}\,d\mu=0$ for
such $\mu$. Moreover, the quantity
$\mathrm{Cal}(f)=\int_{\mathbb{D}}A_{f,\kappa}\,\omega$ does not depend on the
choice of $\kappa$. For example we have $\mathrm{Cal}(f)=0$ if $f$ is an
Euclidean rotation. Note that $\mathrm{Cal}(f)$ is the usual Calabi invariant
in case $f$ fixes every point in a neighborhood of $\mathbb{S}$ (in that case
$A_{f,\kappa}$ vanishes on a neighborhood of $\mathbb{S}$).

We can also define a real function $\widetilde{\mathrm{Cal}}$ on the universal
covering space $\widetilde{\mathrm{Diff}^{1}_{\omega}(\mathbb{D})}$ of
$\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$ as follows. Every identity isotopy of
$f$ in $\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$, meaning every continuous path
$I=(f_{s})_{s\in[0,1]}$ in $\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$ joining
the identity map of $\mathbb{D}$ to $f$, is homotopic, relative to its
endpoints, to a Hamiltonian isotopy $I^{\prime}=(f^{\prime}_{s})_{s\in[0,1]}$.
It means that there exists a time dependent divergence free vector field
$(X_{s})_{s\in[0,1]}$ of class $C^{1}$ such that for every $z\in\mathbb{D}$ it
holds that ${d\over ds}f^{\prime}_{s}(z)=X_{s}(f^{\prime}_{s}(z))$. There
exists a (uniquely defined) family $(H_{s})_{s\in[0,1]}$ of functions of class
$C^{2}$ vanishing on $\mathbb{S}$, such that for every $s\in[0,1]$, every
$z\in\mathbb{D}$ and every $v\in\mathbb{R}^{2}$ it holds that
$dH_{s}(z).v=\omega(v,X_{s}(z))$. The quantity
$\widetilde{\mathrm{Cal}}(I)=\int_{0}^{1}\left(\int_{\mathbb{D}}H_{s}\,\omega\right)\,ds$
depends only on the homotopy class $[I]$ of $I$ and we can also denote it
$\widetilde{\mathrm{Cal}}([I])$. Let us give a simple example. For every
$\alpha\in\mathbb{R}$, denote $R_{\alpha}$ the rotation of angle $2\pi\alpha$
and $T_{\alpha}$ the isotopy $(R_{s\alpha})_{s\in[0,1]}$. It holds that
$\widetilde{\mathrm{Cal}}([T_{\alpha}])=\pi^{2}\alpha$. As stated in [J],
there is a link between $\mathrm{Cal}$ and $\widetilde{\mathrm{Cal}}$ that we
will explain now. Every identity isotopy $I=(f_{s})_{s\in[0,1]}$ of $f$ in
$\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$ defines by restriction an identity
isotopy $I_{|\mathbb{S}}=(f_{s}{}_{|\mathbb{S}})_{s\in[0,1]}$ of
$f_{|\mathbb{S}}$ in $\mathrm{Diff}^{1}(\mathbb{S})$. Moreover, every homotopy
class $[I]\in\widetilde{\mathrm{Diff}^{1}_{\omega}(\mathbb{D})}$ defines by
restriction a homotopy class
$[I]_{\mathbb{S}}\in\widetilde{\mathrm{Diff}_{*}^{1}(\mathbb{S})}$, where
$\widetilde{\mathrm{Diff}_{*}^{1}(\mathbb{S})}$ is the universal covering
space of the group $\mathrm{Diff}_{*}^{1}(\mathbb{S})$ of orientation
preserving diffeomorphisms of the circle. We have the following equation,

$\widetilde{\mathrm{Cal}}([I])=\mathrm{Cal}(f)+\mathrm{rot}([I]_{\mathbb{S}}),$

where $\mathrm{rot}([I]_{\mathbb{S}})$ is the Poincaré rotation number of
$[I]_{\mathbb{S}}$. In particular, if $\mu$ is a Borel probability measure
invariant by $f_{\Sigma}$, then the asymptotic cycle of $\mu$ defined by an
isotopy $I$ (see Schwartzman [Sc]) is equal to $\rho([I])\varpi\in
H_{1}(\mathbb{S},\mathbb{R})$, where $\varpi$ is the fundamental class of
$\mathbb{S}$ defining its usual orientation.

There is a more dynamical interpretation of the Calabi invariant in the case
of a compactly supported symplectic diffeomorphism of the open disk, due to
Fathi [Fa] and developed by Gambaudo and Ghys [GaGh]. This interpretation is
still valid in this more general situation. Consider the usual angular form

$d\theta={ydx-xdy\over x^{2}+y^{2}}$

and define

$W=\\{(z,z^{\prime})\in\mathbb{D}\times\mathbb{D}\,|z\not=z^{\prime}\\}.$

Consider $f\in\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$. For every identity
isotopy $I=(f_{t})_{t\in[0,1]}$ of $f$ in $\mathrm{Homeo}_{*}(\mathbb{D})$ and
every $(z,z^{\prime})\in W$, one gets a path

$\displaystyle I_{z,z^{\prime}}:[0,1]$ $\displaystyle\to\mathbb{R}^{2}$
$\displaystyle s$ $\displaystyle\mapsto f_{s}(z^{\prime})-f_{s}(z).$

The function

$\displaystyle\mathrm{ang}_{I}:W$ $\displaystyle\to\mathbb{R}$
$\displaystyle(z,z^{\prime})$ $\displaystyle\mapsto{1\over
2\pi}\int_{I_{z,z^{\prime}}}d\theta$

depends only on the homotopy class of $I$ in $\mathrm{Homeo}_{*}(\mathbb{D})$
and is bounded because $f$ is a diffeomorphism of class $C^{1}$. In
particular, one naturally gets a function $\mathrm{ang}_{[I]}$ for every
$[I]\in\widetilde{\mathrm{Diff}^{1}_{\omega}(\mathbb{D})}$.

If $I=(f_{s})_{s\in[0,1])}$ and $I^{\prime}=(f^{\prime}_{s})_{s\in[0,1]}$ are
two identity isotopies in $\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$, one can
define an identity isotopy $II^{\prime}=(f^{\prime\prime}_{s})_{s\in[0,1]}$ in
$\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$ writing:

$f^{\prime\prime}_{s}=\begin{cases}f^{\prime}{}_{2s}&\text{ if $0\leq s\leq
1/2$,}\\\ f_{2s-1}\circ f^{\prime}_{1}&\text{ if $1/2\leq s\leq
1$,}\end{cases}.$

The homotopy class of $I^{\prime}J^{\prime}$ depends only on $[I]$ and
$[I^{\prime}]$ and one gets a group structure on
$\widetilde{\mathrm{Diff}}^{1}_{\omega}(\mathbb{D})$ by setting
$[I][I^{\prime}]=[II^{\prime}]$. Note that for every $n\geq 1$ we have

$\mathrm{ang}_{[I]^{n}}=\sum_{k=0}^{n-1}\mathrm{ang}_{[I]}\circ(f^{k}\times
f^{k}).$

Generalizing a proof due to Shelukhin [Sh] the following equality

$\widetilde{\mathrm{Cal}}([I])=\int_{W}\mathrm{ang}_{[I]}(z,z^{\prime})\,\omega(z)\,\omega(z^{\prime}).$

is proved in [J]. Let us add that this interpretation of the Calabi invariant
has permitted to Gambaudo and Ghys [GaGh] to prove that two elements of
$\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$ fixing every point in a neighborhood
of $\mathbb{S}$ and conjugate in $\mathrm{Homeo}_{*}(\mathbb{D})$ have the
same Calabi invariant. To conclude, just note the following:

  * •

the map $[I]\mapsto\widetilde{\mathrm{Cal}(I)}$ defined on
$\widetilde{\mathrm{Diff}^{1}_{\omega}(\mathbb{D})}$ is a morphism;

  * •

the map $f\mapsto{\mathrm{Cal}(f)}$ defined on
$\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$is a homogeneous quasi-morphism;

  * •

for every $[I]$, $[I^{\prime}]$ in
$\widetilde{\mathrm{Diff}^{1}_{\omega}(\mathbb{D})}$, there is a unique
$k\in\mathbb{Z}$ such that $[I^{\prime}]=[I][T_{k}]$ and we have:

$\mathrm{ang}_{[I^{\prime}]}=\mathrm{ang}_{[I]}+k,\enskip\widetilde{\mathrm{Cal}}([I^{\prime}])=\widetilde{\mathrm{Cal}}([I])+\pi^{2}k.$

  * •

for every $[I]\in\widetilde{\mathrm{Diff}^{1}_{\omega}(\mathbb{D})}$ and every
$p$, $q$ in $\mathbb{Z}$ we have

$\widetilde{\mathrm{Cal}}([I]^{q}[T_{p}])=\widetilde{\mathrm{Cal}}([I])^{q}+\pi^{2}p.$

We will say that $f\in\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$ is an irrational
pseudo-rotation if it fixes $0$ and does not possess any other periodic point.
Using an extension of Poincaré-Birkhoff theorem due to Franks [Fr], we know
that for every lift $[I]\in\widetilde{\mathrm{Diff}^{1}_{\omega}(\mathbb{D})}$
of $f$, there exists $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ such that the
sequence of maps $z\mapsto n^{-1}\mathrm{ang}_{[I]^{n}}(0,z)$ converges
uniformly to the function $z\mapsto\alpha$. We will say that $\alpha$ is the
rotation number of $[I]$ and write $\alpha=\widetilde{\mathrm{rot}}([I])$. Of
course, $\overline{\alpha}=\alpha+\mathbb{Z}\in\mathbb{R}/\mathbb{Z}$ is
independent of the choice of $I$ and we say that $f$ is a pseudo-rotation of
rotation number $\overline{\alpha}$. Note that $\overline{\alpha}$ is the
Poincaré rotation number of $f_{|\mathbb{S}}$.

The goal of the article is to prove the following:

######  Theorem 1.1.

Let $f\in\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$ be an irrational pseudo-
rotation such that $f_{|\mathbb{S}}$ is $C^{1}$-conjugate to a rotation. If
$[I]\in\widetilde{\mathrm{Diff}^{1}_{\omega}(\mathbb{D})}$ is a lift of $f$,
then we have

$\widetilde{\mathrm{Cal}}([I])=\pi^{2}\,\widetilde{\mathrm{rot}}([I])$

or equivalently, we have $\mathrm{Cal}(f)=0.$

It is not difficult to prove that a $C^{1}$ diffeomorphism of $\mathbb{S}$ can
be extended to a $C^{1}$ symplectic diffeomorphism of $\mathbb{D}$. So, it is
sufficient to prove the theorem in case $f$ coincides with a rotation on the
unit circle. This theorem was already known and due to Michael Hutchings (with
very slightly stronger hypothesis), appearing as an easy consequence of a
theorem we will recall now. Suppose that
$f\in\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$ coincides with a rotation
$R_{\alpha}$ in a neighborhood of $\mathbb{S}$. Fix a primitive $\kappa$ of
$\omega$ and note that $A_{f,\kappa}$ vanishes on $\mathbb{S}$ whether
$\alpha$ is rational or irrational. If $O$ is a periodic orbit of $f$, one
proves easily that $\sum_{z\in O}A_{f,\kappa}(z)$ does not depend on the
choice of $\kappa$. So, one can define the mean action of $O$ as being

$\mathrm{act}(O)={1\over\\#O}\sum_{z\in O}A_{f,\kappa}(z).$

One has the following ([H])222The proof is stated for smooth diffeomorphisms
but should be possibly extended supposing a low differentiability condition,
as Michael Hutchings explained to us.

######  Theorem 1.2.

Suppose that $f\in\mathrm{Diff}^{\infty}_{\omega}(\mathbb{D})$ coincides with
a rotation in a neighborhood of $\mathbb{S}$ and denote $\mathcal{O}$ the set
of periodic orbits of $f$. Then it holds that:

  * •

if ${\mathrm{Cal}(f)}<0$, then
$\inf_{O\in\mathcal{O}}\mathrm{act}(O)\leq{\mathrm{Cal}(f)}$,

  * •

if ${\mathrm{Cal}(f)}>0$, then
$\sup_{O\in\mathcal{O}}\mathrm{act}(O)\geq{\mathrm{Cal}(f)}$.

It is easy to prove that if $f\in\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$
coincides with a rotation on $\mathbb{S}$, then we have
$\mathrm{act}(\\{0\\})=0$. Consequently, by Theorem 1.2, it holds that
${\mathrm{Cal}(f)}=0$ if $f$ is an irrational pseudo-rotation. It must be
noticed that Hutchings’ theorem has been recently improved by Pirnapasov [P]
(still in the smooth category) that uses a preliminary extension result. A
much weaker hypothesis on the boundary is needed, even weaker than been a
rotation on the boundary. A nice corollary, is the fact that
$\mathrm{Cal}(f)=0$ for every smooth irrational pseudo-rotation $f$,
regardless of any condition on the boundary.

Let us continue with a nice consequence of Theorem 1.1 . Let
$D\subset\mathbb{D}$ be a closed disk that does not contain $0$ and does not
meet $\mathbb{S}$. One constructs easily
$f^{\prime}\in\mathrm{Diff}^{\infty}_{\omega}(\mathbb{D})$ arbitrarily close
to the identity map for the $C^{\infty}$-topology, fixing every point outside
$D$ and such that $\mathrm{Cal}(f^{\prime})\not=0$. If $[I]$ is a lift of $f$
and $[I^{\prime}]$ is the lift of $f^{\prime}$ such that
$[I^{\prime}]_{\mathbb{S}}$ is the trivial homotopy, then it holds that

$\widetilde{\mathrm{Cal}}([I][I^{\prime}])=\widetilde{\mathrm{Cal}}([I])+\widetilde{\mathrm{Cal}}([I^{\prime}])=\widetilde{\mathrm{rot}}([I])+\mathrm{Cal}(f^{\prime})\not=\widetilde{\mathrm{rot}}({[I]})=\widetilde{\mathrm{rot}}([I][I^{\prime}]).$

It implies that $f\circ f^{\prime}$ is not an irrational pseudo-rotation,
which means that it has at least one periodic orbit different from $\\{0\\}$.
Such a periodic orbit must meet $D$.

The proof of Hutchings is based on a reduction to a problem of contact
geometry and then the applications of methods of Embedded Contact Homology
theory. The previous perturbation result is true in a more general situation.
Asaoka and Irie [AI], using Embedded Contact Homology as well, have proved
that it remains true provide $f\in\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$ has
no periodic point inside $D$. Other striking applications of contact or
symplectic geometry to dynamics on surfaces have appeared recently that use
elaborated tools of contact or symplectic geometry (for instance see
Cristofaro-Gardiner, Humilère, Seyfaddini [CrHS] or Cristofaro-Gardiner,
Prasad, Zhang [CrPZ]). The proof of Theorem 1.1 that will be given in the
present article does not use Floer homology but only generating functions. Can
be hope be able to find proofs of these deep results using such classical
tools?

###  1.2. Idea of the proof

Let us state first the following result of Bramham [Br]:

######  Theorem 1.3.

Every irrational pseudo-rotation
$f\in\mathrm{Diff}^{\infty}_{\omega}(\mathbb{D})$ is the limit, for the
$C^{0}$ topology, of a sequence of finite order $C^{\infty}$ diffeomorphisms.

Write $\overline{\alpha}=\mathrm{rot}(f)$. The proof says something more
precise: if $(q_{n})_{n\geq 0}$ is a sequence of positive integers such that
$(q_{n}\overline{\alpha})_{n\geq 0}$ converges to $0$ in
$\mathbb{R}/\mathbb{Z}$, then there exists a sequence of homeomorphisms
$(f_{n})_{n\geq 0}$ fixing $0$ and satisfying $(f_{n})^{q_{n}}=\mathrm{Id}$,
that converges to $f$ for the $C^{0}$ topology (to obtain a sequence of smooth
approximations one needs a simple additional argument of approximation).
Theorem 1.1 would have been an easy consequence of Theorem 1.3 if the
strongest following properties were true:

  * •

the $f_{n}$ are symplectic diffeomorphisms of class $C^{1}$;

  * •

the sequence $(f_{n})_{n\geq 0}$ converges to $f$ in the $C^{1}$ topology.

Unfortunately, there is no reason why the $f_{n}$ appearing in the
construction of the sequence satisfy these properties.

The original proof of Theorem 1.3 uses pseudoholomorphic curve techniques from
symplectic geometry. In [L2] we succeeded to find a finite dimensional proof
by using generating functions, like in Chaperon’s broken geodesics method
[Ch]. Let us remind the ideas of this last proof. The hypothesis were slightly
different, the result was stated for an irrational pseudo-rotation
$f\in\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$ coinciding with a rotation on
$\mathbb{S}$. This last property permits us to extend our map, also denoted
$f$, to a piecewise $C^{1}$ diffeomorphism of the whole plane, being an
integrable polar twist map with increasing rotation number in an annulus
$\\{z\in\mathbb{R}^{2}\,|\,1\leq|z|\leq r_{0}\\}$ and equal to an irrational
rotation in $\\{z\in\mathbb{R}^{2}\,|\,|z|\geq r_{0}\\}$. Let $I$ be an
identity isotopy of $f_{|\mathbb{D}}$ in
$\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$ that fixes $0$. We write
$\alpha=\mathrm{rot}([I]_{\mathbb{S}})$. One can extend $I$ to an identity
isotopy $(f_{t})_{t\in[0,1]}$ of $f$, also denoted $[I]$, where each $f_{t}$
is an integrable polar map in $\\{z\in\mathbb{R}^{2}\,|\,1\leq|z|\leq
r_{0}\\}$ and a rotation in $\\{z\in\mathbb{R}^{2}\,|\,|z|\geq r_{0}\\}$. The
circle ${\mathbb{S}}$ is accumulated from outside by invariant circles
$S_{a/b}$, such that
$[I]^{b}[T_{-a}]{}_{|S_{a/b}}\in\widetilde{\mathrm{Diff}}^{1}(S_{p/q})$ is the
trivial homotopy. Moreover one can find a sequence
$(a_{n},b_{n})\in\mathbb{Z}\times(\mathbb{N}\setminus\\{0\\})$ such that
$|a_{n}-b_{n}\alpha|\leq b_{n}^{-1}$. The map $f$ being piecewise $C^{1}$, one
can write it as the composition of $m$ maps “close to
$\mathrm{Id}_{\mathbb{R}^{2}}$”, where $m\geq 1$, and then construct a
$m$-periodic family $(h_{i})_{i\in\mathbb{Z}}$ of generating real functions
that are $C^{1}$ with Lipschitz derivatives. One knows that for every $b\geq
1$, the fixed point set of $f^{b}$ corresponds to the singular point set of a
$A$-Lipshitz vector field $\zeta$ defined on a $2mb$-dimensional space
$E_{b}$, furnished with a natural scalar product. This vector field is the
gradient flow of a function ${\bf h}$ defined in terms of the $h_{i}$,
$i\in\mathbb{Z}$, and the constant $A$ is independent of $b$. In particular
each circle $S_{a/b}\subset\mathbb{R}^{2}$ corresponds to a curve
$\Sigma_{a}\subset E_{b}$ of singularities of $\zeta$. A fundamental result is
the fact that $\Sigma_{a}$ bounds a disk $\Delta_{a}\subset E_{b}$ that
contains the singular point corresponding to the fixed point $0$ and that is
invariant by the flow and by a natural $\mathbb{Z}/b\mathbb{Z}$ action on
$E_{b}$. Moreover the dynamics of the flow of $\zeta$ on $\Delta_{a}$ is
north-south and the non trivial orbits have the same energy. This energy can
be explicitly computed and is small if $a/b$ is a convergent of $\alpha$.
Using the independence of the Lipshitz constant $A$, one can deduce that
$\zeta$ is “uniformly small” on $\Delta_{a}$. The disk $\Delta_{a}$ projects
homeomorphically on $D_{a/b}$ by an explicit map $q_{1}$. The
$\mathbb{Z}/b\mathbb{Z}$ action on $\Delta_{a}$ defines, by projection on
$D_{a/b}$, a homeomorphism $\widehat{f}$ of order $b$ that coincides with $f$
on $S_{a/b}$. Using was has been said above, in particular the fact that
$\zeta$ is small on $\Delta_{a}$, one can prove that $\widehat{f}$ is
uniformly close to $f$ on the disk $D_{a/b}$ if $a/b$ is a convergent of
$\alpha$. This is the way we prove the approximation result.

The orbits of $\zeta_{|\Delta_{a}}$ define a radial foliation on $\Delta_{a}$
that projects by $q_{1}$ onto a topological radial foliation
${\mathcal{F}}_{1}$ of $S_{a/b}$ that is invariant by $\widehat{f}$. It is a
natural to ask wether one can compute the Calabi invariant of $f$ by using the
fact that $\widehat{f}$ is arbitrarily close to $f$. The main problem is that
$\widehat{f}$ and $\mathcal{F}_{1}$ are continuous. There is no
differentiability anymore, and differentiability is crucial while dealing with
Calabi invariants, regardless of the approach we choose. Nevertheless, the
fact that we are dealing with Lipshitz maps close to $\mathrm{Id}$ (the maps
$f_{1}$,…, $f_{m}$ appearing in the decomposition of $f$) permit us by to
state some quantitative results. The three key-points that will used to get
the proof are related to the foliation $\mathcal{F}_{1}$ :

  1. (1)

the energy of the non trivial orbits of $\zeta_{|\Delta_{a}}$ is bounded by
$K|a-b\alpha|$, where $K$ does not depend on $a$ and $b$;

  2. (2)

the leaves of $\mathcal{F}_{1}$ are Brouwer lines of
$I^{b}T_{-a}{}_{|S_{a/b}}$;

  3. (3)

the winding distance between $\mathcal{F}_{1}$ and $f^{-b}(\mathcal{F}_{1})$
is bounded by $4mb$.

Let us precise these three points.

The non trivial orbits of $\zeta_{|\Delta_{a}}$ have the same energy. This
quantity measures the area sweeped by a leaf of $\mathcal{F}_{1}$ along the
isotopy $I^{b}T_{-a}{}_{|S_{a/b}}$.

By saying that the leaves of $\mathcal{F}_{1}$ are Brouwer lines, we mean that
they are pushed on the left along the isotopy $I^{b}T_{-a}{}_{|S_{a/b}}$.
Equivalently, we can say the following: let $\tilde{D}_{a/b}$ be the universal
covering space of $D_{a/b}^{*}=D_{a/b}\setminus(S_{a/b}\cup\\{0\\})$ and
$\tilde{I}=(\tilde{f}_{s})_{s\in[0,1]}$ the identity isotopy on
$\tilde{D}_{a/b}$ that lifts $I_{|D_{a/b}^{*}}$. Denote
$\tilde{\mathcal{F}_{1}}$ the (non singular) foliation of $\tilde{D}_{a/b}$
that lifts ${\mathcal{F}_{1}}_{|D^{*}_{a/b}}$. Every leaf $\tilde{\phi}$ of
$\tilde{\mathcal{F}_{1}}$ is an oriented topological line of $\tilde{D}_{a/b}$
that separates its complement into two components, one on the left and one one
the right. Let $T$ be the generator of the group of covering automorphisms,
such that $T(\tilde{\phi})$ is on the left of $\tilde{\phi}$. The assertion
(2) states that $\tilde{f}^{b}\circ T^{-a}(\tilde{\phi})$ is on the left of
$\tilde{\phi}$ for every leaf $\tilde{\phi}$. The assertion (1) states that
the area of the domain between $\tilde{\phi}$ and $\tilde{f}^{b}\circ
T^{-a}(\tilde{\phi})$, which is equal to the energy of the corresponding orbit
of $\zeta$, is bounded by $K|a-b\alpha|$.

The assertion (3) says that the foliation $f^{-b}(\mathcal{F}_{1})$ winds
relative to $\mathcal{F}_{1}$ no more than $4mq$. A precise definition will be
given in the next section. Just say that if $\mathcal{F}$ is a radial
foliation of class $C^{1}$, there exists $K>0$ such that for every $b>0$ the
winding distance between the foliations $\mathcal{F}$ and
$f^{-b}(\mathcal{F})$ is bounded by $Kb$. The foliation that appears in our
construction depends on $a$ and $b$, is not differentiable, possess
interesting dynamical properties stated in (1) and (2) and nevertheless
satisfies a “differential-like” property.

The properties (1), (2) and (3) will permit us to bound the Calabi invariant
of each map $f_{|D_{a/b}}$. Note that we will use the third definition, as the
map is only piecewise $C^{1}$ on $D_{a/b}$. The Calabi invariant will be small
if $a/b$ is a convergent that is close to $\alpha$. By a limit process, we
will prove that $\mathrm{Cal}(f_{|\mathbb{D}})=0$.

The rest of the article is divided into three sections.

Section 2 will be dedicated to the study of topological radial foliations
defined on a disk. We will introduce different objects on the set of such
foliations, in particular the winding distance between two foliations. We will
see how to compute the Calabi invariant of a symplectic diffeomorphism by
using such a foliation.

In section 3 we will recall the construction done in [L2] and quickly
explained in the introduction. In particular we will define $\mathcal{F}_{1}$
and prove the properties (1), (2) and (3) stated above.

We will prove Theorem 1.1 in the short section 4.

I would like to thank Benoit Joly, Frédéric Le Roux and Sobhan Seyfaddini for
so useful talks. I am particulary indepted to John Franks for the many
conversations we had on this subject some years ago and would like to thank
him warmly.

##  2\. Radial foliations and Calabi invariant

We will introduce a formal setting to study radial foliations on a disk, based
on a “discretization of the angles”, that seems pertinent for studying
topological foliations. Classical objects related to disk homeomorphisms, like
rotation numbers or linking numbers, will be investigated in this formalism.
One can look at Bechara [Be] for similar questions in a differentiable
context.

###  2.1. The space of radial foliation

Let $D$ be an open disk of the form $D=\\{z\in\mathbb{C}\,|\,|z|<r\\}$, where
$r>0$. We will set $D^{*}=D\setminus\\{0\\}$ and will denote $\tilde{D}$ the
universal covering space of $D^{*}$. We will consider the sets

$W=\\{(z,z^{\prime})\in D^{*}\times D^{*}\,|\,z\not=z^{\prime}\\}$

and

$\tilde{W}=\\{(\tilde{z},\tilde{z}^{\prime})\in\tilde{D}\times\tilde{D}\,|\,\tilde{z}\not=\tilde{z}^{\prime}\\}.$

A radial foliation is an oriented topological foliation on $D^{*}$ such that
every leaf $\phi$ is a ray, meaning that it satisfies

$\alpha(\phi)=\\{0\\},\enskip\omega(\phi)\subset\partial
D=\\{z\in\mathbb{C}\,|\,|z|=r\\}.$

In other words, $\phi$ tends to $0$ in the past and to $\partial D$ in the
future. We denote ${\mathfrak{F}}$ the set of radial foliations. The group
$\mathrm{Homeo}_{*}(D^{*})$ of homeomorphisms of $D^{*}$ that are isotopic to
the identity, acts naturally on ${\mathfrak{F}}$: if
$\mathcal{F}\in\mathfrak{F}$ and $f\in\mathrm{Homeo}_{*}(D^{*})$, then the
foliation $f(\mathcal{F})$, whose leaves are the images by $f$ of the leaves
of $\mathcal{F}$, is a radial foliation. Moreover $\mathrm{Homeo}_{*}(D^{*})$
acts transitively on $\mathfrak{F}$. If $\mathrm{Homeo}_{*}(D^{*})$ is endowed
with the $C^{0}$ topology, meaning the compact-open topology applied to maps
and the inverse maps, the stabilizer $\mathrm{Homeo}_{*,\mathcal{F}}(D^{*})$
of $\mathcal{F}\in\mathfrak{F}$ is a closed subgroup of
$\mathrm{Homeo}_{*}(D^{*})$. Consequently, the map

$\displaystyle\mathrm{Homeo}_{*}(D^{*})/\mathrm{Homeo}_{*,\mathcal{F}}(D^{*})$
$\displaystyle\to{\mathfrak{F}}$ $\displaystyle
f\,\mathrm{Homeo}_{*,\mathcal{F}}(D^{*})$ $\displaystyle\mapsto
f(\mathcal{F})$

is bijective and ${\mathfrak{F}}$ can be furnished with a natural $C^{0}$
topology. It is the topology, induced from the quotient topology defined on
$\mathrm{Homeo}_{*}(D^{*})/\mathrm{Homeo}_{*,\mathcal{F}}(D^{*})$ by this
identification map. Note that this topology does not depend on $\mathcal{F}$.
It is well known that the fondamental groups of $\mathrm{Homeo}_{*}(D^{*})$
and $\mathrm{Homeo}_{*,\mathcal{F}}(D^{*})$ are infinite cyclic and that the
morphism
$i_{*}:\pi_{1}(\mathrm{Homeo}_{*,\mathcal{F}}(D^{*}),\mathrm{Id})\to\pi_{1}(\mathrm{Homeo}_{*}(D^{*}),\mathrm{Id})$
induced by the inclusion map
$i:\mathrm{Homeo}_{*,\mathcal{F}}(D^{*})\to\mathrm{Homeo}_{*}(D^{*})$ is
bijective. It implies that ${\mathfrak{F}}$ is simply connected. In the whole
article, when $\mathcal{F}$ is a radial foliation, the notation
$\tilde{\mathcal{F}}$ means the lift of $\mathcal{F}$ to the universal
covering space $\tilde{D}$. Note that $\tilde{\mathcal{F}}$ is a trivial
foliation. In particular, every leaf $\tilde{\phi}$ of $\tilde{\mathcal{F}}$
is an oriented line of $\tilde{D}$ that separates $\tilde{D}$ into two
connected open sets, the component of $\tilde{D}\setminus\tilde{\phi}$ lying
on the left of $\tilde{\phi}$ will be denoted $L(\tilde{\phi})$ and the
component lying on the right will be denoted $R(\tilde{\phi})$. We get a total
order $\preceq_{\mathcal{F}}$ on the set of leaves of $\tilde{\mathcal{F}}$ as
follows:

$\tilde{\phi}\preceq_{\mathcal{F}}\tilde{\phi}^{\prime}\Longleftrightarrow
R(\tilde{\phi})\subset R(\tilde{\phi}^{\prime}).$

We can also define a partial order $\leq_{{\mathcal{F}}}$ on $\tilde{D}$:
denote $\tilde{\phi}_{\tilde{z}}$ the leaf of $\tilde{\mathcal{F}}$ that
contains $\tilde{z}$ and write $\tilde{z}<_{{\mathcal{F}}}\tilde{z}^{\prime}$
if $\tilde{z}\not=\tilde{z}^{\prime}$, if
$\tilde{\phi}_{\tilde{z}}=\tilde{\phi}_{\tilde{z}^{\prime}}$ and if the
segment of $\tilde{\phi}_{\tilde{z}}$ beginning at $\tilde{z}$ and ending at
$\tilde{z}^{\prime}$ inherits the orientation of $\tilde{\phi}_{\tilde{z}}$.

###  2.2. Topological angles in the universal covering space

Consider the natural projection

$\displaystyle\pi:\mathbb{Z}$ $\displaystyle\to\mathbb{Z}/4\mathbb{Z}$
$\displaystyle k$ $\displaystyle\mapsto\dot{k}=k+4\mathbb{Z}.$

If $\mathbb{Z}/4\mathbb{Z}$ is endowed with the topology whose open sets are

$\emptyset,\\{\dot{1}\\},\\{\dot{3}\\},\\{\dot{1},\dot{3}\\},\\{\dot{1},\dot{2},\dot{3}\\},\\{\dot{3},\dot{0},\dot{1}\\},\\{\dot{0},\dot{1},\dot{2},\,\dot{3}\\},$

and $\mathbb{Z}$ with the topology generated by the sets $2k+1$ and
$\\{2k+1,2k,2k+1\\}$, $k\in\mathbb{Z}$, then $\pi$ is a covering map333This
topology on $\mathbb{Z}$ is usually called the digital line topology on
$\mathbb{Z}$. Note that both sets $\mathbb{Z}/4\mathbb{Z}$ and $\mathbb{Z}$
are non Hausdorff but path connected.

If $k$, $l$ are two integers, we will define

$\lambda(k,l)=\begin{dcases}0&\text{ if $k=l$,}\\\ \\#(k,l)\cap
4\mathbb{Z}+(\\#\\{k,l\\}\cap 4\mathbb{Z})/2&\text{ if $k<l$,}\\\
-\\#(l,k)\cap 4\mathbb{Z}-(\\#\\{l,k\\}\cap 4\mathbb{Z})/2&\text{ if
$k>l$.}\end{dcases}$

Note that for every integers $k,l,m$ we have

$\lambda(k,l)+\lambda(l,m)=\lambda(k,m).$

The quantity $\lambda(k,l)$ measures the “algebraic intersection number” of a
continuous path $\gamma:[s_{0},s_{1}]\to\mathbb{Z}/4\mathbb{Z}$ with
$\\{\dot{0}\\}$, if $\gamma$ is lifted to a path
$\hat{\gamma}:[s_{0},s_{1}]\to\mathbb{Z}$ joining $k$ to $l$.

For every $(\tilde{z},\tilde{z}^{\prime})\in\tilde{W}$, we can define a
continuous function
$\theta_{\tilde{z},\tilde{z}^{\prime}}:{\mathfrak{F}}\to\mathbb{Z}/4\mathbb{Z}$
as follows:

$\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})=\begin{cases}\dot{0}&\text{
if $\tilde{z}^{\prime}>_{\mathcal{F}}\tilde{z}$,}\\\ \dot{1}&\text{ if
$\phi_{\tilde{z}^{\prime}}\succ_{\mathcal{F}}\phi_{\tilde{z}}$,}\\\
\dot{2}&\text{ if $\tilde{z}^{\prime}<_{\mathcal{F}}\tilde{z}^{\prime}$,}\\\
\dot{3}&\text{ if
$\phi_{\tilde{z}^{\prime}}\prec_{\mathcal{F}}\phi_{\tilde{z}}$.}\end{cases}$

The space $\mathfrak{F}$ being simply connected, the Lifting Theorem asserts
that there exists a continuous function,
$\hat{\theta}_{\tilde{z},\tilde{z}^{\prime}}:{\mathfrak{F}}\to\mathbb{Z}$,
uniquely defined up to an additive constant in $4\mathbb{Z}$, such that
$\pi\circ\hat{\theta}_{\tilde{z},\tilde{z}^{\prime}}=\theta_{\tilde{z},\tilde{z}^{\prime}}.$

Note that

  * •

$\theta_{\tilde{z}^{\prime},\tilde{z}}(\mathcal{F})=\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})+\dot{2}$,
for every $\mathcal{F}\in\mathfrak{F}$,

  * •

$\hat{\theta}_{\tilde{z},\tilde{z}^{\prime}}$ and
$\hat{\theta}_{\tilde{z}^{\prime},\tilde{z}}$ can be chosen such that
$\hat{\theta}_{\tilde{z}^{\prime},\tilde{z}}(\mathcal{F})=\hat{\theta}_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})+2$
for every $\mathcal{F}\in\mathfrak{F}$,

  * •

$\theta_{\tilde{f}(\tilde{z}),\tilde{f}(\tilde{z}^{\prime})}(f(\mathcal{F}))=\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})$,
for every $\mathcal{F}\in\mathfrak{F}$ and every
$f\in\mathrm{Homeo}_{*}(D^{*})$,

  * •

$\hat{\theta}_{\tilde{f}(\tilde{z}),\tilde{f}(\tilde{z}^{\prime})}$ and
$\hat{\theta}_{\tilde{z},\tilde{z}^{\prime}}$ can be chosen such that
$\hat{\theta}_{\tilde{f}(\tilde{z}),\tilde{f}(\tilde{z}^{\prime})}(f(\mathcal{F}))=\hat{\theta}_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})$,
for every $\mathcal{F}\in\mathfrak{F}$.

In particular, if $\mathcal{F}$ and $\mathcal{F}$ are two radial foliations,
the numbers

$\hat{\tau}(\tilde{z},\tilde{z}^{\prime},\mathcal{F},\mathcal{F}^{\prime})=\hat{\theta}_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}^{\prime})-\hat{\theta}_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})$

and

$\lambda(\tilde{z},\tilde{z}^{\prime},\mathcal{F},\mathcal{F}^{\prime})=\lambda(\hat{\theta}_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}),\hat{\theta}_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}^{\prime}))$

do not depend on the choice of the lift $\hat{\theta}$.

Suppose that $(\tilde{z},\tilde{z}^{\prime})\in\tilde{W}$, that
$f\in\mathrm{Homeo}_{*}({D^{*}})$, that $\tilde{f}$ is a lift of $f$ and that
$\mathcal{F}$, $\mathcal{F}^{\prime}$, $\mathcal{F}^{\prime\prime}$ belong to
$\mathfrak{F}$. The following results are immediate:

  * •

$|\lambda(\tilde{z},\tilde{z}^{\prime},\mathcal{F},\mathcal{F}^{\prime})|\leq|\hat{\tau}(\tilde{z},\tilde{z}^{\prime},\mathcal{F},\mathcal{F}^{\prime})|$,

  * •

$\hat{\tau}(\tilde{z}^{\prime},\tilde{z},\mathcal{F},\mathcal{F}^{\prime})=\hat{\tau}(\tilde{z},\tilde{z}^{\prime},\mathcal{F},\mathcal{F}^{\prime})$,

  * •

$\hat{\tau}(\tilde{z},\tilde{z}^{\prime},\mathcal{F},\mathcal{F}^{\prime})+\hat{\tau}(\tilde{z},\tilde{z}^{\prime},\mathcal{F}^{\prime},\mathcal{F}^{\prime\prime})=\hat{\tau}(\tilde{z},\tilde{z}^{\prime},\mathcal{F},\mathcal{F}^{\prime\prime})$,

  * •

$\lambda(\tilde{z},\tilde{z}^{\prime},\mathcal{F},\mathcal{F}^{\prime})+\lambda(\tilde{z},\tilde{z}^{\prime},\mathcal{F}^{\prime},\mathcal{F}^{\prime\prime})=\lambda(\tilde{z},\tilde{z}^{\prime},\mathcal{F},\mathcal{F}^{\prime\prime})$,

  * •

$\hat{\tau}(\tilde{z},\tilde{z}^{\prime},\mathcal{F}^{\prime},\mathcal{F})=-\hat{\tau}(\tilde{z},\tilde{z}^{\prime},\mathcal{F},\mathcal{F}^{\prime})$,

  * •

$\lambda(\tilde{z},\tilde{z}^{\prime},\mathcal{F}^{\prime},\mathcal{F})=-\lambda(\tilde{z},\tilde{z}^{\prime},\mathcal{F},\mathcal{F}^{\prime})$,

  * •

$\hat{\tau}(\tilde{f}(\tilde{z}),\tilde{f}(\tilde{z}^{\prime}),f(\mathcal{F}),f(\mathcal{F}^{\prime}))=\hat{\tau}(\tilde{z},\tilde{z}^{\prime},\mathcal{F},\mathcal{F}^{\prime})$,

  * •

$\lambda(\tilde{f}(\tilde{z}),\tilde{f}(\tilde{z}^{\prime}),f(\mathcal{F}),f(\mathcal{F}^{\prime}))=\lambda(\tilde{z},\tilde{z}^{\prime},\mathcal{F},\mathcal{F}^{\prime})$.

The second assertion indicates that
$(\tilde{z},\tilde{z}^{\prime})\mapsto\hat{\tau}(\tilde{z}^{\prime},\tilde{z},\mathcal{F},\mathcal{F}^{\prime})$
is symmetric. The next result precises the lack of symmetry of
$(\tilde{z},\tilde{z}^{\prime})\mapsto\lambda(\tilde{z}^{\prime},\tilde{z},\mathcal{F},\mathcal{F}^{\prime})$.
For every $(\tilde{z},\tilde{z}^{\prime})\in\tilde{W}$, define a function
$\delta_{\tilde{z},\tilde{z}^{\prime}}:{\mathfrak{F}}\to\\{-1/2,0,1/2\\}$ as
follows:

$\delta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})=\begin{cases}0&\text{ if
$\tilde{\phi}_{\tilde{z}^{\prime}}=\tilde{\phi}_{\tilde{z}}$,}\\\ 1/2&\text{
if $\tilde{\phi}_{\tilde{z}^{\prime}}\succ_{\mathcal{F}}\phi_{\tilde{z}}$,}\\\
-1/2&\text{ if
$\tilde{\phi}_{\tilde{z}^{\prime}}\prec_{\mathcal{F}}\phi_{\tilde{z}}$.}\end{cases}$

The next result express that
$\lambda(\tilde{z},\tilde{z}^{\prime},\mathcal{F},\mathcal{F}^{\prime})-\lambda(\tilde{z}^{\prime},\tilde{z},\mathcal{F},\mathcal{F}^{\prime})$
is the “algebraic intersection number” of a “well-oriented” continuous path
$\gamma:[s_{0},s_{1}]\to\mathbb{Z}/4\mathbb{Z}$ joining
$\hat{\theta}_{\tilde{z}^{\prime},\tilde{z}}(\mathcal{F})$ to
$\hat{\theta}_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}^{\prime})$ with
$\\{\dot{0}\\}-\\{\dot{2}\\}$.

######  Lemma 2.1.

For every $(\tilde{z},\tilde{z}^{\prime})\in\tilde{W}$ and every
$\mathcal{F}$, $\mathcal{F}^{\prime}$ in $\mathfrak{F}$, it holds that

$\lambda(\tilde{z},\tilde{z}^{\prime},\mathcal{F},\mathcal{F}^{\prime})-\lambda(\tilde{z}^{\prime},\tilde{z},\mathcal{F},\mathcal{F}^{\prime})=\delta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}^{\prime})-\delta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}).$

###### Proof.

Recall that

$\lambda(\tilde{z},\tilde{z}^{\prime},\mathcal{F},\mathcal{F}^{\prime})=\lambda(\hat{\theta}_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}),\hat{\theta}_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}^{\prime}))$

and that one can suppose that

$\hat{\theta}_{\tilde{z}^{\prime},\tilde{z}}(\mathcal{F})=\hat{\theta}_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})+2,\enskip\hat{\theta}_{\tilde{z}^{\prime},\tilde{z}}(\mathcal{F}^{\prime})=\hat{\theta}_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}^{\prime})+2.$

Now observe that

$\lambda(\tilde{z},\tilde{z}^{\prime},\mathcal{F},\mathcal{F}^{\prime})-\lambda(\tilde{z}^{\prime},\tilde{z},\mathcal{F},\mathcal{F}^{\prime})=\begin{cases}0&\text{
if
$\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})=\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}^{\prime})$,}\\\
0&\text{ if $\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})=\dot{0}$ and
$\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}^{\prime})=\dot{2}$,}\\\
0&\text{ if $\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})=\dot{2}$ and
$\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}^{\prime})=\dot{0}$,}\\\
-1&\text{ if $\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})=\dot{1}$ and
$\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}^{\prime})=\dot{3}$,}\\\
1&\text{ if $\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})=\dot{3}$ and
$\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}^{\prime})=\dot{1}$,}\\\
-1/2&\text{ if $\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})=\dot{1}$
and $\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}^{\prime})=\dot{0}$,}\\\
1/2&\text{ if $\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})=\dot{0}$ and
$\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}^{\prime})=\dot{1}$,}\\\
-1/2&\text{ if $\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})=\dot{0}$
and $\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}^{\prime})=\dot{3}$,}\\\
1/2&\text{ if $\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})=\dot{3}$ and
$\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}^{\prime})=\dot{0}$,}\\\
-1/2&\text{ if $\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})=\dot{1}$
and $\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}^{\prime})=\dot{2}$,}\\\
1/2&\text{ if $\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})=\dot{2}$ and
$\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}^{\prime})=\dot{1}$,}\\\
-1/2&\text{ if $\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})=\dot{2}$
and $\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}^{\prime})=\dot{3}$,}\\\
1/2&\text{ if $\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})=\dot{3}$ and
$\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}^{\prime})=\dot{2}$.}\end{cases}$

∎

The next result will be useful later:

######  Lemma 2.2.

For every compact set $\tilde{K}\subset\tilde{W}$ and every
$\mathcal{F},\mathcal{F}^{\prime}$ in $\mathfrak{F}$, there exists $M>0$ such
that for every $(\tilde{z},\tilde{z}^{\prime})\in\tilde{K}$ it holds that

$|\hat{\tau}(\tilde{z},\tilde{z}^{\prime},\mathcal{F},\mathcal{F}^{\prime})|\leq
M.$

###### Proof.

For every $(\tilde{z},\tilde{z}^{\prime})\in\tilde{W}$, there exists a
neighborhood
$\tilde{d}_{\tilde{z},\tilde{z}^{\prime}}\times\tilde{d}^{\prime}_{\tilde{z},\tilde{z}^{\prime}}\subset\tilde{W}$
of $(\tilde{z},\tilde{z}^{\prime})$ where
$\tilde{d}_{\tilde{z},\tilde{z}^{\prime}}$ and
$\tilde{d}^{\prime}_{\tilde{z},\tilde{z}^{\prime}}$ are topological closed
disks. One can cover $\tilde{K}$ by a finite family of such neighborhoods. So
it is sufficient to prove the result when
$\tilde{K}=\tilde{d}\times\tilde{d}^{\prime}$ is the product of two
topological closed disks. The map

$\displaystyle\tilde{d}\times\tilde{d}^{\prime}\times{\mathfrak{F}}$
$\displaystyle\to\mathbb{Z}/4\mathbb{Z}$
$\displaystyle(\tilde{z},\tilde{z}^{\prime},\mathcal{F})$
$\displaystyle\to\theta_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})$

being continuous and the space
$\tilde{d}\times\tilde{d}^{\prime}\times{\mathfrak{F}}$ being simply
connected, one can lift this last map to a continuous map

$\displaystyle\tilde{d}\times\tilde{d}^{\prime}\times{\mathfrak{F}}$
$\displaystyle\to\mathbb{Z}$
$\displaystyle(\tilde{z},\tilde{z}^{\prime},\mathcal{F})$
$\displaystyle\to\hat{\theta}_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}).$

For every $\mathcal{F},\mathcal{F}^{\prime}$ in $\mathfrak{F}$, the function
$(\tilde{z},\tilde{z}^{\prime})\mapsto\hat{\theta}_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F}^{\prime})-\hat{\theta}_{\tilde{z},\tilde{z}^{\prime}}(\mathcal{F})=\tau(\tilde{z},\tilde{z}^{\prime},\mathcal{F},\mathcal{F}^{\prime})$
is continuous and so is bounded on the compact set
$\tilde{d}\times\tilde{d}^{\prime}$. This last affirmation comes from the fact
that for every $l\geq 1$, the set
$\left\\{(\tilde{z},\tilde{z}^{\prime})\in\tilde{d}\times\tilde{d}^{\prime}\,|\enskip|\tau(\tilde{z},\tilde{z}^{\prime},\mathcal{F},\mathcal{F}^{\prime})|<2l\right\\}$
is an open subset of $\tilde{d}\times\tilde{d}^{\prime}$ as the preimage of an
open set by a continuous map. ∎

Let us conclude with the following:

######  Lemma 2.3.

The function
$d:{\mathfrak{F}}\times{\mathfrak{F}}\to\mathbb{N}\cup\\{+\infty\\}$ is an
extended distance, where

$d(\mathcal{F},\mathcal{F}^{\prime})=\sup_{(\tilde{z},\tilde{z}^{\prime})\in\tilde{W}}|\hat{\tau}(\tilde{z},\tilde{z}^{\prime},\mathcal{F},{\mathcal{F}}^{\prime})|.$

Moreover the equality
$d(f(\mathcal{F}),f(\mathcal{F}^{\prime}))=d(\mathcal{F},\mathcal{F}^{\prime})$
holds for every $f\in\mathrm{Homeo}_{*}({D^{*}})$.

###### Proof.

One proves immediately that $d$ is symmetric and satisfies the triangular
inequality. The fact that $d(\mathcal{F},\mathcal{F})=0$ for every
$\mathcal{F}\in\mathfrak{F}$ is also obvious. It remains to prove that
$d(\mathcal{F},\mathcal{F}^{\prime})\not=0$ if $\mathcal{F}$ and
$\mathcal{F}^{\prime}$ are distinct radial foliations. In that case, one can
find $(\tilde{z},\tilde{z}^{\prime})\in\tilde{W}$ such that $\tilde{z}$ and
$\tilde{z}^{\prime}$ belong to the same leaf of $\widetilde{\mathcal{F}}$ and
to different leaves of $\widetilde{\mathcal{F}^{\prime}}$. Consequently,
$\theta(\tilde{z},\tilde{z}^{\prime},\mathcal{F})\in\\{\dot{0},\dot{2}\\}$ and
$\theta(\tilde{z},\tilde{z}^{\prime},\mathcal{F}^{\prime})\in\\{\dot{1},\dot{3}\\}$.
It implies that
$\hat{\theta}(\tilde{z},\tilde{z}^{\prime},\mathcal{F})\not=\hat{\theta}(\tilde{z},\tilde{z}^{\prime},\mathcal{F}^{\prime})$
if $\hat{\theta}$ is a lift of $\theta$.

The equality
$d(\tilde{f}(\mathcal{F}),\tilde{f}(\mathcal{F}^{\prime}))=d(\mathcal{F},\mathcal{F}^{\prime})$
if $f\in\mathrm{Homeo}_{*}({D^{*}})$ is a direct consequence of the equality

$\hat{\tau}(\tilde{f}(\tilde{z}),\tilde{f}(\tilde{z}^{\prime}),f(\mathcal{F}),f(\mathcal{F}^{\prime}))=\hat{\tau}(\tilde{z},\tilde{z}^{\prime},\mathcal{F},\mathcal{F}^{\prime}).$

∎

We will call $d$ the winding distance between two radial foliations.

###  2.3.  Topological angles in the annulus

Let $\mathcal{F}$ be a radial foliation and $f$ a homeomorphism of $D^{*}$
isotopic to the identity. Let $I=(f_{s})_{s\in[0,1]}$ be an identity isotopy
of $f$ in $\mathrm{Homeo}_{*}(D^{*})$ and
$\tilde{I}=(\tilde{f}_{s})_{s\in[0,1]}$ the lifted identity isotopy to
$\tilde{D}$. The function

$s\in[0,1]\mapsto\theta_{\tilde{f}_{s}(\tilde{z}),\tilde{f}_{s}(\tilde{z}^{\prime})}(\mathcal{F})=\theta_{\tilde{z},\tilde{z}^{\prime}}(f_{s}^{-1}(\mathcal{F}))\in\mathbb{Z}/4\mathbb{Z}$

can be lifted to a function

$s\in[0,1]\mapsto\hat{\theta}_{\tilde{z},\tilde{z}^{\prime}}(f_{s}^{-1}(\mathcal{F}))\in\mathbb{Z}$

and the difference between the value at $1$ and the value at $0$ of this last
map is nothing but
$\hat{\tau}(\tilde{z},\tilde{z},{\mathcal{F}},f^{-1}(\mathcal{F}))$. Of course
it depends neither on the choice of $I$, nor on the choice of the lift
$\tilde{I}$ of $I$. Suppose now that $\tilde{z}$ and $\tilde{z}^{\prime}$
project onto two different points of $D^{*}$. Note that if $k$ is large
enough, then for every $s\in[0,1]$, it holds that

$\theta_{\tilde{f}_{s}(\tilde{z}),\tilde{f}_{s}(T^{k}(\tilde{z}^{\prime}))}(\mathcal{F})=\dot{1},\enskip\theta_{\tilde{f}_{s}(\tilde{z}),\tilde{f}_{s}(T^{-k}(\tilde{z}^{\prime}))}(\mathcal{F})=\dot{3},$

and so the functions

$s\mapsto\hat{\theta}_{\tilde{z},T^{k}(\tilde{z}^{\prime})}(f_{s}^{-1}(\mathcal{F})),\enskip
s\mapsto\hat{\theta}_{\tilde{z},T^{-k}(\tilde{z}^{\prime})}(f_{s}^{-1}(\mathcal{F}))$

are constant, which means that

$\hat{\tau}(\tilde{z},T^{k}(\tilde{z}^{\prime}),\mathcal{F},f^{-1}({\mathcal{F}}))=\hat{\tau}(\tilde{z},T^{-k}(\tilde{z}^{\prime}),\mathcal{F},f^{-1}({\mathcal{F}}))=0.$

Consequently, if $\mathcal{F}$ and $\mathcal{F}^{\prime}$ are two radial
foliations, then for every $(z,z^{\prime})\in W$, one can define

$\displaystyle\overline{\hat{\tau}}(z,z^{\prime},\mathcal{F},{\mathcal{F}}^{\prime})$
$\displaystyle=\sum_{k\in\mathbb{Z}}|\hat{\tau}(\tilde{z},T^{k}(\tilde{z}^{\prime}),\mathcal{F},{\mathcal{F}}^{\prime})|,$
$\displaystyle\hat{\tau}(z,z^{\prime},\mathcal{F},{\mathcal{F}}^{\prime})$
$\displaystyle=\sum_{k\in\mathbb{Z}}\hat{\tau}(\tilde{z},T^{k}(\tilde{z}^{\prime}),\mathcal{F},{\mathcal{F}}^{\prime}),$
$\displaystyle\lambda(z,z^{\prime},\mathcal{F},{\mathcal{F}}^{\prime})$
$\displaystyle=\sum_{k\in\mathbb{Z}}\lambda(\tilde{z},T^{k}(\tilde{z}^{\prime}),\mathcal{F},{\mathcal{F}}^{\prime}),$

each sum being independent of the choice of the lifts
$\tilde{z},\tilde{z}^{\prime}$ of $z,z^{\prime}$.

Suppose that $(z,z^{\prime})\in W$, that $f\in\mathrm{Homeo}_{*}({D^{*}})$ and
that $\mathcal{F}$, $\mathcal{F}^{\prime}$, $\mathcal{F}^{\prime\prime}$
belong to $\mathfrak{F}$. The following results are immediate

  * •

$|\hat{\tau}(z,z^{\prime},\mathcal{F}^{\prime},\mathcal{F})|\leq\overline{\hat{\tau}}(z,z^{\prime},\mathcal{F},\mathcal{F}^{\prime})$,

  * •

$|\lambda(z,z^{\prime},\mathcal{F}^{\prime},\mathcal{F})|\leq\overline{\hat{\tau}}(z,z^{\prime},\mathcal{F},\mathcal{F}^{\prime})$,

  * •

$\overline{\hat{\tau}}(z^{\prime},z,\mathcal{F},\mathcal{F}^{\prime})=\overline{\hat{\tau}}(z,z^{\prime},\mathcal{F},\mathcal{F}^{\prime})$,

  * •

$\hat{\tau}(z^{\prime},z,\mathcal{F},\mathcal{F}^{\prime})=\hat{\tau}(z,z^{\prime},\mathcal{F},\mathcal{F}^{\prime})$,

  * •

$\overline{\hat{\tau}}(z,z^{\prime},\mathcal{F}^{\prime},\mathcal{F})=\overline{\hat{\tau}}(z,z^{\prime},\mathcal{F},\mathcal{F}^{\prime})$,

  * •

$\hat{\tau}(z,z^{\prime},\mathcal{F},\mathcal{F}^{\prime})+\hat{\tau}(z,z^{\prime},\mathcal{F}^{\prime},\mathcal{F}^{\prime\prime})=\hat{\tau}(z,z^{\prime},\mathcal{F},\mathcal{F}^{\prime\prime})$,

  * •

$\lambda(z,z^{\prime},\mathcal{F},\mathcal{F}^{\prime})+\lambda(z,z^{\prime},\mathcal{F}^{\prime},\mathcal{F}^{\prime\prime})=\lambda(z,z^{\prime},\mathcal{F},\mathcal{F}^{\prime\prime})$,

  * •

$\hat{\tau}(z,z^{\prime},\mathcal{F}^{\prime},\mathcal{F})=-\hat{\tau}(z,z^{\prime},\mathcal{F},\mathcal{F}^{\prime})$,

  * •

$\lambda(z,z^{\prime},\mathcal{F}^{\prime},\mathcal{F})=-\lambda(z,z^{\prime},\mathcal{F},\mathcal{F}^{\prime})$,

  * •

$\hat{\tau}(f(z),f(z^{\prime}),f(\mathcal{F}),f(\mathcal{F}^{\prime}))=\hat{\tau}(z,z^{\prime},\mathcal{F},\mathcal{F}^{\prime})$,

  * •

$\lambda(f(z),f(z^{\prime}),f(\mathcal{F}),f(\mathcal{F}^{\prime}))=\lambda(z,z^{\prime},\mathcal{F},\mathcal{F}^{\prime})$.

Note that we have the following:

######  Lemma 2.4.

For every compact subset $K\subset W$, for every
$\mathcal{F},\mathcal{F}^{\prime}$ in $\mathfrak{F}$, there exists $M>0$ such
that for every $(z,z^{\prime})\in K$ it holds that

$\overline{\hat{\tau}}(z,z^{\prime},\mathcal{F},\mathcal{F}^{\prime})\leq M.$

###### Proof.

Like in the proof of Lemma 2.2, it is sufficient to study the case where
$K=d\times d^{\prime}$ is the product of two topological closed disks. Choose
a lift $\tilde{d}$ of $d$ and a lift $\tilde{d}^{\prime}$ of $d^{\prime}$ in
$\tilde{D}$. Choose $f\in\mathrm{Homeo}_{*}(D^{*})$ such that
$\mathcal{F}^{\prime}=f^{-1}(\mathcal{F})$ and an identity isotopy
$(f_{s})_{s\in[0,1]}$ of $f$. Lift this isotopy to an isotopy identity
$(\tilde{f}_{s})_{s\in[0,1]}$ on $\tilde{D}$. There exists $k_{0}>0$ such that
if $k\geq k_{0}$, then for every $s\in[0,1]$, every $\tilde{z}\in\tilde{d}$
and every $\tilde{z}^{\prime}\in\tilde{d}^{\prime}$ it holds that

$\theta_{\tilde{f}_{s}(\tilde{z}),\tilde{f}_{s}(T^{k}(\tilde{z}^{\prime}))}(\mathcal{F})=\dot{1},\enskip\theta_{\tilde{f}_{s}(\tilde{z}),\tilde{f}_{s}(T^{-k}(\tilde{z}^{\prime}))}(\mathcal{F})=\dot{3},$

and it implies that

$\hat{\tau}(\tilde{z},T^{k}(\tilde{z}^{\prime}),\mathcal{F},{\mathcal{F}}^{\prime})=\hat{\tau}(\tilde{z},T^{-k}(\tilde{z}^{\prime}),\mathcal{F},{\mathcal{F}}^{\prime})=0.$

To conclude, it remains to apply Lemma 2.2 to the compact sets
$\tilde{d}\times T^{k}(\tilde{d}^{\prime})$, $|k|<k_{0}$. ∎

Let us conclude this section by stating and analogous of Lemma 2.1. Define
first what is a displacement function. For every
$f\in\mathrm{Homeo}_{*}(D^{*})$, every lift $\tilde{f}$ of $f$ to $\tilde{D}$
and every ray $\phi$ we can define a function
$m_{\tilde{f},\phi}:D^{*}\to\mathbb{Z}$ as follows: we choose a lift
$\tilde{\phi}$ of $\phi$, then for every $z\in\mathbb{D}$, we consider the
lift $\tilde{z}$ of $z$ such that $\tilde{z}\in\overline{L(\tilde{\phi})}\cap
R(T(\tilde{\phi}))$ and we denote $m_{\tilde{f},\phi}(z)$ the integer $m$ such
that $\tilde{f}(\tilde{z})\in\overline{L(T^{m}(\tilde{\phi}))}\cap
R(T^{m+1}(\tilde{\phi}))$, noting that it does not depend on the choice of
$\tilde{\phi}$. Observe that
$m_{T^{k}\circ\tilde{f},\phi}=m_{\tilde{f},\phi}+k$, for every
$k\in\mathbb{Z}$. One proves easily the following result:

######  Lemma 2.5.

For every compact subset $K\subset D^{*}$ and every rays $\phi,\phi^{\prime}$,
there exists $M>0$ such that for every $f\in\mathrm{Homeo}_{*}(D^{*})$ and
every lift $\tilde{f}$ of $f$ to $\tilde{D}$, it holds that

$z\in K\enskip\mathrm{and}\enskip f(z)\in
K\Rightarrow|m_{\tilde{f},\phi}(z)-m_{\tilde{f},\phi^{\prime}}(z)|\leq M.$

The following result precises the lack of symmetry of
$(z,z^{\prime})\mapsto\lambda(z,z^{\prime},\mathcal{F},{\mathcal{F}}^{\prime})$.

######  Lemma 2.6.

Fix $\mathcal{F}$, $\mathcal{F}^{\prime}$ in $\mathfrak{F}$. Choose a leaf
$\phi$ of $\mathcal{F}$, a lift $\tilde{\phi}$ of $\phi$ to $\tilde{D}$ and a
map $f\in\mathrm{Homeo}_{*}(D^{*})$ such that
$\mathcal{F}^{\prime}=f^{-1}({\mathcal{F}})$. For every $z\in D^{*}$, denote
$\tilde{z}^{*}$ the unique lift of $z$ such that
$\tilde{z}\in\overline{L(\tilde{\phi})}\cap R(T(\phi))$. Then, for every
$(z,z^{\prime})\in W$, we have

$\lambda(z,z^{\prime},\mathcal{F},\mathcal{F}^{\prime})-\lambda(z^{\prime},z,\mathcal{F},\mathcal{F}^{\prime}))=m_{\tilde{f},\phi}(z^{\prime})-m_{\tilde{f},\phi}(z)+\delta_{\widetilde{f(z)}^{*},\widetilde{f(z^{\prime})}^{*}}(\mathcal{F})-\delta_{\tilde{z}^{*},\tilde{z}^{*}}(\mathcal{F}).$

###### Proof.

Fix $(z,z^{\prime})\in W$. During the proof, we will lighten the notations by
writing

$\tilde{z}=\tilde{z}^{*},\enskip\tilde{z}^{\prime}=\tilde{z}^{\prime}{}^{*},\enskip
m=m_{\tilde{f},\phi}(z),\enskip m^{\prime}=m_{\tilde{f},\phi}(z^{\prime}).$

By using Lemma 2.1 we get

$\displaystyle\enskip\enskip\enskip\enskip\enskip\enskip\lambda(z,z^{\prime},\mathcal{F},f^{-1}({\mathcal{F}}))-\lambda(z^{\prime},z,\mathcal{F},f^{-1}({\mathcal{F}}))$
$\displaystyle=\sum_{k\in\mathbb{Z}}\lambda(\tilde{z},T^{k}(\tilde{z}^{\prime}),\mathcal{F},f^{-1}({\mathcal{F}}))-\sum_{k\in\mathbb{Z}}\lambda(\tilde{z}^{\prime},T^{k}(\tilde{z}),\mathcal{F},f^{-1}({\mathcal{F}}))$
$\displaystyle=\sum_{k\in\mathbb{Z}}\lambda(\tilde{z},T^{k}(\tilde{z}^{\prime}),\mathcal{F},f^{-1}({\mathcal{F}}))-\sum_{k\in\mathbb{Z}}\lambda(T^{k}(\tilde{z}^{\prime}),\tilde{z},\mathcal{F},f^{-1}({\mathcal{F}}))$
$\displaystyle=\sum_{k\in\mathbb{Z}}\lambda(\tilde{z},T^{k}(\tilde{z}^{\prime}),\mathcal{F},f^{-1}({\mathcal{F}}))-\lambda(T^{k}(\tilde{z}^{\prime}),\tilde{z},\mathcal{F},f^{-1}({\mathcal{F}}))$
$\displaystyle=\sum_{k\in\mathbb{Z}}\delta_{\tilde{z},T^{k}(\tilde{z}^{\prime})}(f^{-1}(\mathcal{F}))-\delta_{\tilde{z},T^{k}(\tilde{z}^{\prime})}(\mathcal{F})$
$\displaystyle=\sum_{k\in\mathbb{Z}}\delta_{\tilde{f}(\tilde{z}),T^{k}(\tilde{f}(\tilde{z}^{\prime}))}(\mathcal{F})-\delta_{\tilde{z},T^{k}(\tilde{z}^{\prime})}(\mathcal{F})$
$\displaystyle=\sum_{k\in\mathbb{Z}}\delta_{\tilde{f}(T^{-m}(\tilde{z})),T^{k}(\tilde{f}(T^{-m}(\tilde{z}^{\prime})))}(\mathcal{F})-\delta_{\tilde{z},T^{k}(\tilde{z}^{\prime})}(\mathcal{F})$
$\displaystyle=\sum_{k\in\mathbb{Z}}\delta_{\tilde{f}(T^{-m}(\tilde{z})),T^{k+m^{\prime}-m}(\tilde{f}(T^{-m^{\prime}}(\tilde{z}^{\prime})))}(\mathcal{F})-\delta_{\tilde{z},T^{k}(\tilde{z}^{\prime})}(\mathcal{F}).$

Recalling that $\tilde{z}=\tilde{z}^{*}$ and
$\tilde{z}^{\prime}=\tilde{z}^{\prime}{}^{*}$, we have

$\delta_{\tilde{z},T^{k}(\tilde{z}^{\prime})}(\mathcal{F})=\begin{cases}-1/2&\text{
if $k<0$}\\\ 1/2&\text{ if $k>0$}\end{cases}$

and

$\delta_{\tilde{f}(T^{-m}(\tilde{z})),T^{k+m^{\prime}-m}(\tilde{f}(T^{-m^{\prime}}(\tilde{z}^{\prime})))}(\mathcal{F})=\begin{cases}-1/2&\text{
if $k+m^{\prime}-m<0$}\\\ 1/2&\text{ if $k+m^{\prime}-m>0$}.\end{cases}$

So we deduce that

$\delta_{\tilde{f}(T^{-m}(\tilde{z})),T^{k+m^{\prime}-m}(\tilde{f}(T^{-m^{\prime}}(\tilde{z}^{\prime})))}(\mathcal{F})-\delta_{\tilde{z},T^{k}(\tilde{z}^{\prime})}(\mathcal{F})=\begin{cases}0&\text{
if $k<\min(0,m-m^{\prime})$}\\\ 0&\text{ if $k>\max(0,m-m^{\prime})$}\\\
1&\text{ if $m-m^{\prime}<k<0$}\\\ -1&\text{ if $0<k<m-m^{\prime}$}\\\
-\delta_{\tilde{z}^{*},\tilde{z}^{\prime*}}(\mathcal{F})-1/2&\text{ if
$k=0<m-m^{\prime}$}\\\
-\delta_{\tilde{z}^{*},\tilde{z}^{\prime*}}(\mathcal{F})+1/2&\text{ if
$k=0>m-m^{\prime}$}\\\
\delta_{\widetilde{f(\tilde{z})}^{*},\widetilde{f(\tilde{z}^{\prime})}^{*}}(\mathcal{F})+1/2&\text{
if $k=m-m^{\prime}<0$}\\\
\delta_{\widetilde{f(\tilde{z})}^{*},\widetilde{f(\tilde{z}^{\prime})}^{*}}(\mathcal{F})-1/2&\text{
if $k=m-m^{\prime}>0$}\\\
\delta_{\widetilde{f(\tilde{z})}^{*},\widetilde{f(\tilde{z}^{\prime})}^{*}}(\mathcal{F})-\check{\theta}_{\tilde{z}^{*},\tilde{z}^{\prime*}}(\mathcal{F})&\text{
if $k=0=m-m^{\prime}$}\end{cases}$

and consequently, that

$\sum_{k\in\mathbb{Z}}\delta_{\tilde{f}(T^{-m}(\tilde{z})),T^{k+m^{\prime}-m}(\tilde{f}(T^{-m^{\prime}}(\tilde{z}^{\prime})))}(\mathcal{F})-\delta_{\tilde{z},T^{k}(\tilde{z}^{\prime})}(\mathcal{F})=m^{\prime}-m+\delta_{\widetilde{f(z)}^{*},\widetilde{f(z^{\prime})}^{*}}(\mathcal{F})-\delta_{\tilde{z}^{*},\tilde{z}^{\prime}{}^{*}}(\mathcal{F}).$

∎

######  Remarks.

1\. The fonction
$\lambda_{f,\mathcal{F}}(z,z^{\prime})\mapsto\lambda(z,z^{\prime},\mathcal{F},f^{-1}({\mathcal{F}}))$
depends only on the foliations $\mathcal{F}$ and $f^{-1}(\mathcal{F})$ (and
not explicitely in $f$) and the fonction $z\mapsto m_{\tilde{f},\phi}(z)$
depends only on the ray $\phi$ and on $\tilde{f}$.

2\. The equality proven in Lemma 2.6 can be written

$\Lambda_{\tilde{f},\mathcal{F},\phi}(z,z^{\prime})-\Lambda_{\tilde{f},\mathcal{F},\phi}(z^{\prime},z)=\delta_{\widetilde{f(z)}^{*},\widetilde{f(z^{\prime})}^{*}}(\mathcal{F})-\delta_{\tilde{z}^{*},\tilde{z}^{*}}(\mathcal{F}),$

where

$\Lambda_{\tilde{f},\mathcal{F},\phi}(z,z^{\prime})=\lambda_{f,\mathcal{F}}(z,z^{\prime})+m_{\tilde{f},\phi}(z).$

3\. For every $k\in\mathbb{Z}$ it holds that
$\Lambda_{T^{k}\circ\tilde{f},\mathcal{F},\phi}=\Lambda_{\tilde{f},\mathcal{F},\phi}+k.$

###  2.4. Rotation number and linking number.

We will see how to define rotation numbers and self linking numbers within
this formalism.

######  Definition 2.7.

Fix $f\in\mathrm{Homeo}_{*}(D^{*})$ and a lift $\tilde{f}$ to $\tilde{D}$. Say
that $z\in D^{*}$ has a rotation number
$\mathrm{rot}_{\tilde{f}}(z)\in\mathbb{R}$ if:

  1. (1)

there exists a compact set $K\subset D^{*}$ such that $\\#\\{n\geq
0\,|\,f^{n}(z)\in K\\}=+\infty$;

  2. (2)

if $\phi$ is a ray and if $K\subset D^{*}$ is a compact set containing $z$,
then for every $\varepsilon>0$, there exists $n_{0}\geq 0$ such that for every
$n\geq n_{0}$ it holds that

$f^{n}(z)\in K\Rightarrow\left|{1\over
n}\sum_{i=0}^{n-1}m_{\tilde{f},\phi}(f^{i}(z))-\mathrm{rot}_{\tilde{f}}(z)\right|\leq\varepsilon.$

######  Remark.

Note that
$\sum_{i=0}^{n-1}m_{\tilde{f},\phi}(f^{i}(z))=m_{\tilde{f}^{n},\phi}(z)$.
Using Lemma 2.5, one deduces that if the second assertion is true for a ray
$\phi$, it is true for every ray $\phi^{\prime}$.

######  Definition 2.8.

Fix $f\in\mathrm{Homeo}^{*}(D^{*})$ and a lift $\tilde{f}$ to $\tilde{D}$. Say
that $(z,z^{\prime})\in W$ has a linking number
$\mathrm{link}_{\tilde{f}}(z,z^{\prime})\in\mathbb{R}$ if

  1. (1)

the points $z$ and $z^{\prime}$ have a rotation number;

  2. (2)

there exists a compact set $K\subset W$ such that $\\#\\{n\geq
0\,|\,(f^{n}(z),f^{n}(z^{\prime}))\in K\\}=+\infty$;

  3. (3)

if $\mathcal{F}$ is a radial foliation, if $\phi$ is leaf of $\mathcal{F}$ and
if $K\subset W$ is a compact set containing $(z,z^{\prime})$, then for every
$\varepsilon>0$, there exists $n_{0}\geq 0$ such that for every $n\geq n_{0}$
it holds that

$(f^{n}(z),f^{n}(z^{\prime}))\in K\Rightarrow\left|{1\over
n}\sum_{i=0}^{n-1}{\Lambda}_{\tilde{f},\mathcal{F},\phi}(f^{i}(z),f^{i}(z^{\prime}))-\mathrm{link}_{\tilde{f}}(z,z^{\prime})\right|\leq\varepsilon.$

######  Remarks.

1\. Note that

$\sum_{i=0}^{n-1}{\Lambda}_{\tilde{f},\mathcal{F},\phi}(f^{i}(z),f^{i}(z^{\prime}))={\Lambda}_{\tilde{f}^{n},\mathcal{F},\phi}(z,z^{\prime}).$

2\. As explained in the remark following the definition of the rotation
number, we know that if the third assertion is true for a leaf $\phi$, it is
true for every other leaf. Let us explain now, why if the third assertion is
true for a foliation $\mathcal{F}\in\mathfrak{F}$, it is true for every other
foliation $\mathcal{F}^{\prime}\in\mathfrak{F}$.444This assertion is moreless
what is used by Gambaudo and Ghys to prove in [GaGh] that two symplectic
diffeomorphisms of $\mathbb{D}$ that are $C^{0}$ conjugate have the same
Calabi invariant. We have

$\displaystyle\enskip\left|\lambda(z,z^{\prime},{\mathcal{F}},f^{-n}({\mathcal{F}}))-\lambda(z,z^{\prime},{\mathcal{F}}^{\prime},f^{-n}({\mathcal{F}^{\prime}}))\right|$
$\displaystyle=$
$\displaystyle\enskip\left|\lambda(z,z^{\prime},{\mathcal{F}},{\mathcal{F}}^{\prime})+\lambda(z,z^{\prime},{\mathcal{F}}^{\prime},f^{-n}({\mathcal{F}}))-\lambda(z,z^{\prime},{\mathcal{F}}^{\prime},f^{-n}({\mathcal{F}^{\prime}}))\right|$
$\displaystyle=$
$\displaystyle\enskip\left|\lambda(z,z^{\prime},{\mathcal{F}},{\mathcal{F}}^{\prime})-\lambda(z,z^{\prime},f^{-n}({\mathcal{F}}),f^{-n}({\mathcal{F}^{\prime}}))\right|$
$\displaystyle=$
$\displaystyle\enskip\left|\lambda(z,z^{\prime},{\mathcal{F}},{\mathcal{F}}^{\prime})-\lambda(f^{n}(z),f^{n}(z^{\prime}),{\mathcal{F}},{\mathcal{F}}^{\prime})\right|$
$\displaystyle\leq$
$\displaystyle\enskip\overline{\hat{\tau}}(z,z^{\prime},{\mathcal{F}},{\mathcal{F}}^{\prime})+\overline{\hat{\tau}}(f^{n}(z),f^{n}(z^{\prime}),{\mathcal{F}},{\mathcal{F}}^{\prime}).$

It remains to apply Lemma 2.4 : for every compact set $K\subset D^{*}$, there
exists $M>0$ such that for every point $(z,z^{\prime})$ satisfying
$(z,z^{\prime})\in K$ and $(f^{n}(z),f^{n}(z^{\prime}))\in K$, we have
$\overline{\hat{\tau}}(z,z^{\prime},{\mathcal{F}},{\mathcal{F}}^{\prime})\leq
M$ and
$\overline{\hat{\tau}}(f^{n}(z),f^{n}(z^{\prime}),{\mathcal{F}},{\mathcal{F}}^{\prime})\leq
M$.

3\. By the remark following Lemma 2.6, if
$\mathrm{link}_{\tilde{f}}(z,z^{\prime})$ exists, then
$\mathrm{link}_{\tilde{f}}(z^{\prime},z)$ exists and
$\mathrm{link}_{\tilde{f}}(z^{\prime},z)=\mathrm{link}_{\tilde{f}}(z,z^{\prime})$.
Indeed, we have

$\left|{\Lambda}_{\tilde{f}^{n},\mathcal{F},\phi}(z,z^{\prime})-{\Lambda}_{\tilde{f}^{n},\mathcal{F},\phi}(z^{\prime},z)\right|=\left|\delta_{\widetilde{f^{n}(z)}^{*},\widetilde{f(^{n}z^{\prime})}^{*}}(\mathcal{F})-\delta_{\tilde{z}^{*},\tilde{z}^{*}}(\mathcal{F})\right|\leq
1.$

######  Definition 2.9.

Fix $f\in\mathrm{Homeo}_{*}(D^{*})$ and a lift $\tilde{f}$ to $\tilde{D}$.
Suppose that $f$ lets invariant a finite Borel measure $\mu$.

  1. (1)

Say that $\mu$ has a rotation number
$\mathrm{rot}_{\tilde{f}}(\mu)\in\mathbb{R}$ if $\mu$-almost every point $z$
has a rotation number and if the function $\mathrm{rot}_{\tilde{f}}$ is
$\mu$-integrable, and in that case set

$\mathrm{rot}_{\tilde{f}}(\mu)=\int_{\mathbb{D}^{*}}\mathrm{rot}_{\tilde{f}}\,d\mu.$

  2. (2)

Say that $\mu$ has a self-linking number
$\mathrm{link}_{\tilde{f}}(\mu)\in\mathbb{R}$, if

     * •

$\mu$ is non atomic,

     * •

$\mu$ has a rotation number,

     * •

$\mu\times\mu$-almost every couple $(z,z^{\prime})\in W$ has a linking number
and the function $\mathrm{link}_{\widetilde{f}}$ is $\mu\times\mu$-integrable,
and in that case set

$\mathrm{link}_{\tilde{f}}(\mu)=\int_{W}\mathrm{link}_{\tilde{f}}\,d\mu\times
d\mu.$

######  Remarks.

1\. If there exists a ray $\phi$ such that $m_{\tilde{f},\phi}$ is
$\mu$-integrable, then by Birkhoff Ergodic Theorem, one knows that $\mu$ has a
rotation number and it holds that

$\mathrm{rot}_{\tilde{f}}(\mu)=\int_{D^{*}}m_{\tilde{f},\phi}\,d\mu.$

2\. If there exists a radial foliation $\mathcal{F}$ and a leaf $\phi$ of
$\mathcal{F}$ such that $m_{\tilde{f},\phi}$ is $\mu$-integrable and
${\lambda}_{f,\mathcal{F}}$ is $\mu\times\mu$-integrable, then $\mu$ has a
self-linking number and it holds that

$\mathrm{link}_{\tilde{f}}(\mu)=\int_{W}{\Lambda}_{\tilde{f},\mathcal{F},\phi}\,d\mu\times
d\mu.$

Suppose that $f\in\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$. Define
$\mathbb{D}^{*}=\mathbb{D}\setminus(\mathbb{S}\cup\\{0\\})$. Choose an
identity isotopy $I=(f_{s})_{s\in[0,1]}$ of $f$ in
$\mathrm{Homeo}_{*}(\mathbb{D})$ that fixes $0$ and write $\tilde{f}$ for the
lift of $f_{|\mathbb{D}^{*}}$ naturally defined by the restriction of $I$ to
$\mathbb{D}^{*}$. Extend the isotopy to a family $(f_{s})_{s\in\mathbb{R}}$
such that $f_{s+1}=f_{s}\circ f$ for every $s\in[0,1]$. Denote $\mu_{\omega}$
the finite measure naturally defined by $\omega$. Consider the Euclidean
radial foliation $\mathcal{F}_{*}$ on $\mathbb{D}^{*}$, whose leaves are the
paths $(0,1)\ni t\mapsto te^{2i\pi\alpha}$, $\alpha\in[0,1)$. Note now that
for every leaf $\phi$ of $\mathcal{F}_{*}$, every $n\geq 1$ and every $z\in
D^{*}$ it holds that

$\left|m_{\tilde{f}^{n},\phi}(z)-\mathrm{ang}_{I^{n}}(0,z)\right|\leq 1.$

Moreover, the function $z\mapsto\mathrm{ang}_{I}(0,z)$ is bounded because $f$
is a $C^{1}$ diffeomorphism. It implies that $m_{\tilde{f},\phi}$ is bounded
and so, $\mu_{\omega_{\omega}}$ has a rotation number (according to Definition
2.9) and we have

$\mathrm{rot}_{\tilde{f}}(\mu)=\int_{D^{*}}m_{\tilde{f},\phi}\,d\mu_{\omega}=\int_{D^{*}}\mathrm{ang}_{I}(0,z)\,d\mu_{\omega}.$

Note now that for every $n\geq 1$ and every $(z,z^{\prime})\in W$ it holds
that

$\left|{\Lambda}_{\tilde{f}^{n},\mathcal{F}_{*},\phi}(z,z^{\prime})-\mathrm{ang}_{I}^{n}(z,z^{\prime})\right|\leq
1+1=2.$

To get this inequality, one must consider a frame at $z$ moving with time: at
time $n$ it is the image of the original frame at $z$ by the rotation
$R_{\mathrm{ang}_{I^{n}}(0,z)}$. Indeed, we have the two following properties:

  * •

$\left|m_{\tilde{f}^{n},\phi}(z)-\mathrm{ang}_{I^{n}}(0,z)\right|\leq 1$;

  * •

the difference between the variation of angle of the vector
$f_{s}(z^{\prime})-f_{s}(z)$, $s\in[0,n]$, in the moving frame and
${\lambda}_{f^{n},\mathcal{F}_{*}}(z,z^{\prime})$ is smaller than $1$.

One deduces that $\mu_{\omega}$ has a self-linking number (according to
Definition 2.9) and we have

$\mathrm{link}_{\tilde{f}}(\mu_{\omega})=\int_{W}{\Lambda}_{\tilde{f},\mathcal{F},\phi}\,d\mu_{\omega}\times
d\mu_{\omega}=\int_{W}\mathrm{ang}_{I}^{n}(z,z^{\prime})\,d\mu_{\omega}\times
d\mu_{\omega}=\widetilde{\mathrm{Cal}}(I).$

Let us conclude with a proposition, that will useful to prove Theorem 1.1, and
that summarize what has been done in this section.

######  Proposition 2.10.

Suppose that $f\in\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$ fixes $0$. Let $I$
be an identity isotopy of $f$ and $\tilde{f}$ be the lift of
$f_{|\mathbb{D}^{*}}$ naturally defined by $[I]$. If
$\mathcal{F}\in\mathfrak{F}$ is a radial foliation and $\phi$ is a leaf of
$\mathcal{F}$ such that $m_{\tilde{f},\phi}$ is $\mu_{\omega}$-integrable and
${\lambda}_{f,\mathcal{F}}$ is $\mu_{\omega}\times\mu_{\omega}$-integrable,
then it holds that

$\
\widetilde{\mathrm{Cal}}(I)=\int_{W}{\Lambda}_{\tilde{f},\mathcal{F},\phi}\,d\mu_{\omega}\times
d\mu_{\omega}.$

##  3\. Construction of a good radial foliation for an irrational pseudo-
rotation.

The three first sections of this chapter come from [L2]. All proofs can be
found there. The fourth one, concerning properties of projected foliations is
mainly new. The last proposition of the fourth section and the isotopy defined
in the last section already appeared in [L1] but in a slightly different
context (twist maps instead of untwisted maps).

###  3.1. Generating functions

Let us denote $\pi_{1}:(x,y)\mapsto x$ and $\pi_{2}:(x,y)\mapsto y$ the two
projections defined on the Euclidean plane $\mathbb{R}^{2}$. An orientation
preserving homeomorphism $f$ of $\mathbb{R}^{2}$ will be called untwisted if
the map

$(x,y)\mapsto(\pi_{1}(f(x,y)),y)$

is a homeomorphism, which means that there exist two continuous functions
$g,g^{\prime}$ on $\mathbb{R}^{2}$ such that

$f(x,y)=(X,Y)\Leftrightarrow\begin{cases}x=g(X,y),\\\
Y=g^{\prime}(X,y).\end{cases}$

In this case, the maps $X\mapsto g(X,y)$ and $y\mapsto g^{\prime}(X,y)$ are
orientation preserving homeomorphisms of $\mathbb{R}$. If moreover, $f$ is
area preserving, the continuous form $xdy+YdX$ is exact: there exists a
$C^{1}$ function $h:\mathbb{R}^{2}\to\mathbb{R}$ such that

$\displaystyle g={\partial h\over\partial y},\enskip\displaystyle
g^{\prime}={\partial h\over\partial X}.$

The function $h$, defined up to an additive constant, is a generating function
of $f$.

We can precise the definition by saying that $f$ is a $K$ Lipschitz untwisted
homeomorphism, where $K\geq 1$, if

i) $f$ is untwisted;

ii) $f$ is $K$ bi-Lipschitz;

iii) the maps $X\mapsto g(X,y)$ and $y\mapsto g^{\prime}(X,y)$ are $K$ bi-
Lipschitz;

iv) the maps $y\mapsto g(X,y)$ and $X\mapsto g^{\prime}(X,y)$ are $K$
Lipschitz.

If $f$ is a diffeomorphism of $\mathbb{R}^{2}$, denote $\mathrm{Jac}(f)(z)$
the Jacobian matrix at a point $z$. One proves easily, that for every $K>1$,
there exists a neighborhood $\mathcal{U}$ of the identity matrix in the space
of square matrices of order $2$, such that every $C^{1}$ diffeomorphism
satisfying $\mathrm{Jac}(f)(z)\in\mathcal{U}$, for every $z\in\mathbb{R}^{2}$,
is a $K$ Lipschitz untwisted homeomorphism.

Suppose that $f$ is a $C^{1}$ orientation preserving diffeomorphism of
$\mathbb{D}$ that fixes $0$ and coincides with a rotation $R_{\alpha}$ on
$\mathbb{S}$. Fix $\beta>\alpha$. We can extend our map to a homeomorphism of
the whole plane (also denoted $f$) such that:

$f(z)=\begin{cases}R_{\alpha+r-1}(z)&\text{ if \enskip$1\leq|z|\leq
1+\beta-\alpha,$}\\\ R_{\beta}(z)&\text{ if \enskip$|z|\geq
1+\beta-\alpha$.}\end{cases}$

Using the fact that the group of orientation preserving $C^{1}$
diffeomorphisms of $\mathbb{D}$ (and the group of symplectic diffeomorphisms
of $\mathbb{D}$) that fix $0$ and every point of $\mathbb{S}$, when furnished
with the $C^{1}$ topology, is path connected, one can prove the following:

######  Proposition 3.1.

For every $K>1$, one can find a decomposition $f=f_{m}\circ\dots\circ f_{1}$,
where each $f_{i}$ is a $K$ Lipschitz untwisted homeomorphism that fixes $0$
and induces a rotation on every circle of origin ${0}$ and radius $r\geq 1$.
Moreover, if $f$ is area preserving, one can suppose that each $f_{i}$
preserves the area.

We fix $K>1$ and a decomposition $f=f_{m}\circ\dots\circ f_{1}$ given by
Proposition 3.1. We define two families $(g_{i})_{1\leq i\leq m}$,
$(g^{\prime}_{i})_{1\leq i\leq m}$ of continuous maps as follows

$f_{i}(x,y)=(X,Y)\Leftrightarrow\begin{cases}x=g_{i}(X,y),\\\
Y=g^{\prime}_{i}(X,y),\end{cases}$

For every $i\in\\{1,\dots,m\\}$ one can find an identity isotopy
$I_{i}=(f_{i,s})_{s\in[0,1]}$ where each $f_{i,s}$ is an untwisted map such
that

$f_{i,s}(x,y)=(X,Y)\Leftrightarrow\begin{cases}x=(1-s)X+sg_{i}(X,y),\\\
Y=(1-s)y+sg^{\prime}_{i}(X,y),\end{cases}$

and a natural isotopy $I=I_{m}\circ\dots\circ I_{1}$ of $f$. Note that if each
$f_{i}$ is area preserving, one can find a family $(h_{i})_{1\leq i\leq m}$ of
$C^{1}$ maps, such that

$\displaystyle g_{i}={\partial h_{i}\over\partial y},\enskip\displaystyle
g^{\prime}_{i}={\partial h_{i}\over\partial X}.$

In that case every map $f_{i,s}$, $1\leq i\leq m$, $s\in[0,1]$, is area
preserving and it holds that

$(X,y)\mapsto(1-s)Xy+sh_{i}(X,y)$

is a generating function of $f_{i,s}$.

###  3.2. The vector field associated to a decomposition

We consider in this section a $C^{1}$ orientation preserving diffeomorphism
$f$ of $\mathbb{D}$ that fixes $0$ and coincides with a rotation $R_{\alpha}$
on $\mathbb{S}$. We suppose that it is extended and then decomposed into
untwisted maps as in Section 3.1. We keep the same notations. We extend the
families

$(f_{i})_{1\leq i\leq m},\enskip(g_{i})_{1\leq i\leq
m},\enskip(g^{\prime}_{i})_{1\leq i\leq m},$

to $m$ periodic families

$(f_{i})_{i\in\mathbb{Z}},\enskip(g_{i})_{i\in\mathbb{Z}},\enskip(g^{\prime}_{i})_{i\in\mathbb{Z}},$

and the family $(h_{i})_{1\leq i\leq m}$ to a $m$ periodic family
$(h_{i})_{i\in\mathbb{Z}}$ in case the $f_{i}$ are area preserving.

We fix an integer $b\geq 1$ and consider the finite dimensional vector space

$E_{b}=\left\\{{\bf
z}=(z_{i})_{i\in{\mathbb{Z}}}\in(\mathbb{R}^{2})^{\mathbb{Z}}\enskip\enskip|\enskip
z_{i+mb}=z_{i},\enskip\mathrm{for\enskip all}\enskip
i\in{\mathbb{Z}}\right\\},$

furnished with the scalar product

$\left\langle(z_{i})_{i\in{\mathbb{Z}}},(z^{\prime}_{i})_{i\in{\mathbb{Z}}}\right\rangle=\sum_{0<i\leq
mb}x_{i}x^{\prime}_{i}+y_{i}y^{\prime}_{i},$

where $z_{i}=(x_{i},y_{i})$ and
$z^{\prime}_{i}=(x^{\prime}_{i},y^{\prime}_{i})$. We define on $E_{b}$ a
vector field $\zeta=(\zeta_{i})_{i\in\mathbb{Z}}$ by writing

${\bf\zeta}_{i}({\bf z})=({\bf\xi}_{i}({\bf z}),{\bf\eta}_{i}({\bf
z}))=\left(y_{i}-g^{\prime}_{i-1}(x_{i},y_{i-1}),\enskip
x_{i}-g_{i}(x_{i+1},y_{i})\right).$

Observe that $\zeta$ is invariant by the ($b$ periodic) shift

$\displaystyle{\varphi:E_{b}}$ | $\displaystyle{{}\to E_{b}\enskip,}$
---|---
$\displaystyle{(z_{i})_{i\in{\mathbb{Z}}}}$ | $\displaystyle{{}\mapsto(z_{i+m})_{i\in{\mathbb{Z}}}.}$

Let us state some facts about $\zeta$.

######  Lemma 3.2.

The vector field is $A$ Lipschitz, where $A=\sqrt{6K^{2}+3}$.

One deduces that the associated differential system

$\begin{cases}\dot{x}_{i}=y_{i}-g^{\prime}_{i-1}(x_{i},y_{i-1}),\\\
\dot{y}_{i}=x_{i}-g_{i}(x_{i+1},y_{i}),\end{cases}$

defines a flow on $E$. We will denote by ${\bf z}^{t}$ the image at time $t$
of a point ${\bf z}\in E$ by this flow. As an application of Gronwall’s Lemma,
one gets:

######  Lemma 3.3.

For every $({\bf z},{\bf z}^{\prime})\in E_{b}$ and every $t\in\mathbb{R}$,
one has

$e^{-A|t|}\|{\bf z}-{\bf z}^{\prime}\|\leq\|{\bf z}^{t}-{\bf
z}^{\prime}{}^{t}\|\leq e^{A|t|}\|{\bf z}-{\bf z}^{\prime}\|$

and

$e^{-A|t|}\|\zeta({\bf z})\|\leq\|\zeta({\bf z}^{t})\|\leq
e^{A|t|}\|\zeta({\bf z})\|.$

In the case where the $f_{i}$ are area preserving, observe that $\zeta$ is the
gradient vector field of the function

${\bf h}:{\bf z}\mapsto\sum_{0<i\leq mb}x_{i}y_{i}-h_{i-1}(x_{i},y_{i-1})$

and that ${\bf h}$ is invariant by $\varphi$. One can define the energy of an
orbit $({\bf z}^{t})_{t\in\mathbb{R}}$ to be

$\int_{-\infty}^{+\infty}\|\zeta({\bf z}^{t})\|^{2}\,dt=\lim_{t\to+\infty}{\bf
h}({\bf z^{t}})-\lim_{t\to-\infty}{\bf h}({\bf z^{t}}).$

As a consequence of Lemma 3.2 it holds that

######  Lemma 3.4.

For every ${\bf z}\in E_{b}$, one has

$\|\zeta({\bf z})\|^{2}\leq A\int_{-\infty}^{+\infty}\|\zeta({\bf
z}^{t})\|^{2}\,dt=A\left(\lim_{t\to+\infty}{\bf h}({\bf
z^{t}})-\lim_{t\to-\infty}{\bf h}({\bf z^{t}})\right).$

For every $i\in\mathbb{Z}$, define the maps
$Q_{i},\,P_{i},\,Q^{\prime}_{i}:E_{b}\to\mathbb{R}^{2}$, where

$Q_{i}({\bf z})=(g_{i}(x_{i+1},y_{i}),y_{i}),\enskip P_{i}({\bf
z})=(x_{i},y_{i}),\enskip Q^{\prime}_{i}({\bf
z})=(x_{i},g^{\prime}_{i-1}(x_{i},y_{i-1})).$

Let us state the main properties of these maps:

  * •

$f_{i}\circ Q_{i}=Q^{\prime}_{i+1},$

  * •

$\zeta_{i}=J\circ(Q^{\prime}_{i}-Q_{i})$ where $J(x,y)=(-y,x)$,

  * •

${\bf z}\in E$ is a singularity of $\zeta$ if and only if $Q_{i}({\bf
z})=Q^{\prime}_{i}({\bf z})$ for every $i\in\mathbb{Z}$,

  * •

if ${\bf z}\in E$ is a singularity of $\zeta$ then $Q_{i}({\bf z})=P_{i}({\bf
z})=Q^{\prime}_{i}({\bf z})$ for every $i\in\mathbb{Z}$,

  * •

$Q_{1}$ induces a bijection between the singular set of $\zeta$ and the fixed
point set of $f^{b}$,

  * •

the sequence ${\bf 0}=(0)_{i\in\mathbb{Z}}$ is a singular point of $\zeta$
that is sent onto $0$ by each $Q_{i}$, $P_{i}$ or $Q^{\prime}_{i}$,

  * •

$\zeta$ is $C^{1}$ in a neighborhood of ${\bf 0}$.

###  3.3. The case of an irrational pseudo-rotation

In this section we keep the notation of Section 3.2 but we suppose than $f$ is
an irrational pseudo-rotation. We suppose moreover than
$\beta\not\in\mathbb{Q}$ and that $(\alpha,\beta)\cap\mathbb{Z}=\emptyset$.
The extension $f$ is a piecewise $C^{1}$ area preserving transformation that
satisfies the following properties:

\- $0$ is the unique fixed point of $f$;

\- there is no periodic point of period $b$ if
$(b\alpha,b\beta)\cap\mathbb{Z}=\emptyset$;

\- if $(b\alpha,b\beta)\cap\mathbb{Z}\not=\emptyset$, the set of periodic
points of period $b$ can be written $\bigcup_{\alpha<a/b<\beta}S_{a/b}$, where
$S_{a/b}$ is the circle of center $0$ and radius $1+a/b-\alpha$.

In that case, we have the additional following properties for the vector
$\zeta$ defined on $E_{b}$:

  * •

the singular set consists of the constant sequence ${\bf 0}$ and of finitely
many smooth closed curves
$(\Sigma_{a})_{a\in(b\alpha,b\beta)\cap{\mathbb{Z}}}$;

  * •

the curve $\Sigma_{a}$ is sent homeomorphically onto $S_{a/b}$ by each
$Q_{i}$, $P_{i}$ or $Q^{\prime}_{i}$;

  * •

$\xi$ is $C^{\infty}$ in a neighborhood of $\Sigma_{a}$.

We fix an integer $b\geq 2$ such that
$(b\alpha,b\beta)\cap{\mathbb{Z}}\not=\emptyset$. If ${\bf z}$ and ${\bf
z^{\prime}}$ are two singular points of $\zeta$, the quantity ${\bf h}({\bf
z})-{\bf h}({\bf 0})$ is the difference of action between the two
corresponding fixed points of $f^{b}$. A computation gives us:

######  Lemma 3.5.

For every ${\bf z}\in\Sigma_{a}$, one has

${\bf h}({\bf z})-{\bf h}({\bf
0})=\pi(a-b\alpha)\left(1+(a/b-\alpha)+{(a/b-\alpha)^{2}\over 3}\right).$

We will denote

$C(a,b)=\pi(a-b\alpha)\left(1+(a/b-\alpha)+{(a/b-\alpha)^{2}\over 3}\right).$

Now let us state the fundamental result of [L2]:

######  Proposition 3.6.

The curve $\Sigma_{a}$ bounds a topological disk $\Delta_{a}\subset E_{b}$
that satisfies the following:

i) $\Delta_{a}$ contains the constant sequence ${\bf 0}$;

ii) $\Delta_{a}$ is invariant by $\varphi$;

iii) each projection ${\bf z}\mapsto(x_{i},y_{i-1})$, $i\in\mathbb{Z}$, is one
to one on $\Delta_{a}$;

iv) each projection ${\bf z}\mapsto(x_{i},y_{i})$, $i\in\mathbb{Z}$, is one to
one on $\Delta_{a}$;

v) $\Delta_{a}$ is invariant by the flow of $\zeta$;

vi) for every ${\bf z}\in\Delta_{a}^{*}=\Delta_{a}\setminus(\\{{\bf
0}\\}\cup\Sigma_{a})$, one has $\lim_{t\to-\infty}{\bf z}^{t}={\bf 0}$ and
$\lim_{t\to+\infty}d({\bf z}^{t},\Sigma_{a})=0$.

Let us explain more precisely what is proved in [L2]. In what follows, the
function $\mathrm{sign}$ assigns $+1$ to a positive number and $-1$ to a
negative number.

Let us consider the set

$V=\\{{\bf z}\in E_{b}\enskip|\,x_{i}\not=0\enskip\mathrm{and}\enskip
y_{i}\not=0\enskip{\rm for\enskip all}\enskip i\in\mathbb{Z}\\}$

and the function $L$ on $V$ defined by the formula

$\displaystyle{L({\bf z})}$ | $\displaystyle{{}={1\over 4}\sum_{0<i\leq mb}\mathrm{sign}(x_{i})\left(\mathrm{sign}(y_{i})-\mathrm{sign}(y_{i-1})\right),}$
---|---
| $\displaystyle{{}={1\over 4}\sum_{0<i\leq
mb}\mathrm{sign}(y_{i})\left(\mathrm{sign}(x_{i})-\mathrm{sign}(x_{i+1})\right).}$

It extends continuously to the open set

$V^{\prime}=\\{{\bf z}\in E_{b}\enskip|\enskip x_{i}=0\Rightarrow
y_{i-1}y_{i}>0,\enskip y_{i}=0\Rightarrow x_{i}x_{i+1}>0\\}.$

It is integer valued and takes its values in $\\{-[mb/2],\dots,[mb/2]\\}$,
where the notation $[x]$ denotes the integer part of a real number $x$. The
assertion iii) and iv) are immediate consequences of the following fact:

######  Proposition 3.7.

If ${\bf z}$ and ${\bf z^{\prime}}$ are two different points of $\Delta_{a}$,
then ${\bf z}-{\bf z^{\prime}}\in V^{\prime}$ and $L({\bf z}-{\bf
z^{\prime}})=a$.

The fundamental result that permits to construct $\Delta_{a}$ is the
following:

######  Proposition 3.8.

If ${\bf z}$, ${\bf z}$ are two distinct points of $E_{b}$ satisfying ${\bf
z}^{\prime}-{\bf z}\not\in V^{\prime}$, then there exists $\varepsilon>0$ such
that for every $t\in(0,\varepsilon]$, it holds that:

${\bf z^{\prime}}^{-t}-{\bf z}^{-t}\in V^{\prime},\enskip{\bf
z^{\prime}}^{t}-{\bf z}^{t}\in V^{\prime},\enskip L({\bf z^{\prime}}^{-t}-{\bf
z}^{-t})<L({\bf z^{\prime}}^{t}-{\bf z}^{t}).$

This result admits an infinitesimal version (see [L1]):

######  Proposition 3.9.

If ${\bf z}\in E_{b}$ is non singular and satisfies $\zeta({\bf z})\not\in W$,
then there exists $\varepsilon>0$ such that for every $t\in(0,\varepsilon]$,
it holds that:

$\zeta({\bf z^{\prime}}^{-t})\in V^{\prime},\enskip\zeta({\bf
z^{\prime}}^{t})\in V^{\prime},\enskip L(\zeta({\bf
z^{\prime}}^{-t}))<L(\zeta({\bf z^{\prime}}^{t}))$

and

${\bf z^{\prime}}^{-t}-{\bf z}\in V^{\prime},\enskip{\bf z^{\prime}}^{t}-{\bf
z}\in V^{\prime},\enskip L({\bf z^{\prime}}^{-t}-{\bf z})<L({\bf
z^{\prime}}^{t}-{\bf z}).$

As an immediate corollary of the second assertion, one gets the following
result (not explicitely stated in [L2])

######  Corollary 3.10.

If ${\bf z}\in\Delta_{a}^{*}$, then $\zeta({\bf z})\in V^{\prime}$ and
$L(\zeta({\bf z}))=a$.

The assertion iii) tells us that the maps $Q_{i}|_{\Delta_{a}}$ and
$Q^{\prime}_{i}|_{\Delta_{a}}$, $i\in\mathbb{Z}$, induce homeomorphisms from
$\Delta_{a}$ to $D_{a/b}$ denoted respectively $q_{i}$ and $q^{\prime}_{i}$
555 We do not refer to $a$ to lighten the notations.. The assertion iv) tells
us that the maps $P_{i}|_{\Delta_{a}}$, $i\in\mathbb{Z}$, induce
homeomorphisms from $\Delta_{a}$ to $D_{a/b}$ denoted $p_{i}$. Note that
$p_{i}\circ q_{i}^{-1}$ is a homeomorphism which let invariant every
horizontal segment of $D_{a/b}$ and induces on this segment an increasing
homeomorphism and similarly $p_{i}\circ q^{\prime}{}_{i}^{-1}$ is a
homeomorphism which let invariant every vertical segment of $D_{a/b}$ and
induces on this segment an increasing homeomorphism. We denote ${\mathcal{F}}$
the radial foliation defined on $\Delta_{a}^{*}$ whose leaves are the gradient
lines of $\zeta$ included in $\Delta_{a}^{*}$.

For every $s\in[0,1]$, and every $i\in\mathbb{Z}$ we define

$q^{s}_{i}=(1-s)q_{i}+sp_{i},\enskip
q^{\prime}{}^{s}_{i}=(1-s)p_{i}+sq^{\prime}_{i}.$

The maps $q^{s}_{i}$ and $q^{\prime}{}^{s}_{i}$ are homeomorphisms from
$\Delta_{a}$ to $D_{a/b}$ (all coinciding on $\Sigma_{a}$) and send
$\mathcal{F}$ onto a radial foliation of $D_{a/b}$.

###  3.4. Projected radial foliations

Recall that $d$ denotes the winding distance between two radial foliations.
Let us begin with the following result:

######  Lemma 3.11.

For every $s_{1},s_{2}$ in $[0,1]$ and every $i\in\mathbb{Z}$, we have

$d(q_{i}^{s_{1}}(\mathcal{F}),q_{i}^{s_{2}}(\mathcal{F}))\leq 2,\enskip
d(q^{\prime}{}_{i}^{s_{1}}(\mathcal{F}),q^{\prime}{}_{i}^{s_{2}}(\mathcal{F}))\leq
2.$

###### Proof.

We fix $i\in\mathbb{Z}$ and will prove the first inequality, the proof of the
second one being similar. The leaves of $q_{i}^{s}(\mathcal{F})$ are the paths
$t\mapsto q_{i}^{s}({\bf z}^{t})$, ${\bf z}\in\Delta_{a}^{*}$. Note that the
map $t\mapsto\pi_{2}(q_{i}^{s}({\bf z}^{t}))$ is $C^{1}$ and that

${d\over dt}\pi_{2}\circ q^{s}_{i}({\bf z}^{t})_{|t=0}=\eta_{i}({\bf
z})=x_{i}-g_{i}(x_{i+1},y_{i}).$

Say that $z\in D_{a/b}^{*}$ is horizontal  if it can be written $z=q_{i}({\bf
z})$, where $x_{i}-g_{i}(x_{i+1},y_{i})=0$. Denote $H_{i}$ the set of
horizontal points666 if $\eta_{i}({\bf z})=0$, then $\xi_{i}({\bf z})\not=0$,
so every leaf of $p_{i}(\mathcal{F})$ is a $C^{1}$ embedded line and $H_{i}$
is nothing but the set of points where the foliation $p_{i}(\mathcal{F})$ is
horizontal.. For every point $z\in D_{a/b}^{*}$, define $J(z)$ to be equal to
$\\{z\\}$ if $z$ is horizontal and to the largest horizontal interval
contained in $D_{a/b}^{*}\setminus H_{i}$ and containing $z$, if $z$ is not
horizontal. The sign of $\eta_{i}(q_{i}^{-1}(z^{\prime}))$ does not depend on
the choice of $z^{\prime}\in J(z)$. We orient $J(z)$ with $x$ increasing if
this sign is positive and with $x$ decreasing if this sign is negative. Note
that the path $s\mapsto u_{i}^{s}(z)=q_{i}^{s}\circ q_{i}^{-1}(z)$ is constant
if $z$ is horizontal and is an oriented segment of $J(z)$ inheriting the same
orientation if $z$ is not horizontal. The fact that $\displaystyle{d\over
dt}\pi_{2}\circ q^{s}_{i}({\bf z}^{t})_{|t=0}=\eta_{i}({\bf z})$, for every
$s\in[0,1]$ and every ${\bf z}\in\Delta_{a}^{*}$, implies that if $z$ is not
horizontal, the oriented interval $J(z)$ is transverse to the foliations
$q_{i}^{s}(\mathcal{F})$, $s\in[0,1]$, crossing locally every leaf from the
right to the left.

Denote $\tilde{D}_{a/b}$ the universal covering space of $D_{a/b}^{*}$ and for
every $\tilde{z}\in\tilde{D}_{a/b}$ that lifts $z\in D_{a/b}^{*}$, denote
$\tilde{J}(\tilde{z})$ the lift of $J(z)$ containing $\tilde{z}$, with the
induced orientation in case $z\not\in H_{i}$. Write $\tilde{H}_{i}$ for the
lift of $H_{i}$, write $\widetilde{q_{i}^{s}(\mathcal{F})}$ for the lift of
$q_{i}^{s}(\mathcal{F})$ and denote $(\tilde{u}_{i}^{s})_{s\in[0,1]}$ the
identity isotopy that lifts $(u_{i}^{s})_{s\in[0,1]}$. Suppose that
$\tilde{z}\in\tilde{D}_{a/b}$ is not in $\tilde{H}_{i}$. For every
$s_{0}\in[0,1]$, the oriented interval $\tilde{I}(\tilde{z})$ is transverse to
the foliation $\widetilde{q_{i}^{s_{0}}(\mathcal{F})}$, crossing locally every
leaf from the right to the left. Moreover if $0\leq s_{1}<s_{2}$, then
$\tilde{u}_{i}^{s_{2}}(\tilde{z})$ follows $\tilde{u}_{i}^{s_{1}}(\tilde{z})$
on $\tilde{J}(\tilde{z})$. Consequently it holds that:

$0\leq s_{0}\leq 1\enskip\mathrm{and}\enskip 0\leq s_{1}<s_{2}\leq
1\Rightarrow\theta_{\tilde{u}_{i}^{s_{1}}(\tilde{z}),\tilde{u}_{i}^{s_{2}}(\tilde{z})}(q_{i}^{s_{0}}(\mathcal{F}))=\dot{1}.$

Now, let us fix two different points $\tilde{z}_{0}$ and $\tilde{z}_{1}$ in
$\tilde{D}_{a/b}$ and $s\in[0,1]$. Observe that the three sets

$\displaystyle{\\{(\tilde{z}^{\prime}_{0},\tilde{z}^{\prime}_{1})\in\tilde{J}(\tilde{z}_{0})\times\tilde{J}(\tilde{z}_{1})\,}$ | $\displaystyle{{}|\,\,\tilde{z}^{\prime}_{0}\not=\tilde{z}^{\prime}_{1},\,\theta_{\tilde{z}^{\prime}_{0},\tilde{z}^{\prime}_{1}}(q_{i}^{s}(\mathcal{F}))=\dot{1}\\},}$
---|---
$\displaystyle{\\{(\tilde{z}^{\prime}_{0},\tilde{z}^{\prime}_{1})\in\tilde{J}(\tilde{z}_{0})\times\tilde{J}(\tilde{z}_{1})\,}$ | $\displaystyle{{}|\,\,\tilde{z}^{\prime}_{0}\not=\tilde{z}^{\prime}_{1},\,\theta_{\tilde{z}^{\prime}_{0},\tilde{z}^{\prime}_{1}}(q_{i}^{s}(\mathcal{F}))=\dot{3}\\},}$
$\displaystyle{\\{(\tilde{z}^{\prime}_{0},\tilde{z}^{\prime}_{1})\in\tilde{J}(\tilde{z}_{0})\times\tilde{J}(\tilde{z}_{1})\,}$ | $\displaystyle{{}|\,\,\tilde{z}^{\prime}_{0}\not=\tilde{z}^{\prime}_{1},\,\theta_{\tilde{z}^{\prime}_{0},\tilde{z}^{\prime}_{1}}(q_{i}^{s}(\mathcal{F}))\in\\{\dot{0},\dot{2}\\}\\}}$

are connected, whether the points belong to $\tilde{H}_{i}$ or not. Indeed,
the paths $\tilde{J}(\tilde{z}_{0})$ and $\tilde{J}(\tilde{z}_{1})$ in
$\tilde{D}_{a/b}$ draw intervals of leaves of the foliation
$\widetilde{q_{i}^{s}(\mathcal{F})}$ whose intersection is an interval of
leaves if not empty. In particular the set of couples
$(\tilde{z}^{\prime}_{0},\tilde{z}^{\prime}_{1})\in\tilde{J}(\tilde{z}_{0})\times\tilde{J}(\tilde{z}_{1})$
such that $\tilde{z}^{\prime}_{0}$ and $\tilde{z}^{\prime}_{1}$ are distinct
and belong to the same leaf of $\widetilde{q_{i}^{s}(\mathcal{F})}$ is
connected: it is an interval (possibly empty) if
$\tilde{J}(\tilde{z}_{0})\cap\tilde{J}(\tilde{z}_{1})=\emptyset$, it is empty
if $\tilde{J}(\tilde{z}_{0})\cap\tilde{J}(\tilde{z}_{1})\not=\emptyset$
(because the intersection is a “horizontal” path in $\tilde{D}_{a/b}$ ). The
map
$(\tilde{z}^{\prime}_{0},\tilde{z}^{\prime}_{1})\mapsto\theta_{\tilde{z}^{\prime}_{0},\tilde{z}^{\prime}_{1}}(q_{i}^{s}(\mathcal{F}))$
being continuous on $\tilde{W}$, takes at most three values on
$\tilde{J}(\tilde{z}_{0})\times\tilde{J}(\tilde{z}_{1})$ (either $\dot{0}$ or
$\dot{2}$ is missing).

In particular, if $0\leq s_{1}<s_{2}\leq 1$, the map

$s\in[s_{1},s_{2}]\mapsto\theta_{\tilde{u}_{i}^{s}\circ(\tilde{u}_{i}^{s_{1}})^{-1}(\tilde{z}_{0}),\,\tilde{u}_{i}^{s}\circ(\tilde{u}_{i}^{s_{1}})^{-1}(\tilde{z}_{1})}(q_{i}^{s_{2}}(\mathcal{F}))=\theta_{\tilde{z}_{0},\tilde{z}_{1}}(u_{i}^{s_{1}}\circ(u_{i}^{s})^{-1}\circ
q_{i}^{s_{2}}(\mathcal{F}))\in\mathbb{Z}/4\mathbb{Z}$

takes at most three values and is lifted to a map

$s\in[s_{1},s_{2}]\mapsto\hat{\theta}_{\tilde{z}_{0},\tilde{z}_{1}}(u_{i}^{s_{1}}\circ(u_{i}^{s})^{-1}\circ
q_{i}^{s_{2}}(\mathcal{F}))\in\mathbb{Z}$

taking at most three values, which implies that

$|\hat{\tau}(\tilde{z}_{0},\tilde{z}_{1},q_{i}^{s_{2}}(\mathcal{F}),q_{i}^{s_{1}}(\mathcal{F}))|\leq
2.$

∎

The homeomorphism $\tilde{u}_{i}^{s}$ sends the foliation
$\widetilde{q_{i}(\mathcal{F})}$ onto the foliation
$\widetilde{q_{i}^{s}(\mathcal{F})}$. The leaves of
$\widetilde{q_{i}(\mathcal{F})}$ are pushed on the left by the isotopy
$(\tilde{u}_{i}^{s})_{s\in[0,1]}$. More precisely, for every $0\leq
s_{1}<s_{2}\leq 1$ and every leaf $\tilde{\phi}$ of
$\widetilde{q_{i}(\mathcal{F})}$, it holds that:

  * •

$L(\tilde{u}_{i}^{s_{2}}(\tilde{\phi}))\subset
L(\tilde{u}_{i}^{s_{1}}(\tilde{\phi}))$,

  * •

if $\tilde{z}\in\tilde{\phi}$ belongs to $\tilde{H}_{i}$, then
$\tilde{z}\in\tilde{u}_{i}^{s_{1}}(\tilde{\phi})\cap\tilde{u}_{i}^{s_{2}}(\tilde{\phi})$,

  * •

if $\tilde{z}\in\tilde{\phi}$ does not belong to $\tilde{H}_{i}$, then
$\tilde{u}_{i}^{s_{2}}(\tilde{z})\in L(\tilde{u}_{i}^{s_{1}}(\tilde{\phi}))$.

Similarly, for every ${\bf z}\in\Delta_{a}^{*}$, the map
$t\mapsto\pi_{1}(q^{\prime}{}_{i}^{s}({\bf z}^{t}))$ is $C^{1}$ and we have

${d\over dt}\pi_{1}\circ q^{\prime}{}^{s}_{i}({\bf z}^{t})_{|t=0}=\xi_{i}({\bf
z})=y_{i}-g_{i-1}(x_{i},y_{i-1}).$

Say that $z\in D_{a/b}^{*}$ is vertical if it can be written
$z=q^{\prime}_{i}({\bf z})$, where $y_{i}-g_{i-1}(x_{i},y_{i-1})=0$. Denote
$V_{i}$ the set of vertical points and $\tilde{V}_{i}$ its lift in
$\tilde{D}_{a/b}$. Define $u^{\prime}{}_{i}^{s}=q^{\prime}{}_{i}^{s}\circ
p_{i}^{-1}$ and lift the isotopy $(u^{\prime}{}_{i}^{s})_{s\in[0,1]}$ to an
identity isotopy $(\tilde{u}^{\prime}{}_{i}^{s})_{s\in[0,1]}$. Write
$\widetilde{q^{\prime}{}_{i}^{s}(\mathcal{F})}$ for the lift of
$q^{\prime}{}_{i}^{s}(\mathcal{F})$. Then, for every $0\leq s_{1}<s_{2}\leq 1$
and every every leaf $\tilde{\phi}$ of
$\widetilde{q^{\prime}{}_{i}(\mathcal{F})}$, it holds that:

  * •

$L(\tilde{u}^{\prime}{}_{i}^{s_{2}}(\tilde{\phi}))\subset
L(\tilde{u}^{\prime}{}_{i}^{s_{1}}(\tilde{\phi}))$,

  * •

if $\tilde{z}\in\tilde{\phi}$ belongs to $\tilde{V}_{i}$, then
$\tilde{z}\in\tilde{u}^{\prime}{}_{i}^{s_{1}}(\tilde{\phi})\cap\tilde{u}^{\prime}{}_{i}^{s_{2}}(\tilde{\phi})$,

  * •

if $\tilde{z}\in\tilde{\phi}$ does not belong to $\tilde{V}_{i}$, then
$\tilde{u}^{\prime}{}_{i}^{s^{\prime}}(\tilde{z})\in
L(\tilde{u}^{\prime}{}_{i}^{s}(\tilde{\phi}))$.

To conclude, consider the family $(v_{i}^{s})_{s\in[0,2]}$ where

$v_{i}(s)=\begin{cases}u_{i}^{s}&\text{ if $0\leq s\leq 1$,}\\\
u^{\prime}{}_{i}^{s-1}\circ u_{i}^{1}=q^{\prime}{}_{i}^{s-1}\circ
q_{i}^{-1}&\text{ if $1\leq s\leq 2$,}\end{cases}$

is an isotopy from $\mathrm{Id}$ to $q^{\prime}_{i}\circ q^{-1}_{i}$. The
isotopy $\left(v_{i}^{s}{}_{|D_{a/b}^{*}}\right)_{s\in[0,2]}$ can be lifted to
an identity isotopy $\left(\tilde{v}_{i}^{s}\right)_{s\in[0,2]}$ such that

$\tilde{v}_{i}(s)=\begin{cases}\tilde{u}_{i}^{s}&\text{ if $0\leq s\leq
1$,}\\\ \tilde{u}^{\prime}{}_{i}^{s-1}\circ\tilde{u}_{i}^{1}&\text{ if $1\leq
s\leq 2$.}\end{cases}$

The map $\tilde{v}_{i}^{s}$ sends the foliation
$\widetilde{q_{i}(\mathcal{F})}$ onto the foliation
$\widetilde{q^{s}_{i}(\mathcal{F})}$ if $0\leq s\leq 1$ and onto the foliation
$\widetilde{q^{\prime}{}_{i}^{s-1}(\mathcal{F})}$ if $1\leq s\leq 2$.
Moreover:

  * •

for every $0\leq s_{1}<1<s_{2}\leq 2$ and every leaf $\tilde{\phi}$ of
$\widetilde{\mathcal{F}}_{i}^{0}$, it holds that
$\overline{L(\tilde{v}_{i}^{s_{2}}(\tilde{\phi}))}\subset
L(\tilde{v}_{i}^{s_{1}}(\tilde{\phi})$.

Indeed, the sets $H_{i}$ and $V_{i}$ are disjoint.

###  3.5. Construction of a good isotopy

One gets a $m$-periodic family of homeomorphisms
$(\widehat{f}_{i})_{i\in\mathbb{Z}}$ of $D_{a/b}$ by writing:

$\widehat{f}_{i}=q_{i+1}\circ q_{i}^{-1}.$

Its periodicity comes from the equalities

$\widehat{f}_{i+m}=q_{i+m+1}\circ
q_{i+m}^{-1}=q_{i+1}\circ\varphi_{|\Delta_{a}}\circ(q_{i}\circ\varphi_{|\Delta_{a}})^{-1}$

Moreover $\widehat{f}=\widehat{f}_{m}\circ\dots\circ\widehat{f}_{1}$ has order
$q$ because

$\widehat{f}=q_{m+1}\circ q_{1}^{-1}=q_{1}\circ(\varphi_{|\Delta_{a}})\circ
q_{1}^{-1}.$

or equivalently because

$\widehat{f}^{b}=\widehat{f}_{mb}\circ\dots\circ\widehat{f}_{1}=q_{mb+1}\circ
q_{1}^{-1}=\mathrm{Id}_{D_{a/b}}.$

Note also that $\widehat{f}_{i}$ fixes $0$ and coincides with $f_{i}$ on
$S_{a/b}$ because

$f_{i}|_{D_{a/b}}=q^{\prime}_{i+1}\circ q_{i}^{-1}.$

Let us define an isotopy $\check{I}=(\check{f}_{s})_{s\in[0,mb]}$ from
$\mathrm{Id}$ to $f^{b}$ in the following way:

If $s\in[2k,2k+1]$, $0\leq k<mb$, then

$\displaystyle\check{f}_{s}$ $\displaystyle=f_{mq}\circ\dots\circ
f_{mb-k+1}\circ q_{mb-k+1}^{s-2k}\circ q_{1}^{-1}$
$\displaystyle=f_{mq}\circ\dots\circ f_{mb-k+1}\circ q_{mb-k+1}^{s-2k}\circ
q^{-1}_{mq-k}\circ\widehat{f}_{mb-k-1}\circ\dots\circ\widehat{f}_{1}.$

If $s\in[2k+1,2k+2]$, $0\leq k<mq$, then

$\displaystyle\check{f}_{s}$ $\displaystyle=f_{mb}\circ\dots\circ
f_{mb-k+1}\circ q^{\prime}{}_{mb-k+1}^{s-2k-1}\circ q_{1}^{-1}$
$\displaystyle=f_{mq}\circ\dots\circ f_{mb-k+1}\circ
q^{\prime}{}_{mb-k+1}^{s-2k-1}\circ
q^{-1}_{mb-k}\circ\widehat{f}_{mb-k-1}\circ\dots\circ\widehat{f}_{1}.$

Note that

$\displaystyle\check{f}_{2k}$ $\displaystyle=f_{mb}\circ\dots\circ
f_{mb-k+1}\circ\widehat{f}_{mb-k}\circ\dots\circ\widehat{f}_{1},$
$\displaystyle\check{f}_{2k+1}$ $\displaystyle=f_{mb}\circ\dots\circ
f_{mb-k+1}\circ p_{mb-k+1}\circ
q_{mb-k}^{-1}\circ\widehat{f}_{mb-k-1}\circ\dots\circ\widehat{f}_{1}.$

In particular, we have

$\check{f}_{s}(q_{1}({\mathcal{F}}))=\begin{cases}f_{mb}\circ\dots\circ
f_{mb-k+1}(q_{i}^{s-2k}({\mathcal{F}}))&\text{ if $s\in[2k,2k+1]$,}\\\
f_{mb}\circ\dots\circ
f_{mb-k+1}(q^{\prime}{}_{i}^{s-2k-1}({\mathcal{F}}))&\text{ if
$s\in[2k+1,2k+2]$.}\end{cases}$

######  Proposition 3.12.

For every $s_{1}$, $s_{2}$ in $[0,2mb]$ we have

$d\left(\check{f}_{s_{1}}(q_{1}({\mathcal{F}})),\check{f}_{s_{2}}(q_{1}({\mathcal{F}}))\right)\leq
2|s_{2}-s_{1}|+4.$

Moreover, if $s_{1}$ and $s_{2}$ are integers we have

$d\left(\check{f}_{s_{1}}(q_{1}({\mathcal{F}})),\check{f}_{s_{2}}(q_{1}({\mathcal{F}}))\right)\leq
2|s_{2}-s_{1}|.$

###### Proof.

We will use Lemma 3.11. Note that for every $s_{1}$, $s_{2}$ in $[2k,2k+1]$ we
have

$\displaystyle
d\left(\check{f}_{s_{1}}(q_{1}({\mathcal{F}})),\check{f}_{s_{2}}(q_{1}({\mathcal{F}})\right)$
$\displaystyle=d\left(f_{mb}\circ\dots\circ
f_{mq-k+1}(q_{i}^{s_{1}-2k}({\mathcal{F}})),f_{mb}\circ\dots\circ
f_{mb-k+1}(q_{i}^{s_{2}-2k}({\mathcal{F}}))\right)$
$\displaystyle=d\left(q_{i}^{s_{1}-2k}({\mathcal{F}}),q_{i}^{s_{2}-2k}({\mathcal{F}})\right)$
$\displaystyle\leq 2,$

and similarly for every $s_{1}$, $s_{2}$ in $[2k+1,2k+2]$ we have

$\displaystyle
d\left(\check{f}_{s_{1}}(q_{1}({\mathcal{F}})),\check{f}_{s_{2}}(q_{1}({\mathcal{F}})\right)$
$\displaystyle=d\left(f_{mb}\circ\dots\circ
f_{mb-k+1}(q^{\prime}{}_{i}^{s_{1}-2k-1}({\mathcal{F}})),f_{mb}\circ\dots\circ
f_{mb-k+1}(q^{\prime}{}_{i}^{s_{2}-2k-1}({\mathcal{F}})\right)$
$\displaystyle=d\left(q^{\prime}{}_{i}^{s_{1}-2k-1}({\mathcal{F}}),q^{\prime}{}_{i}^{s_{2}-2k-1}({\mathcal{F}}))\right)$
$\displaystyle\leq 2.$

We easily deduce Proposition 3.12. ∎

We immediately deduce

######  Corollary 3.13.

It holds that

$d\left(f^{b}(q_{1}({\mathcal{F}})),q_{1}({\mathcal{F}})\right)\leq 4mb.$

Let us conclude this section by the following result

######  Proposition 3.14.

If $\tilde{\check{I}}=(\tilde{\check{f}}_{s})_{s\in[0,2mb]}$ is the identity
isotopy that lifts
$\left(\check{f}_{s}{}_{|D_{a/b}^{*}}\right)_{s\in[0,2mb]}$, then every leaf
$\tilde{\phi}$ of $\widetilde{q_{1}({\mathcal{F}})}$ is a Brouwer line of
$\tilde{\check{f}}_{2mb}$.

###### Proof.

Let $\phi$ be a leaf of $\mathcal{F}$, then for every $s\in[0,2mq]$, we have

$\check{f}_{s}(q_{1}(\phi))=\begin{cases}f_{mq}\circ\dots\circ f_{mb-k+1}\circ
q_{mb-k+1}^{s-2k}(\phi)&\text{if $s\in[2k,2k+1]$, $0\leq k<mb$}\\\
f_{mq}\circ\dots\circ f_{mb-k+1}\circ
q^{\prime}{}_{mb-k+1}^{s-2k}(\phi)&\text{if $s\in[2k+1,2k+2]$, $0\leq
k<mq$}\end{cases}$

Equivalently, it holds that

$\check{f}_{s}(q_{1}(\phi))=f_{mq}\circ\dots\circ f_{mb-k+1}\circ
v_{mb-k+1}^{s-2k}(q_{mb-k+1}(\phi))$

if $s\in[2k,2k+2]$, $0\leq k<mb$.

We deduce that for every leaf $\tilde{\phi}_{1}$ of
$\widetilde{q_{1}({\mathcal{F}})}$, it holds that
$L(\tilde{\check{f}}_{s_{2}}(\tilde{\phi}_{1}))\subset
L(\tilde{\check{f}}_{s_{1}}(\tilde{\phi}_{1}))$ if $0\leq s_{1}<s_{2}\leq 2mb$
and that $\overline{L(\tilde{\check{f}}_{2k+2}(\tilde{\phi}_{1}))}\subset
L(\tilde{\check{f}}_{2k}(\tilde{\phi}_{1}))$ if $0\leq k<mb$. In particular we
have $\overline{L(\tilde{\check{f}}_{2mb}(\tilde{\phi}))}\subset
L(\tilde{\phi}_{1})$

∎

##  4\. Proof of Theorem 1.1

###### Proof of Theorem 1.1.

Let $f\in\mathrm{Diff}^{1}_{\omega}(\mathbb{D})$ be an irrational pseudo-
rotation that coincides with a rotation on $\mathbb{S}$. We want to prove that
$\mathrm{Cal}(f)=0$. We extended our map to the whole plane and decompose it
in twist maps as explained is Section 3. In particular we get a natural
identity isotopy $I$ such that $\mathrm{rot}(I_{|\mathbb{S}})=\alpha$, where
$\alpha\in\mathbb{R}\setminus\mathbb{Q}$ and we want to prove that
$\widetilde{\mathrm{Cal}}(I)=\pi^{2}\alpha$. We use the results of Section 3,
keeping the same notations. If
$(a,b)\in\mathbb{Z}\times\mathbb{N}\setminus\\{0\\}$, satisfies
$a\in(b\alpha,b\beta)$, then $f_{|D_{a/b}}$ is a piecewise diffeomorphism of
class $C^{1}$. It is easy to see that $\widetilde{\mathrm{Cal}}(I_{|D_{a/b}})$
is well-defined, if one refers to the third definition given in the
introduction. It can be explicitly computed, we have

$\displaystyle\widetilde{\mathrm{Cal}}(I_{|D_{a/b}})$
$\displaystyle=\widetilde{\mathrm{Cal}}(I)+2\int_{1}^{1+a/b-\alpha}(\pi
r^{2})(\alpha+r-1)2\pi r\,dr$
$\displaystyle=\widetilde{\mathrm{Cal}}(I)+4\pi^{2}\int_{1}^{1+a/b-\alpha}r^{3}(\alpha+r-1)dr.$

Of course it holds that

$\lim_{{a/b}\to\alpha}\widetilde{\mathrm{Cal}}(I_{|D_{a/b}})=\widetilde{\mathrm{Cal}}(I_{|\mathbb{D}}).$

We have constructed a radial foliation $\mathcal{F}$ on $\Delta_{a}^{*}\subset
E_{b}$. We define ${\mathcal{F}}_{1}=q_{1}(\mathcal{F})$ and denote
$\tilde{\mathcal{F}}_{1}$ the lift of ${\mathcal{F}}_{1}$ to
$\tilde{D}_{a/b}$. We define $W_{a/b}=\\{(z,z^{\prime})\in
D_{a/b}^{*}\,|\,z\not=z^{\prime}\\}$. Note that the area of $D_{a/b}$ is

$A(a,b)=\pi(1+a/b-\alpha)^{2}$

and recall that the difference of the value of ${\bf h}$ between the points of
$\Sigma_{a}$ and $\\{0\\}$ is

$C(a,b)=\pi(a-b\alpha)\left(1+(a/b-\alpha)+{(a/b-\alpha)^{2}\over 3}\right).$

######  Lemma 4.1.

The following bound holds

$\mu_{\omega}\times\mu_{\omega}\left(\\{(z,z^{\prime})\in
W_{a/b}\,|\,\overline{\hat{\tau}}(z,z^{\prime},{\mathcal{F}}_{1},f^{-b}({\mathcal{F}}_{1}))\not=0\\}\right)\leq
2\,A(a,b)\,C(a,b).$

###### Proof.

Fix $z\in D_{a/b}^{*}$ and choose a lift $\tilde{z}\in\tilde{D}_{a,b}$. The
set

$O(\tilde{z})=\overline{L({\widetilde{\check{f}}_{2mq}{}^{-1}}(\tilde{\phi}_{\tilde{z}}))\cap
R(\tilde{\phi}_{\tilde{z}}})$

has measure $C(a,b)$ (for the lifted measure $\tilde{\mu}_{\omega}$) and
projects onto a subset $O(z)\subset D_{a/b}$ satisfying
$\mu_{\omega}(O(z))\leq C(a,b)$. Moreover, we have

$\tilde{z}^{\prime}\in\tilde{O}(\tilde{z})\Longleftrightarrow\tilde{\check{I}}(\tilde{z}^{\prime})\cap\tilde{\phi}_{\tilde{z}}\not=\emptyset,\enskip
z^{\prime}\in
O(z)\Longleftrightarrow\check{I}(z^{\prime})\cap\phi_{z}\not=\emptyset.$

Note that $\tilde{\check{I}}(\tilde{z})$ and
$\tilde{\check{I}}(\tilde{z}^{\prime})$ meet a common leaf if and only if
$\tilde{z}^{\prime}\in\tilde{O}(\tilde{z})$ or
$\tilde{z}\in\tilde{O}(\tilde{z}^{\prime})$ and that ${\check{I}}(z)$ and
${\check{I}}(z^{\prime})$ meet a common leaf if and only if $z^{\prime}\in
O(z)$ or $z\in O(z^{\prime}).$ Obviously, $I(z)$ and $I(z^{\prime})$ meet a
common leaf if
$\overline{\hat{\tau}}(z,z^{\prime},{\mathcal{F}}_{1},f^{-b}({\mathcal{F}}_{1}))\not=0$
and so, it holds that

$\displaystyle\enskip\enskip\enskip\mu_{\omega}\times\mu_{\omega}\left(\\{(z,z^{\prime})\in
W_{a/b}\,|\,\hat{\tau}(z,z^{\prime},{\mathcal{F}}_{1},f^{-b}({\mathcal{F}}_{1}))\not=0\\}\right)$
$\displaystyle\leq\int_{D_{a/b}^{*}}\mu_{\omega}(O(z))\,d\mu_{\omega}(z)+\int_{D_{a/b}^{*}}\mu_{\omega}(O(z^{\prime}))\,d\mu_{\omega}(z^{\prime})$
$\displaystyle\leq 2A(a,b)\,C(a,b).$

∎

We can be more precise:

######  Lemma 4.2.

The following bound holds

$\int_{W_{a/b}}\overline{\hat{\tau}}(z,z^{\prime},{\mathcal{F}}_{1},f^{-b}({\mathcal{F}}_{1}))\,d\mu_{\omega}(z)\,d\mu_{\omega}(z^{\prime})\leq
8mb\,A(a,b)\,C(a,b).$

###### Proof.

We keep the same notations as in Lemma 4.1. For every $(z,z^{\prime})\in
W_{a/b}$ define

$\nu(z,z^{\prime})=\\#\\{k\in\mathbb{Z}\,|\,\overline{\hat{\tau}}(\tilde{z},T^{-k}(\tilde{z}^{\prime}),{\mathcal{F}}_{1},f^{-b}({\mathcal{F}}_{1}))\not=0\\}$

where $\tilde{z},\tilde{z}^{\prime}$ are given lifts of $z,z^{\prime}$. Now,
for every $z\in D_{a/b}^{*}$, define a function
$\gamma_{z}:D_{a/b}^{*}\to\mathbb{N}$ assigning to every point $z^{\prime}\in
D_{a/b}^{*}$ the number of its lifts that are in $O(\tilde{z})$, where
$\tilde{z}$ is a given lift of $z$. Suppose that
$\gamma_{z}(z^{\prime})\not=0$, or equivalently that $z^{\prime}\in O(z)$.
Then, for every $l\in\\{1,\dots,\gamma_{z}(z^{\prime})\\}$ there exists a
unique lift $\tilde{z}^{\prime}\in\tilde{O}(\tilde{z})$ of $z^{\prime}$ such
that $\tilde{I}(\tilde{z}^{\prime})$ meets the leaves
$T^{r}(\tilde{\phi}_{\tilde{z}})$, $0\leq r<l$. Note also that

$\int_{D_{a/b}^{*}}\gamma_{z}(z^{\prime})\,d\mu_{\omega}(z^{\prime})=\tilde{\mu}_{\omega}(\tilde{O}(\tilde{z}))=C(a,b)$

and that

$\nu(z,z^{\prime})\leq\gamma_{z}(z^{\prime})+\gamma_{z^{\prime}}(z).$

To conclude, we refer to Corollary 3.13 that says that for every
$(\tilde{z},\tilde{z}^{\prime})\in W_{a/b}$, it holds that
$|\hat{\tau}(\tilde{z},\tilde{z}^{\prime},{\mathcal{F}}_{1},f^{-b}({\mathcal{F}}_{1}))|\leq
4mb$. ∎

Note also that for every couple $(a,b)$ defined as above, it holds that
$\tilde{\check{f}}_{2mb}=(\tilde{f}_{|\tilde{D}_{a/b}})^{b}-T^{a}$ (or
equivalently that $[\check{I}]=[I]_{|D_{a/b}}^{b}[T_{-a}]_{|D_{a/b}}$). Indeed
both maps lift $f^{b}$ and coincide on $S_{a/b}$. If $\phi$ is a leaf of
${\mathcal{F}}_{1}$, then $m_{\tilde{\check{f}}_{2mb},\phi}$ is non negative
and

$\int_{D^{*}_{a/b}}m_{\tilde{\check{f}}_{2mb},\phi}\,d\mu_{\omega}=C(a,b).$

We deduce that $\mu_{\omega}$ has a rotation vector (which was obviously
known) and that

$\mathrm{rot}_{\tilde{\check{f}}_{2mb}}(\mu_{\omega})=\int_{D^{*}_{a/b}}m_{\tilde{\check{f}}_{2mb},\phi}\,d\mu_{\omega}=C(a,b).$

We deduce from Lemma 4.2 that $\lambda_{f^{b},{\mathcal{F}}_{1}}$ is
integrable on $W_{a/b}$ for the measure $\mu_{\omega}\times\mu_{\omega}$ and
that

$\left|\int_{W_{a,b}}\lambda_{f^{b},{\mathcal{F}}_{1}}\,d\mu_{\omega}(z)\,d\mu_{\omega}(z^{\prime})\right|\leq
8mb\,A(a,b)\,C(a,b).$

According to the results of Section 2, we deduce that $\mu_{\omega}$ has a
self linking number for $\tilde{\check{f}}_{2mb}$ and that

$\left|\mathrm{link}_{\tilde{\check{f}}_{2mb}}(\mu_{\omega})-\int_{D_{a/b}^{*}}\mathrm{rot}_{\tilde{\check{f}}_{2mb}}(z)\,d\mu_{\omega}(z)\,d\mu_{\omega}(z^{\prime})\right|\leq
8mb\,A(a,b)\,C(a,b).$

But we know that $\tilde{\check{f}}_{2mb}=\check{f}^{b}\circ T^{-a}$ and so we
obtain

$\left|b\,\mathrm{link}_{\tilde{f}_{|\tilde{D}_{a/b}}}(\mu_{\omega})-aA(a,b)^{2}-A(a,b)\int_{D_{a/b}^{*}}\mathrm{rot}_{\tilde{\check{f}}_{2mq}}(z)\,d\mu_{\omega}(z)\right|\leq
8mb\,A(a,b)\,C(a,b),$

which can be written

$\left|\mathrm{link}_{\tilde{f}_{|\tilde{D}_{a/b}}}(\mu_{\omega})-{a\over
b}A(a,b)^{2}-{A(a,b)\,C(a,b)\over b}\right|\leq 8mA(a,b)C(a,b).$

One can find a sequence $(a_{n},b_{n})_{n\geq 0}$ such that
$\lim_{n\to+\infty}a_{n}-b_{n}\alpha=0$. Writing

$\left|\mathrm{link}_{\tilde{f}_{|\tilde{D}_{a_{n}/b_{n}}}}(\mu_{\omega})-{a_{n}\over
b_{n}}A(a_{n},b_{n})^{2}-{A(a_{n},b_{n})\,C(a_{n},b_{n})\over b}\right|\leq
8mA^{2}(a_{n},b_{n})\,C(a_{n},b_{n})$

and letting $n$ tend to $+\infty$, one obtains

$\mathrm{link}_{\tilde{f}_{|\tilde{\mathbb{D}}}}(\mu_{\omega})-\pi^{2}\alpha=0.$

But we know that
$\mathrm{link}_{\tilde{f}_{|\tilde{\mathbb{D}}}}(\mu_{\omega})=\widetilde{\mathrm{Cal}}(I)$,
so we can conclude. ∎

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