# The asymptotic mean action and the asymptotic linking number for pseudo-
rotations

David Bechara Senior  David Bechara Senior, RWTH Aachen, Jakobstrasse 2,
Aachen 52064, Germany<EMAIL_ADDRESS>,  Patrice Le Calvez
Patrice Le Calvez, Institut de Mathématiques de Jussieu-Paris Rive Gauche,
IMJ-PRG, Sorbonne Université, Université Paris-Cité, CNRS, F-75005, Paris,
France<EMAIL_ADDRESS>and  Abror Pirnapasov  Abror Pirnapasov,
University of Maryland, 4176 Campus Drive - William E. Kirwan Hall College
Park, MD 20742<EMAIL_ADDRESS>
###### Abstract.

We establish that the asymptotic mean action and the asymptotic linking number
of irrational pseudo-rotations remain well-defined everywhere and constant for
every $C^{1}$ irrational pseudo-rotation that behaves as a rotation on the
boundary. As a consequence, we demonstrate that the isotopy of irrational
pseudo-rotations with a positive rotation number is a right-handed isotopy in
the sense of Ghys.

##  1\. Introduction and Main Results

###  1.1. The asymptotic mean action of an irrational pseudo-rotation

We consider $\mathbb{D}\subset\mathbbm{R}^{2}$ the unit closed two disk
centered at the origin with polar coordinates $(r,\theta)$. Let $f$ be an
area-preserving smooth diffeomorphism of $\mathbb{D}$ isotopic to
$\mathrm{id}$ via an area-preserving isotopy
$\tilde{f}=\\{f_{t}\\}_{t\in[0,1]}$. Let $\beta$ be a primitive of the
standard symplectic form $\omega=\frac{r}{\pi}dr\wedge d\theta.$

The action of $f$ with respect to $\beta$ and the given isotopy is the unique
smooth function $a_{\tilde{f}}(x):\mathbb{D}\to\mathbb{R}$ satisfying:

(1.1) $\begin{cases}da_{\tilde{f}}=f^{*}\beta-\beta\\\
a_{\tilde{f}}(x)=\int_{\tilde{f}(x)}\beta\qquad\text{for every}\
x\in\partial\mathbb{D},\end{cases}$

where $\tilde{f}(x)$ is the path $t\mapsto f_{t}(x)$.

We define the corresponding Calabi invariant by

$\mathrm{CAL}(\tilde{f}):=\int_{\mathbb{D}}a_{\tilde{f}}\ \omega.$

The quantity $a^{\infty}_{\tilde{f}}(x)$ is called the asymptotic mean action
of $x$ with respect to $f$, whenever following limit exists

$a^{\infty}_{\tilde{f}}(x):=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^{n-1}a_{\tilde{f}}({f}^{i}(x)).$

A straightforward application of Birkhoffs’ ergodic theorem implies that
$a^{\infty}_{\tilde{f}}(x)$ is well defined for Lebesgue a.e.
$x\in\mathbb{D}$, furthermore it is independant of the primitive $\beta$ of
$\omega$. If $x$ is periodic or $x\in\partial\mathbb{D}$, then
$a^{\infty}_{\tilde{f}}(x)$ is well defined. We will denote $\mathrm{Per}(f)$
the set of periodic points of $f$. Moreover $a^{\infty}_{\tilde{f}}(x)$ is
equal to $\rho(\tilde{f}\big{|}_{\partial\mathbb{D}})$ for every
$x\in\partial\mathbb{D}$ where $\rho(\tilde{f}\big{|}_{\partial\mathbb{D}})$
is the rotation number of the isotopy
$\tilde{f}\big{|}_{\partial\mathbb{D}}=\\{f_{t}\big{|}_{\partial\mathbb{D}}\\}_{t\in[0,1]}$
restricted to the boundary of the disc $\partial\mathbb{D}$.

In [Hut16] Hutchings discovered a very interesting relation between the Calabi
invariant and the asymptotic mean action of periodic points of smooth area-
preserving disk maps which are a rotation on the boundary.

######  Theorem 1.1.

[Hut16] Let $f$ be a smooth area-preserving diffeomorphism of $\mathbb{D}$
that is a rigid rotation near the boundary and isotopic to $\mathrm{id}$ via
$\tilde{f}$. If

(1.2)
$\displaystyle\mathrm{CAL}(\tilde{f})<\rho(\tilde{f}\big{|}_{\partial\mathbb{D}}),$

then

(1.3)
$\displaystyle\inf\left\\{a^{\infty}_{\tilde{f}}(x)\,|\,x\in\mathrm{Per}(f)\right\\}\leq\mathrm{CAL}(\tilde{f}).$

Weiler [Wei21] established a similar result for area-preserving
diffeomorphisms of the closed annulus. The third author [Pir21] proved Theorem
1.1 without the assumption of the map being a rotation near the boundary by
using preliminary extensions. The third author and Prasad [PP22] proved the
validity of inequality (1.3) for generic Hamiltonian maps of surfaces with
boundaries by proving the generic equidistribution theorem for Hamiltonian
diffeomorphisms of surfaces with boundary. Recently, in [NW23], Nelson and
Weiler established a nongeneric version of Theorem 1.1 on higher-genus
surfaces but with stronger assumptions.

For the particular case of irrational pseudo-rotations the third author showed
that the Calabi invariant of smooth irrational pseudo-rotations is equal to
the rotation number(cf.[Pir21]), this was proven by using the general case of
Theorem 1.1. This result was subsequently reaffirmed by the second author
using finite-dimensional methods for $C^{1}$ irrational pseudo-rotations which
are conjugated to a rotation on the boundary.

The following question, posed in [BSHS21] in the context of three-dimensional
Reeb dynamics, addresses the relationship between the asymptotic mean action
of periodic points and the asymptotic mean action of all points:

######  Question 1.2.

Does the following equality hold for all area-preserving disk maps?

(1.4) $\overline{\big{\\{}a^{\infty}_{\tilde{f}}(x):\
x\in\mathbb{D}\big{\\}}}=\overline{\big{\\{}a^{\infty}_{\tilde{f}}(x):\
x\in\mathrm{Per}(f)\big{\\}}}$

Equality (1.4) implies Theorem 1.1 because the set
$\overline{\big{\\{}a^{\infty}_{\tilde{f}}(x):\ x\in\mathbb{D}\big{\\}}}$
contains the Calabi invariant. In his thesis [BS23], the first author showed
that the equality (1.4) holds for all diffeomorphisms generated by an
autonomous Hamiltonian of the disc.

######  Definition 1.3.

An area-preserving diffeomorphism $f:\mathbb{D}\to\mathbb{D}$ is called an
irrational pseudo-rotation if $\\#\mathrm{Per}(f)=1$.

In the following theorem, we affirmatively answer question 1.2 for $C^{1}$
irrational pseudo-rotations, which are conjugate to a rotation when restricted
to the boundary.

######  Theorem 1.4.

If $f:\mathbb{D}\to\mathbb{D}$ is a $C^{1}$ irrational pseudo-rotation that is
$C^{1}$ conjugate to a rotation on the boundary, then the sequence of
functions

$\frac{\sum_{i=0}^{n-1}a_{\tilde{f}}({f}^{i}(x))}{n}$

uniformly converges to $\rho(\tilde{f})$. In particular, the asymptotic mean
action $a^{\infty}_{\tilde{f}}(x)={\rho(\tilde{f})}$, where $\rho(\tilde{f})$
is the Poincare rotation number of the isotopy restricted to the boundary of
the disk.

######  Remark 1.5.

We prove Theorem 1.4 for irrational pseudo-rotations that fix $0$ and coincide
with a rotation on the boundary. This is merely a technical condition because
if $x\in\mathbb{D}$ is a fixed point of an irrational pseudo-rotation $f$,
then there exists a compactly supported Hamiltonian $g$ such that $g(0)=x$.
This implies that $g\circ f\circ g^{-1}$ is also an irrational pseudo-rotation
and $0$ is its fixed point. We also note that every $C^{1}$ diffeomorphism of
$\partial{\mathbb{D}}$ can be extended as a $C^{1}$ area preserving
diffeomorphism of $\mathbb{D}$. This is the reason why we can suppose that $f$
is $C^{1}$ conjugate to a rotation on $\partial\mathbb{D}$ in Theorem 1.3.

In his thesis [BS23], the first author proved Theorem 1.4 for $C^{0}$-rigid
irrational pseudo-rotation using Joly’s winding number bound [Lemma 5.3
in[Jol21]] for $C^{0}$-rigid irrational pseudo-rotations. This statement also
holds for minimally ergodic irrational pseudo-rotations [FK04], which are
irrational pseudo-rotations having only three ergodic invariant measures: the
Dirac measure at the fixed point, the Lebesgue measure on the boundary, and
the Lebesgue measure on the disc. Together with the Calabi identity result in
[Pir21], minimally ergodicity implies that the integral of the action function
with respect to any invariant probability measure is equal to the rotation
number. This implication leads to the conclusion of Theorem 1.4 using a
Krylov-Bogolyubov type argument. In the proof of Theorem 1.4, we also prove
that the integral of the action function with respect to any invariant measure
is equal to the rotation number using Le Calvez’s finite-dimensional methods.

###  1.2. The asymptotic orbit linking number of an irrational pseudo-rotation

Let $f\in\text{Diff}^{1}(\mathbb{D},\omega_{0})$ and let
$\tilde{f}=\\{f_{t}\\}_{t\in[0,1]}$ be a Hamiltonian isotopy such that
$f^{0}=\mathrm{id}$ and $f^{1}=f$. Here,
$\text{Diff}^{1}(\mathbb{D},\omega_{0})$ is the set of $C^{1}$ diffeomorphisms
of $\mathbb{D}$ which preserves $\omega_{0}$.

######  Definition 1.6.

The winding number $W_{\tilde{f}}(x,y)$ of a pair of distinct points
$x,y\in\mathbb{D}$ is the real number

$W_{\tilde{f}}(x,y):=\frac{{\theta}(1)-{\theta}(0)}{2\pi},$

where $\theta:[0,1]\rightarrow\mathbbm{R}$ is a continuous function such that

(1.5) $\frac{f_{t}(y)-f_{t}(x)}{||f_{t}(y)-f_{t}(x)||}=e^{i\theta(t)}.$

The value of $W_{\tilde{f}}(x,y)$ does not depend on the isotopy joining $f$
to the identity if isotopies have the same rotation number on the boundary.
This holds because any two paths joining $f$ to the identity are homotopic if
the isotopies have the same rotation number when restricted to the boundary,
which implies that the winding number is the same. The map $W_{\tilde{f}}$ is
smooth and bounded on $(\mathbb{D}\times\mathbb{D})\setminus\Delta$, where
$\Delta$ denotes the diagonal $\\{(x,x)|x\in\mathbb{D}\\}$ in
$\mathbb{D}\times\mathbb{D}$. To see this, let $K$ be indeed the blow-up of
the diagonal in $\mathbb{D}\times\mathbb{D}$. In more concrete terms, $K$ is
the compact set formed by replacing $\Delta$ with the unit tangent bundle
$T^{1}\mathbb{D}$). Of course $\mathbb{D}\times\mathbb{D}\setminus\Delta$
naturally embeds into $K$ as a dense open set. To be more precise a sequence
$(x_{n},y_{n})\subset\mathbb{D}\times\mathbb{D}\setminus\Delta$ converges to a
point $(x,\xi)\in T^{1}\mathbb{D}\simeq\mathbb{D}\times S^{1}$ whenever
$x_{n},y_{n}\rightarrow x$ and if the sequence of oriented vectors
$\frac{y_{n}-x_{n}}{||y_{n}-x_{n}||}=e^{i\theta_{n}}\in\mathbb{S}^{1}$
converges to $\xi$. We assert that if $f$ is of class $C^{1}$, the function
$W_{\tilde{f}}$ extends to $K$ as a continuous function, and in particular,
$W_{\tilde{f}}$ is bounded. This is, in fact, an extension of the definition
of $W_{\tilde{f}}$ to element the form $(x,\xi)\in T^{1}\mathbb{D}$. Let
$(f_{t})$ be the isotopy between the identity and $f$ among the
$C^{1}$-diffeomorphisms of $\text{Diff}^{1}(\mathbb{D},\omega_{0})$. We then
define $W_{\tilde{f}}(x,\xi)$ as the variation in the angle of the vector
$df_{t}(\xi)\in\mathbbm{R}^{2}$, images of $\xi$ under the differential
$df_{t}$, as $t$ varies from $0$ to $1$. This indeed defines the desired
continuous extension of $W_{\tilde{f}}$ to $K$.

Define the set $\mathbb{O}(\mathbb{D})\subset\mathbb{D}\times\mathbb{D}$ as
follows:

$\mathbb{O}(\mathbb{D})=\\{(x,y)\in{\mathbb{D}}\times{\mathbb{D}}\,|\enskip
f^{i}(x)\not=f^{j}(y)\enskip\mathrm{for\enskip
every\enskip}(i,i^{\prime})\in\mathbb{Z}\times\mathbb{Z}\\}.$

######  Definition 1.7.

Suppose $(x,y)\in\mathbb{O}(\mathbb{D})$, the number

$L_{\tilde{f}}^{\infty}(x,y):=\lim_{n\to\infty}\frac{\sum_{i=0}^{n-1}\sum_{j=0}^{n-1}W_{\tilde{f}}(f^{i}(x),f^{j}(y))}{n^{2}}$

is called the asymptotic orbit linking number of $(x,y)$ if the limit exists.

In the next theorem, we prove that the asymptotic linking number exists and is
equal to the rotation number of the isotopy for every pair of points that are
not in the same orbit of the map.

######  Theorem 1.8.

If $f:\mathbb{D}\to\mathbb{D}$ is a $C^{1}$ irrational pseudo-rotation that is
$C^{1}$ conjugate to a rotation on the boundary and $\tilde{f}$ is an isotopy
from $\mathrm{id}$ to $f$, then the sequence

$\left(\frac{\sum_{i=0}^{n-1}\sum_{j=0}^{n-1}W_{\tilde{f}}(f^{i}(x),f^{j}(y))}{n^{2}}\right)_{n\geq
0}$

converges uniformly to $\rho(\tilde{f})$ for every
$(x,y)\in\mathbb{O}(\mathbb{D})$. In particular, the asymptotic mean linking
number $L_{\tilde{f}}^{\infty}(x,y)$ is defined for every
$(x,y)\in\mathbb{O}(\mathbb{D})$ and equal to $\rho(\tilde{f})$.

######  Definition 1.9.

The isotopy $\tilde{f}$ is called a right-handed isotopy, if

  * •

for every $(x,y)\in\mathbb{O}(\mathbb{D})$ if holds that

$\liminf_{n\to\infty}\frac{1}{n^{2}}\sum_{i=0}^{n-1}\sum_{j=0}^{n-1}W_{\tilde{f}}(f^{i}(x),f^{j}(y))>0.$

  * •

for every $x\in\mathrm{Per}(f)$ if holds that

$\liminf_{n\to\infty}\frac{1}{n^{2}}\left(\sum_{\begin{subarray}{c}i,j=0\\\
f^{i}(x)\neq
f^{j}(x)\end{subarray}}^{n-1}W_{\tilde{f}}(f^{i}(x),f^{j}(x))+\sum_{\begin{subarray}{c}i,j=0\\\
f^{i}(x)=f^{j}(x)\end{subarray}}^{n-1}W_{\tilde{f}}(f^{i}(x),df^{j}(\xi))\right)>0.$

######  Remark 1.10.

The isotopy $\tilde{f}$ is called left-handed if both limits are negative in
Definition 1.9.

In [Ghy09], Ghys introduced right-handed flows and demonstrated that any
finite collection of periodic orbits admits a global surface of section.
Subsequently, in [FH21], Florio and Hryniewicz provided a sufficient condition
for Reeb flows to be right-handed. Prasad [Pra22] showed that volume-
preserving right-handed flows are conformally equivalent to Reeb flows. In her
thesis [Lhu18], Lhuissier showed that definition 1.9 is equivalent to the
definition of right-handedness due to Ghys if one looks at the suspension of
the disk map as a flow (see [Lhu18] Theorem 7.3.2).

######  Corollary 1.11.

If $f:\mathbb{D}\to\mathbb{D}$ is a $C^{1}$ irrational pseudo-rotation that is
$C^{1}$ conjugate to a rotation on the boundary, and $\tilde{f}$ is an area-
preserving isotopy from the identity map $\mathrm{id}$ to $f$ such that the
rotation number of the isotopy is (negative) positive, then the isotopy
$\tilde{f}$ is (left-handed) right-handed.

Finally we proceed to prove Corollary 1.11 using Theorem 1.8.

###### Proof of Corollary 1.11.

It is enough to prove the theorem in the case were the isotopy is right-
handed. Assume that the rotation number of the isotopy is positive. By using
Theorem 1.8, we can conclude that for every $(x,y)\in\mathbb{O}(\mathbb{D})$,
the asymptotic orbit linking number is defined and equal to the rotation
number and by assumption, it is positive. That is

$L_{\tilde{f}}^{\infty}(x,y)=\lim_{n\to\infty}\frac{\sum_{i=0}^{n-1}\sum_{j=0}^{n-1}W_{\tilde{f}}(f^{i}(x),f^{j}(y))}{n^{2}}=\rho(\tilde{f})>0$

for every $(x,y)\in\mathbb{O}(\mathbb{D})$.

Now we check the second condition of Definition 1.9. Since the fixed point $0$
of $f$ is the only periodic point of $f$, the sum in the second condition
simplifies to the following sum:

$\liminf_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}W_{\tilde{f}}(0,df^{i}(\xi))>0.$

But, this is the linearized rotation number of $0$ through the isotopy. By the
classical result of Franks’ [Fra88] and also of the second author [LC01], the
linearized rotation number of the fixed point is also equal to
$\rho(\tilde{f})$ hence it is also positive.

This completes the proof of Theorem 1.11. ∎

Structure of the article: In Section 2, we recall basic notions relevant to
the article, such as weak convergence of probability measures, radial
foliations, and their properties. In Section 3, we introduce radial foliations
and isotopies for irrational pseudo-rotations whose study has been developed
by the second author in [LC23a]; then, we show that the measure between any
leaf of a radial foliation and its image is equal to the rotation number of
the map. By combining this with properties of the foliation, we show that the
integral of the winding number with respect to the product of any two
invariant probability measures is also equal to the rotation number. In
Section 4, we use the integral of the winding number together with a Krylov-
Bogolyubov argument to prove Theorem 1.4 and Theorem 1.8.

### Acknowledgments

AP would like to express his gratitude to Barney Bramham for a lot of
conversations and suggestions about asymptotic mean action. AP is also
grateful to Alberto Abbondandolo, Umberto Hryniewicz, Marco Mazzuchelli, and
Rohil Prasad for many stimulating conversations and suggestions. The work of
DBS is partially supported by the DFG SFB/TRR 191 “Symplectic Structures in
Geometry, Algebra and Dynamics”, Projektnummer 281071066-TRR 191. The work of
AP is partially supported by the DFG Walter Benjamin Fellowship, Projektnummer
518128580, and the ANR CoSyDy “Conformally Symplectic Dynamics beyond
symplectic dynamics” (ANR-CE40-0014).

##  2\. Preliminaries

In this section, we introduce necessary results in ergodic theory and also in
the theory of radial foliations specifically in the case of irrational pseudo-
rotations. In order to state several of the inequalities needed to prove the
main theorems, we require different quantities coming from measurements of
topological angles.

###  2.1. Invariant measures and pointwise ergodic theorem.

Assume $f:\mathbb{D}\to\mathbb{D}$ is a homeomorphism. For $x\in\mathbb{D}$,
let $\delta_{x}$ denote the Dirac measure supported at $x$. We will denote:

$\mu_{n}(x):=\frac{1}{n}\sum^{n-1}_{i=0}\delta_{f^{i}(x)}$

######  Proposition 2.1.

Let $(x_{k})_{k\geq 0}$ and $(y_{k})_{k\geq 0}$ be two sequences in
$\mathbb{D}$ and $(n_{k})_{k\geq 0}$ an increasing sequence of positive
integers. Then there exists a subsequence $(n_{k_{j}})_{j\geq 0}$ such that
the sequences of measures $(\mu_{n_{k_{j}}}(x_{k_{j}}))_{j\geq 0}$ and
$(\mu_{n_{k_{j}}}(y_{k_{j}}))_{j\geq 0}$ converge for the weak* topology and
their limits are invariant probability measures of $f$.

The proof of Proposition 2.1 is very similar to the proof of the Krylov-
Bogolyubov Theorem and is based on the compactness of the space of Borel
probability measures furnished with the weak∗ topology and the fact that every
limit point of a sequence $(\mu_{n_{k}}(x_{k}))_{k\geq 0}$ is invariant. The
next result is straightforward.

######  Proposition 2.2.

If sequences of probability measures $(\mu_{k})_{k\geq 0}$ and
$(\nu_{k})_{k\geq 0}$ converge for the weak∗ topology to $\nu$ and $\mu$, then
the sequence of probability measures $(\mu_{k}\times\nu_{k})_{k\geq 0}$
converges for the weak∗ topology to $\nu\times\mu$.

The following theorem is a direct implication of Templemann’s pointwise
ergodic theorem. For more information, please refer to Theorem 2.6 in Section
2.3.3 of [Sar09].

######  Theorem 2.3.

Assume that $\mu$ and $\nu$ are invariant probability measures of $f$. If
$\phi\in L^{1}(\mathbb{D}\times\mathbb{D},\mu\times\nu)$, then

$S_{n}\phi(z):=\frac{1}{n^{2}}\sum_{i=0}^{n-1}\sum_{j=0}^{n-1}\phi(f^{i}(x),f^{j}(y))$

converges for $(\mu\times\nu)$-almost every
$z=(x,y)\in\mathbb{D}\times\mathbb{D}$.

###  2.2. Properties of radial foliations and topological angles

Let us begin with some notations and definitions, specially regarding radial
foliations on a disk. Let $D:=\\{z\in\mathbbm{R}^{2}\,|\enskip\|z\|<r\\}$ be
an open disk and set $D^{*}=D\setminus\\{0\\}$ the punctured disk, lastly let
us write $\tilde{D}$ for the universal covering space of $D^{*}$. We will
consider the sets

$W=\\{(z,z^{\prime})\in D^{*}\times D^{*}\ |\ z\neq z^{\prime}\\}$

and

$\tilde{W}=\\{(\tilde{z},\tilde{z}^{\prime})\in\tilde{D}\times\tilde{D}\ |\
\tilde{z}\neq\tilde{z}^{\prime}\\}.$

######  Definition 2.4.

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

$\alpha(\phi)=\\{0\\},\qquad\omega(\phi)\subset\overline{D}\setminus
D\quad\text{(here the closure is taken in }\mathbb{R}^{2})$

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
$\mathscr{F}\in\mathfrak{F}$ and $f\in\mathrm{Homeo}_{*}(D^{*})$, then the
foliation $f(\mathscr{F})$, whose leaves are the images by $f$ of the leaves
of $\mathscr{F}$, 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 $\mathrm{Homeo}_{*,\mathscr{F}}(D^{*})$
of $\mathscr{F}\in\mathfrak{F}$ is a closed subgroup of
$\mathrm{Homeo}_{*}(D^{*})$. Consequently, the map

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

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
$\mathrm{Homeo}_{*}(D^{*})/\mathrm{Homeo}_{*,\mathscr{F}}(D^{*})$ by this
identification map. Note that this topology does not depend on $\mathscr{F}$.
It is well known that the fundamental groups of $\mathrm{Homeo}_{*}(D^{*})$
and $\mathrm{Homeo}_{*,\mathscr{F}}(D^{*})$ are infinite cyclic and that the
morphism
$i_{*}:\pi_{1}(\mathrm{Homeo}_{*,\mathscr{F}}(D^{*}),\mathrm{Id})\to\pi_{1}(\mathrm{Homeo}_{*}(D^{*}),\mathrm{Id})$
induced by the inclusion map
$i:\mathrm{Homeo}_{*,\mathscr{F}}(D^{*})\to\mathrm{Homeo}_{*}(D^{*})$ is
bijective. It implies that ${\mathfrak{F}}$ is simply connected. In the whole
article, when $\mathscr{F}$ is a radial foliation, the notation
$\tilde{\mathscr{F}}$ means the lift of $\mathscr{F}$ to the universal
covering space $\tilde{D}$. Note that $\tilde{\mathscr{F}}$ is a trivial
foliation. In particular, every leaf $\tilde{\phi}$ of $\tilde{\mathscr{F}}$
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 $\preceq_{\mathscr{F}}$ on the set of leaves of $\tilde{\mathscr{F}}$ as
follows:

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

We can also define a partial order $\leq_{{\mathscr{F}}}$ on $\tilde{D}$:
denote $\tilde{\phi}_{\tilde{z}}$ the leaf of $\tilde{\mathscr{F}}$ that
contains $\tilde{z}$ and write $\tilde{z}<_{{\mathscr{F}}}\tilde{z}^{\prime}$
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 $\tilde{\phi}_{\tilde{z}}$.

The above formalism of radial foliations allows us to work with a certain
“discretization of angles”, these ideas were introduced by the second author
in [LC23a] and further explained in [LC23b] with specific applications to
twist maps. For the purpose of this article we will introduce only the
necessary definitions without stepping into all of the many properties of this
technique. We now introduce topological angles, these serve as a bridge
between the measurement given by the winding number 1.5 and the information
captured by radial foliations.

Consider the natural projection

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

If $\mathbbm{Z}/4\mathbbm{Z}$ 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 $\mathbbm{Z}$ with the topology generated by the sets $2k+1$ and
$\\{2k+1,2k,2k+1\\}$, $k\in\mathbbm{Z}$, then $\pi$ is a covering map111This
topology is usually called the digital line topology on $\mathbbm{Z}$. It is
also known as Khalimsky topology. Note that both sets
$\mathbbm{Z}/4\mathbbm{Z}$ and $\mathbbm{Z}$ 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\mathbbm{Z}+(\\#\\{k,l\\}\cap 4\mathbbm{Z})/2&\text{ if $k<l$,}\\\
-\\#(l,k)\cap 4\mathbbm{Z}-(\\#\\{l,k\\}\cap 4\mathbbm{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 $\gamma:[s_{0},s_{1}]\to\mathbbm{Z}/4\mathbbm{Z}$ with
$\\{\dot{0}\\}$, if $\gamma$ is lifted to a path
$\hat{\gamma}:[s_{0},s_{1}]\to\mathbbm{Z}$ joining $k$ to $l$. For every
$(\tilde{z},\tilde{z}^{\prime})\in\tilde{W}$, we can define a continuous
function
$\dot{\theta}_{\tilde{z},\tilde{z}^{\prime}}:{\mathfrak{F}}\to\mathbbm{Z}/4\mathbbm{Z}$
as follows:

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

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

  * •

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

  * •

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

  * •

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

  * •

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

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

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

and

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

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

We can define a pseudo-distance on $\mathfrak{F}$, by writing
$d(\mathscr{F},\mathscr{F}^{\prime})=\sup_{(\tilde{z},\tilde{z}^{\prime})\in\tilde{W}}\tau(\tilde{z},\tilde{z}^{\prime},\mathscr{F},\mathscr{F^{\prime}})\in[0,+\infty]$.
We call it the winding distance.

####  2.2.1.  Topological angles in the annulus

Let $\mathscr{F}$ 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

$s\in[0,1]\mapsto\dot{\theta}_{\tilde{f}_{s}(\tilde{z}),\tilde{f}_{s}(\tilde{z}^{\prime})}(\mathscr{F})=\dot{\theta}_{\tilde{z},\tilde{z}^{\prime}}(f_{s}^{-1}(\mathscr{F}))\in\mathbbm{Z}/4\mathbbm{Z}$

can be lifted to a function

$s\in[0,1]\mapsto\theta_{\tilde{z},\tilde{z}^{\prime}}(f_{s}^{-1}(\mathscr{F}))\in\mathbbm{Z}$

and the difference between the value at $1$ and the value at $0$ of this last
map is nothing but
$\tau(\tilde{z},\tilde{z},{\mathscr{F}},f^{-1}(\mathscr{F}))$. 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

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

and so the functions

$s\mapsto\theta_{\tilde{z},T^{k}(\tilde{z}^{\prime})}(f_{s}^{-1}(\mathscr{F})),\enskip
s\mapsto\theta_{\tilde{z},T^{-k}(\tilde{z}^{\prime})}(f_{s}^{-1}(\mathscr{F}))$

are constant, which means that

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

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

$\displaystyle\overline{\tau}(z,z^{\prime},\mathscr{F},{\mathscr{F}}^{\prime})$
$\displaystyle:=\sum_{k\in\mathbbm{Z}}|\tau(\tilde{z},T^{k}(\tilde{z}^{\prime}),\mathscr{F},{\mathscr{F}}^{\prime})|,$
$\displaystyle\tau(z,z^{\prime},\mathscr{F},{\mathscr{F}}^{\prime})$
$\displaystyle:=\sum_{k\in\mathbbm{Z}}\tau(\tilde{z},T^{k}(\tilde{z}^{\prime}),\mathscr{F},{\mathscr{F}}^{\prime}),$
$\displaystyle\lambda(z,z^{\prime},\mathscr{F},{\mathscr{F}}^{\prime})$
$\displaystyle:=\sum_{k\in\mathbbm{Z}}\lambda(\tilde{z},T^{k}(\tilde{z}^{\prime}),\mathscr{F},{\mathscr{F}}^{\prime}),$

each sum being independent of the choice of lifts
$\tilde{z},\tilde{z}^{\prime}$ of $z,z^{\prime}$. These identities implies the
following inequality

(2.1)
$|\lambda(z,z^{\prime},\mathscr{F},\mathscr{F}^{\prime})|\leq\overline{\tau}(z,z^{\prime},\mathscr{F},\mathscr{F}^{\prime}).$

Now let us define 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
$m_{\tilde{f},\phi}:D^{*}\to\mathbbm{Z}$ as follows: we choose a lift
$\tilde{\phi}$ of $\phi$, then for every $z\in D$, 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
$k\in\mathbbm{Z}$.

Denote:

$\lambda_{f,\mathscr{F}}(z,z^{\prime}):=\lambda(z,z^{\prime},\mathscr{F},f^{-1}(\mathscr{F})),$
(2.2)
$\Lambda_{\tilde{f},\mathscr{F},\phi}(z,z^{\prime}):=\lambda_{f,\mathscr{F}}(z,z^{\prime})+m_{\tilde{f},\phi}(z).$

######  Definition 2.5.

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

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

######  Remark 2.6.

The notion $\mathrm{rot}_{\tilde{f}}(z)$ does not depend neither on the choice
of the ray, nor on the choice of the foliation. Both of these claims follow
from the fact that for every compact set $K\subset D^{*}$ the difference in
$m_{\tilde{f},\phi}(z)$ is bounded when changing either the ray of the
foliation or the foliation itself (see Lemma 2.5 in [LC23a]).

It is worth noticing the following identities, which directly follow from the
definition:

(2.3)
$\displaystyle\sum_{i=0}^{n-1}m_{\tilde{f},\phi}(f^{i}(z))=m_{\tilde{f}^{n},\phi}(z),$
(2.4)
$\displaystyle\sum_{i=0}^{n-1}\lambda_{f,\mathscr{F}}(f^{i}(z))=\lambda_{f^{n},\mathscr{F}}(z),$
(2.5)
$\displaystyle\sum_{i=0}^{n-1}\Lambda_{\tilde{f},\mathscr{F},\phi}(f^{i}(z))=\Lambda_{\tilde{f}^{n},\mathscr{F},\phi}(z).$

In the next proposition, we consider the Euclidean radial foliation
$\mathscr{F}_{*}$ on $D^{*}$, whose leaves are the paths

$(0,1)\ni t\to te^{2i\pi\alpha},\ \ \alpha\in[0,1).$

The second author proved the following properties in [LC23a].

######  Proposition 2.7.

Suppose that $(z,z^{\prime})\in W$ and $\phi\in\mathscr{F}_{*}$. The following
inequalities hold.

  1. (1)

$|m_{\tilde{f}^{n},\phi}(z)-W_{\tilde{f}^{n}}(0,z)|\leq 1$,

  2. (2)

$|\Lambda_{\tilde{f}^{n},\mathscr{F}_{*},\phi}(z,z^{\prime})-W_{\tilde{f}^{n}}(z,z^{\prime})|\leq
2$.

where the function $W_{\tilde{f}^{n}}$ is defined on $D\times D$ with the same
formula as defined in Definition 1.6 on $\mathbb{D}\times\mathbb{D}$.

######  Definition 2.8.

Fix $f\in\mathrm{Homeo}_{*}(D^{*})$ and a lift $\tilde{f}$ to $\tilde{D}$.
Consider $\mu$ an $f$-invariant probability measure on $\mathbb{D}^{*}$,we 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 the function
$\mathrm{rot}_{\tilde{f}}$ is $\mu$-integrable. In this case we set

$\mathrm{rot}_{\tilde{f}}(\mu):=\int_{{D}^{*}}\mathrm{rot}_{\tilde{f}}\leavevmode\nobreak\
d\mu.$

The following theorem is an application of property (1) of Proposition 2.7 and
equality (2.3), which tells about the rotation number of the invariant
measure.

######  Theorem 2.9.

Suppose that $f\in\mathrm{Diff}^{1}({D})$ fixes $0$. Let $I$ be an identity
isotopy of $f$ and let $\tilde{f}$ be the lift of $f\big{|}_{{D}^{*}}$ to
$\tilde{D}^{*}$ naturally defined by $\tilde{I}$. If
$\mathscr{F}\in\mathfrak{F}$ is a radial foliation and $\phi$ is a leaf of
$\mathscr{F}$ such that $m_{\tilde{f},\phi}$ is $\mu$-integrable, for some
invariant probability measure $\mu$ such that $\mu(\\{0\\})=0$, then it holds
that

(2.6)
$\mathrm{rot}_{\tilde{f}}(\mu)=\int_{{D}^{*}}W_{\tilde{f}}(0,z)\leavevmode\nobreak\
d\mu(z)=\int_{{D}^{*}}m_{\tilde{f},\phi}(z)\leavevmode\nobreak\ d\mu(z).$

The following theorem is a generalization of Theorem [LC23a]. The proof uses
inequality (2) of Proposition 2.7.

######  Theorem 2.10.

Suppose that $f\in\mathrm{Diff}^{1}({D})$ fixes $0$. Let $I$ be an identity
isotopy of $f$ and let $\tilde{f}$ be the lift of $f\big{|}_{{D}^{*}}$
naturally defined by $\tilde{I}$. If $\mathscr{F}\in\mathfrak{F}$ is a radial
foliation and $\phi$ is a leaf of $\mathscr{F}$ such that $m_{\tilde{f},\phi}$
is $\eta_{1}$ and $\eta_{2}$-integrable and $\lambda_{f,\mathscr{F}}$ is
$\eta$-integrable, then it holds that

(2.7) $\int_{W}W_{\tilde{f}}(z,z^{\prime})\leavevmode\nobreak\
d\eta=\int_{W}\Lambda_{\tilde{f},\mathscr{F},\phi}(z,z^{\prime})\leavevmode\nobreak\
d\eta.$

Here, $\eta$ is an invariant probability measure on $D\times D$, and
$\eta_{1}$ and $\eta_{2}$ are the projections of $\eta$ onto the first and
second factors, respectively such that $\eta(\Delta)=0$ and
$\eta_{1}(\\{0\\})=\eta_{2}(\\{0\\})=0$.

###### Proof.

By the assumptions of the theorem and Birkhoff’s theorem, the sequence
$\frac{m_{\tilde{f}^{n},\phi}(z)}{n}$ converges $\eta_{1}$-almost everywhere
and also $\eta_{2}$-almost everywhere. Moreover, the sequence

$\frac{\lambda_{f^{n},\mathscr{F}}(z,z^{\prime})}{n}$

converges $\eta$-almost everywhere on $W$. Furthermore, we deduce that the
sequence

$\frac{\Lambda_{\tilde{f}^{n},\mathscr{F},\phi}(z,z^{\prime})}{n}$

converges $\eta$-almost everywhere. Consequently, by inequality (2) of
Proposition 2.7, the same holds for the sequence

$\frac{W_{\tilde{f}^{n}}(z,z^{\prime})}{n}$

which converges to the same value as

$\frac{\Lambda_{\tilde{f}^{n},\mathscr{F},\phi}(z,z^{\prime})}{n}.$

Again, by Birkhoff’s theorem, we deduce the equality (2.7). ∎

######  Remark 2.11.

Recall that integrals $\int_{W}W_{\tilde{f}}(z,z^{\prime})\leavevmode\nobreak\
d\eta$ and
$\int_{W}\Lambda_{\tilde{f},\mathscr{F},\phi}(z,z^{\prime})\leavevmode\nobreak\
d\eta$ depend only on the homotopy class of the isotopy with fixed ends.

The following theorem is an implication of Franks result [Fra88]. We could
also deduce it from the more general result of the second author [LC01].

######  Theorem 2.12.

If $f:\mathbb{D}\to\mathbb{D}$ is an irrational pseudo-rotation on
$\mathbb{D}$, then for any invariant probability measure $\mu$ such that
$\mu(\\{0\\})=0$ we have

(2.8) $\int_{\mathbb{D}}W_{\tilde{f}}(0,z)\leavevmode\nobreak\
d\mu(z)=\rho(\tilde{f}).$

Where $\rho(\tilde{f})$ is the rotation number of the irrational pseudo-
rotation $f$.

##  3\. Radial foliations for irrational pseudo-rotations and their properties

Let $f$ be an irrational pseudo rotation and $\tilde{f}$ an identity isotopy
of $f$. If $f$ is extended to the whole plane, the isotopy $\tilde{f}$ defines
a natural lift of $f_{{\mathbb{R}}^{2}\setminus\\{0\\}}$ to the universal
covering space of ${\mathbb{R}^{2}\setminus\\{0\\}}$, also denoted
$\tilde{f}$. If $T$ is the natural generator of the group of covering
transformations, every map $\tilde{f}^{b}\circ T^{-a}$,
$b\in\mathbb{N}\setminus\\{0\\}$, $a\in\mathbb{Z}$ is associated to a certain
identity isotopy of $f^{b}$ denoted $\tilde{f}^{b}\circ T^{-a}$.

In this section, we will explain how after suitably extending $f$ to the whole
plane, we can construct for every couple $(a,b)$ an invariant disk
$\mathbb{D}_{a/b}$ containing $\mathbb{D}$ and a radial foliation
$\mathscr{F}$ on $\mathbb{D}_{a/b}$ such that:

  * -

the winding distance between $\mathscr{F}$ and $f^{-b}(\mathscr{F})$ is
bounded by $Mb$, where $M$ depends only on $f$,

  * -

every leaf of the lifted foliation $\mathscr{F}$ to the universal covering
space of $\mathbb{D}_{a/b}\setminus\\{0\\}$ is sent on its right by
$\tilde{f}^{b}\circ T^{-a}$.

We will use the fact that the rotation number of a probability measure $\mu$
invariant by the original map $f$ is equal to the measure of the strip between
a leaf of $\tilde{\mathscr{F}}$ and its image by $\tilde{f}^{b}\circ T^{-a}$
and the bound above concerning foliations to give an explicit upper bound of
the integral of the winding function of $\tilde{f}^{b}\circ T^{-a}$ with
respect to the product of any two invariant probability measures. Using this
upper bound and using a sequence of convergents of $\tilde{\alpha}$, we will
deduce that the measure of the winding function of $\tilde{f}$ with respect to
the product of these two invariant probability measures is equal to
$\rho(\tilde{f})$.

Assume $f$ is a $C^{1}$ is an irrational pseudo-rotation of $\mathbb{D}$ that
fixes the center $0=(0,0)$ of $\mathbb{D}$ and coincides with a rotation
$R_{\alpha}$ on $\partial\mathbb{D}$. Fix $\beta>\alpha$. We can extend our
map to a homeomorphism of the whole plane (also denoted $f$) such that:

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

We suppose moreover than $\beta\notin\mathbb{Q}$ 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 $(b\alpha,b\beta)\cap\mathbb{Z}\neq\emptyset$, the set of periodic points
of period $b$ can be written $\cup_{\alpha<a/b<\beta}S_{a/b}$, where $S_{a/b}$
is the circle of center $0$ and radius $1+a/b-\alpha$.

We denote the disk centered at $0$ and with radius disk $1+a/b-\alpha$ by
$\mathbb{D}_{a/b}$.

Using generating functions and the decomposition of $f$ in $m$ maps “close to
the identity”, $m\geq 1$, the second author constructed a radial foliation
$\mathscr{F}_{1}$ on $\mathbb{D}_{a/b}^{*}=\mathbb{D}_{a/b}\setminus\\{0\\}$
and an isotopy ${I}=({f}_{s})_{s\in[0,mb]}$ from $\mathrm{id}$ to $f^{b}$ for
every $b$ such that $f^{b}|_{S_{a/b}}=\mathrm{id}$ and $f^{b}(z)\neq z$ if
$1\leq|z|<1+a/b-\alpha$. The isotopy ${I}$ fixes each point of $S_{a/b}$. We
do not provide the construction in detail, but instead utilize the properties
of the foliation and the isotopy. For more details, please refer to [LC23a].

######  Proposition 3.1.

[LC23a, Corollary 3.13] For every $z,z^{\prime}\in\mathbb{D}^{*}_{a/b}$ the
following holds:

${\tau}(\tilde{z},\tilde{z}^{\prime},\mathscr{F}_{1},f^{-b}(\mathscr{F}_{1}))\leq
4mb.$

######  Proposition 3.2.

[LC23a, Proposition 3.14] If $\tilde{{I}}=(\tilde{{f}}_{s})_{s\in[0,2mb]}$ is
the identity isotopy on the universal covering space
$\tilde{\mathbb{D}}_{a/b}$ of $\mathbb{D}_{a/b}^{*}$ that lifts
$({f}_{s}\big{|}_{\mathbb{D}^{*}_{a/b}})_{s\in[0,2mb]}$, then every leaf
$\tilde{\phi}$ of $\tilde{\mathscr{F}_{1}}$ is a Brouwer line of
$\tilde{{f}}_{2mb}$. That is
$\overline{R(\tilde{{f}}_{2mb}(\tilde{\phi}_{\tilde{z}}))}\subset
R(\tilde{\phi}_{\tilde{z}})$ for every $\tilde{z}\in\mathbb{D}^{*}_{a/b}$.

We will prove now the following result. Such kind of result was established in
[LC23a], but with the Lebesgue measure on the whole disk $D_{a/b}$.

######  Lemma 3.3.

We fix $z\in\mathbb{D}^{*}_{a/b}$ and choose a lift
$\tilde{z}\in\widetilde{\mathbb{D}}_{a/b}$. Now we define

$O(\tilde{z}):=\overline{R(\tilde{{f}}_{2mb}^{-1}(\tilde{\phi}_{\tilde{z}}))}\cap
L(\tilde{\phi}_{\tilde{z}}).$

For every invariant probability measure $\mu$ such that $\mu(\\{0\\})=0$ and
$\mu(\mathbb{D})=1$, we have

(3.1) $\tilde{\mu}(O(\tilde{z}))=a-b\alpha.$

Here, the measure $\tilde{\mu}$ is a lift of $\mu$.

######  Remark 3.4.

The number $\mu(O(\tilde{z}))$ is independent of $z$ and its lift; it is an
invariant of the map, meaning it is independent of the isotopy with fixed
ends, and of the invariant measure. In the proof, we show that it is also
independent of the invariant probability measure.

###### Proof.

Without loss of generality, we assume $\mu$ is an invariant probability
measure of $f:\mathbb{R}^{2}\to\mathbb{R}^{2}$ that satisfies the condition
$\mu(\\{0\\})=0$ and $\mu(\mathbb{D})=1$, The set $\tilde{\mathbb{D}}_{a/b}$
has the following form:

$\tilde{\mathbb{D}}_{a/b}=\mathbb{R}\times(0,1-a/b+\alpha].$

In this case the fundamental covering automorphism $T$ can be explicitely
written as

$T(x,y):=(x+2\pi,y).$

Finally define the projection map

$p_{1}:\tilde{\mathbb{D}}_{a/b}\to\mathbb{R}\qquad(x,y)\mapsto p_{1}(x,y)=x.$

By Theorem 2.12, $\mu$ has a rotation number $\alpha$ for the lift $\tilde{f}$
to $\tilde{\mathbb{D}}_{a/b}$ given by the decomposition of $f$. Therefore, it
has a rotation number

$b\alpha-a$

for the lift $\tilde{{f}}_{2mb}=\tilde{f}^{b}\circ{T}^{-a}$ of $f^{b}$ that
fixes the points of the preimage of $S_{a/b}$ by the covering projection. Let
us explain why $a-b\alpha$ is the measure of the domain between a leaf
$\tilde{\phi}$ (included) and $\tilde{{f}}_{2mb}(\tilde{\phi})$ (excluded).

Consider the first coordinate projection $p_{1}$ and the map $\pi_{1}$ that
takes the values $k$ in the domain contained between $T^{k}(\tilde{\phi})$
(included) and $T^{k+1}(\tilde{\phi})$ (excluded).

The real functions $\pi_{1}\circ\tilde{f}_{2mb}-\pi_{1}$,
$p_{1}\circ\tilde{f}_{2mb}-p_{1}$, $p_{1}-\pi_{1}$, being invariant by $T$,
can be seen as defined on $\mathbb{D}^{*}_{a/b}$ and that we will use later
the equality

$\pi_{1}\circ\tilde{f}^{i+1}_{2mb}-\pi_{1}\circ\tilde{f}^{i}_{2mb}=(\pi_{1}\circ\tilde{f}_{2mb}-\pi_{1})\circ
f^{i}_{2mb}$

and similar ones for the other maps.

We have:

$\int_{\mathbb{D}^{*}_{a/b}}(p_{1}\circ\tilde{{f}}_{2mb}-p_{1})\leavevmode\nobreak\
d{\mu}=-(a-b\alpha)$

by definition of the rotation vector. We also know that

(3.2)
$\int_{\mathbb{D}^{*}_{a/b}}(\pi_{1}\circ\tilde{{f}}_{2mb}-\pi_{1})\leavevmode\nobreak\
d{\mu}=-\tilde{\mu}(O(\tilde{z})).$

To see why this holds, note that $\pi_{1}\circ\tilde{{f}}_{2mb}-\pi_{1}$ takes
non positive values. Moreover, for every point $x\in\mathbb{D}^{*}_{a/b}$, the
fact that $(\pi_{1}\circ\tilde{{f}}_{2mb}-\pi_{1})(x)=-k<0$ means that there
exists a lift $\tilde{x}$ of $x$ such that

$T(\tilde{x})\in
L(\tilde{f}^{-1}_{2mb}(\tilde{\phi}_{\tilde{z}})),\,\tilde{x}\in\overline{R(\tilde{f}^{-1}_{2mb}(\tilde{\phi}_{\tilde{z}}))},\dots,\,T^{-k+1}(\tilde{x})\in\overline{R(\tilde{f}^{-1}_{2mb}(\tilde{\phi}_{\tilde{z}}))},\,T^{-k}(\tilde{x})\in\overline{R(\tilde{\phi}_{\tilde{z}})}.$

So we have

$\tilde{\mu}(O(\tilde{z}))=\sum_{k>0}k\mu(\\{x\in\mathbb{D}^{*}_{a/b}\,|\enskip(\pi_{1}\circ\tilde{{f}}_{2mb}-\pi_{1})(x)=-k\\})=-\int_{\mathbb{D}^{*}_{a/b}}(\pi_{1}\circ\tilde{{f}}_{2mb}-\pi_{1})\leavevmode\nobreak\
d{\mu}.$

By Birkhoff’s ergodic theorem for $\mu$-almost every $z$, the sequences

$\frac{\sum_{i=0}^{n-1}(p_{1}\circ\tilde{{f}}_{2mb}-p_{1})\circ
f^{i}_{2mb}(z)}{n}\ \ \mbox{and}\ \
\frac{\sum_{i=0}^{n-1}(\pi_{1}\circ\tilde{{f}}_{2mb}-\pi_{1})\circ
f^{i}_{2mb}(z)}{n}$

converge, the first one to $p_{*}(z)\in\mathbb{R}$ the second one to
$\pi_{*}(z)\in[-\infty,0].$ Moreover,

  * •

$p_{*}$ is integrable and
$\int_{\mathbb{D}^{*}_{a/b}}p_{*}\leavevmode\nobreak\
d\mu=\int_{\mathbb{D}^{*}_{a/b}}p_{1}\circ\tilde{f}_{2m}-p_{1}\leavevmode\nobreak\
d\mu,$

  * •

$\int_{\mathbb{D}^{*}_{a/b}}\pi_{*}\leavevmode\nobreak\
d\mu=\int_{\mathbb{D}^{*}_{a/b}}\pi_{1}\circ\tilde{f}_{2m}-\pi_{1}\leavevmode\nobreak\
d\mu.$

We know that $\mu$-almost every point is recurrent and that the function
$(p_{1}-\pi_{1})$ is locally bounded. This implies $\pi_{*}(z)=p_{*}(z)$ for
$\mu$-almost every $z$ because when we consider the Birkhoff means of the two
functions applied between $0$ and $n_{k}$, where ${f}_{2mb}^{n_{k}}(z)$ is
close to $z$, we find that

$\displaystyle p_{*}(z)-\pi_{*}(z)$
$\displaystyle=\lim_{k\to\infty}\frac{\sum_{i=0}^{n_{k}-1}(p_{1}\circ\tilde{{f}}_{2mb}-p_{1})\circ
f^{i}_{2mb}(z)-(\pi_{1}\circ\tilde{{f}}_{2mb}-\pi_{1})\circ
f^{i}_{2mb}(z)}{n_{k}}$
$\displaystyle=\lim_{k\to\infty}\frac{\sum_{i=0}^{n_{k}-1}(p_{1}\circ\tilde{{f}}_{2mb}^{i+1}-p_{1}\circ\tilde{{f}}_{2mb}^{i})(z)-(\pi_{1}\circ\tilde{{f}}_{2mb}^{i+1}-\pi_{1}\circ\tilde{{f}}_{2mb}^{i})(z)}{n_{k}}$
$\displaystyle=\lim_{k\to\infty}\frac{\sum_{i=0}^{n_{k}-1}(p_{1}\circ\tilde{{f}}_{2mb}^{i+1}-\pi_{1}\circ\tilde{{f}}_{2mb}^{i+1})(z)-(p_{1}\circ\tilde{{f}}_{2mb}^{i}-\pi_{1}\circ\tilde{{f}}_{2mb}^{i})(z)}{n_{k}}$
$\displaystyle=\lim_{k\to\infty}\frac{(p_{1}\circ\tilde{{f}}_{2mb}^{n_{k}}-\pi_{1}\circ\tilde{{f}}_{2mb}^{n_{k}})(z)-(p_{1}-\pi_{1})(z)}{n_{k}}$
$\displaystyle=\lim_{k\to\infty}\frac{(p_{1}-\pi_{1})\circ
f^{n_{k}}_{2mb}(z)-(p_{1}-\pi_{1})(z)}{n_{k}}=0$

for $\mu$-almost every $z$. This completes the proof of Lemma 3.3. ∎

######  Remark 3.5.

We could attempt to prove Lemma 3.3 using the following approach:

$\displaystyle\int_{\mathbb{D}^{*}_{a/b}}(p_{1}\circ\tilde{{f}}_{2mb}-p_{1})\leavevmode\nobreak\
d\mu-\int_{\mathbb{D}^{*}_{a/b}}(\pi_{1}\circ\tilde{{f}}_{2mb}-\pi_{1})\leavevmode\nobreak\
d\mu$
$\displaystyle=\int_{\mathbb{D}^{*}_{a/b}}(p_{1}\circ\tilde{{f}}_{2mb}-\pi_{1}\circ\tilde{{f}}_{2mb})-(p_{1}-\pi_{1}))\leavevmode\nobreak\
d\mu\ $
$\displaystyle=\int_{\mathbb{D}^{*}_{a/b}}((p_{1}-\pi_{1})\circ\tilde{{f}}_{2mb}-(p_{1}-\pi_{1}))\leavevmode\nobreak\
d\mu\ $ $\displaystyle=0.$

This might seem correct at first glance. However, the last equality is not
correct because one does not know if $p_{1}-\pi_{1}$ is integrable, which is
necessary to use the fact that $\mu$ is invariant.

######  Remark 3.6.

Lemma 3.3 holds for any Brouwer radial foliation, not just for the foliation
constructed by the second author in [LC23a].

We now introduce the following notation

$W_{a/b}:=\\{(z,z^{\prime})\in\mathbb{D}^{*}_{a/b}|\ z\neq
z^{{}^{\prime}}\\}.$

The following result is a generalization of Lemma 4.2 in [LC23a] for every
couple of measures instead of the product of the Lebesgue measure by itself.

######  Lemma 3.7.

For any two invariant probability measures $\nu$ and $\mu$ of $f$ such that
$\mu(\\{0\\})=\nu(\\{0\\})=0$ and $\mu(\mathbb{D})=\nu(\mathbb{D})=1$, the
following holds.

$\int_{W_{a/b}}\overline{{\tau}}(z,z^{\prime},\mathscr{F}_{1},f^{-b},(\mathscr{F}_{1}))\leavevmode\nobreak\
d\mu(z)\times d\nu(z^{\prime})\leq 8mb(a-b\alpha).$

######  Remark 3.8.

The proof of Lemma 3.7 is identical to the proof of Lemma 4.2 in [LC23a]. It
relies on Proposition 3.1 and the identity 3.1 for the Lebesgue measure.
However, based on Lemma 3.3, we know that this identity 3.1 holds for any two
invariant probability measures. Therefore, we omit it

The following theorem is a generalization of Theorem 1.1 [LC23a] for any pair
of invariant measures.

######  Theorem 3.9.

For any two invariant probability measures $\nu$ and $\mu$ of
$f\big{|}_{\mathbb{D}}$ such that $\mu(\\{0\\})=\nu(\\{0\\})=0$. We have the
following

(3.3) $\int_{\mathbb{D}\times\mathbb{D}}W_{\tilde{f}}(x,y)\leavevmode\nobreak\
d\mu\times d\nu=\rho(\tilde{f}).$

######  Remark 3.10.

Recall that the right-hand side of the identity 3.3 depends only on the
homotopy class of the isotopy with fixed ends that we choose, similarly the
left-hand side changes accordingly.

###### Proof of Theorem 3.9.

First of all we extend the invariant probability measures $\mu$ and $\nu$ to
$\mathbb{D}_{a/b}$ in the following way:

$\mu(A)=\mu(\mathbb{D}\cap A)\quad\text{and}\quad\nu(A)=\nu(\mathbb{D}\cap A)$

for every Borel set $A$ of $\mathbb{D}_{a/b}$.

Using Lemma 3.7 and inequality (2.1), we establish the integrability of
$\lambda(z,z^{\prime},\mathscr{F}_{1},f^{-b}(\mathscr{F}_{1}))$. In particular
we have

$\left|\int_{W_{a/b}}\lambda(z,z^{\prime},\mathscr{F}_{1},f^{-b}(\mathscr{F}_{1}))\leavevmode\nobreak\
d\mu(z)\times d\nu(z^{\prime})\right|\leq 8mb(a-b\alpha).$

By the last inequality together with equality (2.2), allows us to deduce that

$\left|\int_{W_{a/b}}\Lambda_{\tilde{f}_{2mb},\mathscr{F}_{1},\phi}(z,z^{\prime})-m_{\tilde{f}_{2mb},\phi}(z^{\prime})\leavevmode\nobreak\
d\mu(z)\times d\nu(z^{\prime})\right|\leq 8mb(a-b\alpha)$

if $\phi$ is a leaf of $\mathscr{F}_{1}$.

By Theorem 2.12, we know that $m_{\tilde{f}_{2mb},\phi}(z)$ is integrable, and
this, together with the last inequality, implies the following:

$\left|\int_{W_{a/b}}\Lambda_{\tilde{f}_{2mb},\mathscr{F}_{1},\phi}(z,z^{\prime})\leavevmode\nobreak\
d\mu(z)\times
d\nu(z^{\prime})-\int_{W_{a/b}}m_{\tilde{f}_{2mb},\phi}(z^{\prime})\leavevmode\nobreak\
d\mu(z)\times d\nu(z^{\prime})\right|\leq$ $\leq 8mb(a-b\alpha).$

The fact that $\mu(\\{0\\})=\nu(\\{0\\})=0$ implies that

$\mu\times\nu(\\{(z,z^{\prime})\in\mathbb{D}^{*}_{a/b}\times\mathbb{D}^{*}_{a/b}|\
z=z^{\prime}\\})=0,$

using this together with previous inequality yields

$\left|\int_{\mathbb{D}^{*}_{a/b}\times\mathbb{D}^{*}_{a/b}}\Lambda_{\tilde{{f}}_{2mb},\mathscr{F}_{1},\phi}(z,z^{\prime})\leavevmode\nobreak\
d\mu(z)\times
d\nu(z^{\prime})-\int_{\mathbb{D}^{*}_{a/b}\times\mathbb{D}^{*}_{a/b}}m_{\tilde{f}_{2mb},\phi}(z^{\prime})\leavevmode\nobreak\
d\mu(z)\times d\nu(z^{\prime})\right|\leq$ $\leq 8mb(a-b\alpha).$

Combining this with Theorem 2.9 and 2.10, we can now state the following
result:

$\left|\int_{\mathbb{D}^{*}_{a/b}\times\mathbb{D}^{*}_{a/b}}W_{\tilde{f}_{2mb}}(x,y)\leavevmode\nobreak\
d\mu(z)\times
d\nu(z^{\prime})-\int_{\mathbb{D}^{*}_{a/b}}\text{rot}_{\tilde{f}_{2mb}}(\nu)\leavevmode\nobreak\
d\mu(z)\right|\leq$ $\leq 8mb(a-b\alpha).$

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

$\displaystyle\left|b\int_{\mathbb{D}^{*}_{a/b}\times\mathbb{D}^{*}_{a/b}}W_{\tilde{f}}(x,y)\leavevmode\nobreak\
d\mu(z)\times
d\nu(z^{\prime})-a-b\int_{\mathbb{D}^{*}_{a/b}}\text{rot}_{\tilde{f}}(\nu)\leavevmode\nobreak\
d\mu(z)+a\right|\leq 8bm(a-b\alpha)$

which can be written as

$\left|\int_{\mathbb{D}^{*}_{a/b}\times\mathbb{D}^{*}_{a/b}}W_{\tilde{f}}(x,y)\leavevmode\nobreak\
d\mu(z)\times d\nu(z^{\prime})-\alpha\right|\leq 8m(a-b\alpha).$

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

$\left|\int_{\mathbb{D}^{*}_{a_{n}/b_{n}}\times\mathbb{D}^{*}_{a_{n}/b_{n}}}W_{\tilde{f}}(x,y)\leavevmode\nobreak\
d\mu(z)\times d\nu(z^{\prime})-\alpha\right|\leq 8m(a_{n}-b_{n}\alpha).$

By construction of $\mu$ and $\nu$, we have

$\left|\int_{\mathbb{D}\times\mathbb{D}}W_{\tilde{f}}(x,y)\leavevmode\nobreak\
d\mu(z)\times d\nu(z^{\prime})-\alpha\right|\leq 8m(a_{n}-b_{n}\alpha).$

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

$\int_{\mathbb{D}\times\mathbb{D}}W_{\tilde{f}}(x,y)\leavevmode\nobreak\
d\mu(z)\times d\nu(z^{\prime})-\alpha=0.$

This completes the proof of Theorem 3.9. ∎

##  4\. Proof of Main Results

In this section, we use the last result from the previous section together
with a Krylov-Bogolyubov type argument to prove Theorems 1.4 and 1.8.

In his thesis, the first author established the inequality (4.1) for
$C^{\infty}$ smooth area-preserving diffeomorphisms, but it was not stated as
a theorem. However, one could find it in the proof of Theorem 4.3.2 in [BS23],
which is a generalization of the main argument in [BS21]. By repeating the
same argument, we could also show that it works for $C^{1}$ area-preserving
diffeomorphisms. As the proof remains the same, we omit its details.

######  Theorem 4.1.

[BS23] If $f:\mathbb{D}\to\mathbb{D}$ is a $C^{1}$ area-preserving
diffeomorphism, then

(4.1)
$\Big{|}\frac{a_{\tilde{f}^{n}}(x)-\int_{\mathbb{D}}W_{\tilde{f}^{n}}(x,y)\leavevmode\nobreak\
d\omega(y)}{n}\Big{|}\leq\frac{8}{n}.$

Here $\omega=\frac{r}{\pi}dr\wedge d\theta$.

We will now prove Theorem 1.4 by invoking Theorem 4.1 and Theorem 3.9.

###### Proof of Theorem 1.4.

Theorem 4.1 implies the following claim:

Claim: For any invariant probability measure $\mu$, we have

$\int_{D}a_{\tilde{f}}(x)\leavevmode\nobreak\ d\mu(x)=\rho(\tilde{f}).$

Proof: Using inequality (4.1), we obtain the following inequality:

$-\frac{8}{n}\leq\frac{a_{\tilde{f}^{n}}(x)-\int_{\mathbb{D}}W_{\tilde{f}^{n}}(x,y)\leavevmode\nobreak\
d\omega(y)}{n}\leq\frac{8}{n}.$

First, we integrate both sides of the inequality with respect to the measure
$\mu$, and then apply the following two identities:

(4.2) $\int_{\mathbb{D}}a_{\tilde{f}}(x)\leavevmode\nobreak\
d\mu(x)=\frac{\int_{\mathbb{D}}a_{\tilde{f}^{n}}(x)\leavevmode\nobreak\
d\mu(x)}{n},$ (4.3)
$\int_{\mathbb{D}}\int_{\mathbb{D}}W_{\tilde{f}}(x,y)\leavevmode\nobreak\
d\omega(y)\times
d\mu(x)=\frac{\int_{\mathbb{D}}\int_{\mathbb{D}}W_{\tilde{f}^{n}}(x,y)\leavevmode\nobreak\
d\omega(y)\times d\mu(x)}{n}.$

Therefore, we have:

$-\frac{8}{n}\leq\frac{\int_{\mathbb{D}}a_{\tilde{f}^{n}}\leavevmode\nobreak\
d\mu(x)-\int_{\mathbb{D}}\int_{\mathbb{D}}W_{\tilde{f}^{n}}(x,y)\leavevmode\nobreak\
d\omega(y)\times d\mu(x)}{n}\leq\frac{8}{n}.$

The last inequality together with the equalities (4.2) and (4.3) implies that

$-\frac{8}{n}\leq\int_{\mathbb{D}}a_{\tilde{f}}\leavevmode\nobreak\
d\mu(x)-\int_{\mathbb{D}}\int_{\mathbb{D}}W_{\tilde{f}}(x,y)\leavevmode\nobreak\
d\omega(y)\times d\mu(x)\leq\frac{8}{n}.$

By taking the limit from both sides of the inequality, we obtain:

(4.4) $\int_{\mathbb{D}}a_{\tilde{f}}\leavevmode\nobreak\
d\mu(x)=\int_{\mathbb{D}}\int_{\mathbb{D}}W_{\tilde{f}}(x,y)\leavevmode\nobreak\
d\omega(y)\times d\mu(x).$

We can write the measure $\mu$ in the following form
$b\delta_{0}+(1-b)\mu_{1}$ where $\delta_{0}$ is the Dirac measure at $0$,
$b\in[0,1]$, and the measure $\mu_{1}$ satisfies $\mu_{1}(\\{0\\})=0$. Now we
rewrite the equality (4.4) using it.

$\int_{\mathbb{D}}a_{\tilde{f}}\leavevmode\nobreak\
d\mu(x)=b\int_{\mathbb{D}}\int_{\mathbb{D}}W_{\tilde{f}}(x,y)\leavevmode\nobreak\
d\omega(y)\times
d\delta_{0}(x)+(1-b)\int_{\mathbb{D}}\int_{\mathbb{D}}W_{\tilde{f}}(x,y)\leavevmode\nobreak\
d\omega(y)\times\mu_{1}(x).$

By Theorem 3.9 and the equality (4.4), we have the following.

(4.5) $\int_{\mathbb{D}}a_{\tilde{f}}\leavevmode\nobreak\
d\mu(x)=b\int_{\mathbb{D}}a_{\tilde{f}}\leavevmode\nobreak\
d\delta_{0}(x)+(1-b)\rho(\tilde{f}).$

By combining of Theorem 4.1 and Theorem 2.12, we get
$\int_{\mathbb{D}}a_{\tilde{f}}d\delta_{0}(x)=\rho(\tilde{f})$, together with
the last equality this gives the desired equality
$\int_{\mathbb{D}}a_{\tilde{f}}d\mu(x)=\rho(\tilde{f})$. ∎

Assume the sequence

$\left(\frac{\sum^{n-1}_{i=0}a_{\tilde{f}}(f^{i}(x))}{n}\right)_{n\geq 0}$

does not converge uniformly, so there exist $\epsilon>0$ and a sequence
$x_{{k}}$ such that

(4.6)
$\big{|}\frac{\sum^{n_{k}-1}_{i=0}a_{\tilde{f}}(f^{i}(x_{{k}}))}{n_{k}}-\rho(\tilde{f})\big{|}\geq\epsilon.$

We now choose a subsequence $n_{k_{j}}$ of $n_{k}$ so that

$\mu_{n_{k_{j}}}(x_{k_{j}})\rightharpoonup\mu,$

here, $\mu$ is an invariant probability measure of
$f:\mathbb{D}\to\mathbb{D}$. The definition of weak∗ convergence implies the
following identity:

$\lim_{k_{j}\to\infty}\frac{\sum^{n_{k_{j}}-1}_{i=0}a_{\tilde{f}}(f^{i}(x_{{k_{j}}}))}{n_{k_{j}}}=\lim_{{k_{j}}\to\infty}\int_{\mathbb{D}}a_{\tilde{f}}\
d\mu_{n_{k_{j}}}(x_{k_{j}})=\int_{\mathbb{D}}a_{\tilde{f}}\
d\mu=\rho(\tilde{f}).$

The last equality contradicts the inequality (4.6). This completes a proof of
Theorem 1.4. ∎

We can now prove Theorem 1.8 as an application of Theorem 3.9.

###### Proof of Theorem 1.8.

Let us begin by assuming that the sequence

$\int_{\mathbb{D}\times\mathbb{D}}W_{\tilde{f}}(x,y)\leavevmode\nobreak\
d\mu_{n}(x)\times
d\mu_{n}(y)=\frac{\sum_{i=0}^{n-1}\sum_{j=0}^{n-1}W_{\tilde{f}}(f^{i}(x),f^{j}(y))}{n^{2}}$

does not converge uniformly on $\mathbb{O}(\mathbb{D})$ to $\rho(\tilde{f})$.
This implies the existence of $\epsilon>0$, of an increasing sequence
$(n_{k})_{k\geq 0}$ of positive integers and of a sequence
$(x_{k},y_{k})_{k\geq 0}$ in $\mathbb{D}\times\mathbb{D}$ such that

(4.7)
$\left|\frac{\sum_{i=0}^{n_{k}-1}\sum_{j=0}^{n_{k}-1}W_{\tilde{f}}(f^{i}(x_{{k}}),f^{j}(y_{{k}}))}{n_{k}^{2}}-\rho(\tilde{f})\right|\geq\epsilon.$

By Proposition 2.1, we can find a subsequence $({k_{j}})_{j\geq 0}$ such that

$\mu_{n_{k_{j}}}(x_{k_{j}})\rightharpoonup\mu$

and

$\mu_{n_{k_{j}}}(y_{k_{j}})\rightharpoonup\nu$

where $\mu$ and $\nu$ are invariant probability measures.

We may assume, up to passing to a further sub-sequence that the sequence
$(n_{k_{j}})_{j\in\mathbbm{N}}$ is such that
$\mu_{n_{k_{j}}}(x_{k_{j}})\times\mu_{n_{k_{j}}}(y_{k_{j}})$ weakly converges
in $K$, we denote by $\lambda$ its limit. Here $K$ is the compactification of
$\mathbb{D}\times\mathbb{D}\setminus\Delta$ such that
$W_{\tilde{f}}(z,z^{\prime})$ is well-defined and continuous (see Defition
1.6), where $\Delta$ is the diagonal.

Define $\pi:K\to\mathbb{D}\times\mathbb{D}$ as the (continuous) projection,
which coincides with the inclusion on
$(\mathbb{D}\times\mathbb{D})\setminus\Delta$ and maps $(x,\xi)\mapsto(x,x)$
on $\partial K=T^{1}\mathbb{D}$. We have $\pi_{*}(\lambda)=\mu\times\nu$, and
therefore $\lambda(\partial K)=0$ if and only if $\mu\times\nu(\Delta)=0$.

Define $\rho^{\prime}=\int_{K}W_{\tilde{f}}\,d\lambda$ and note that
$|\rho^{\prime}-\rho(\tilde{f})|\geq\varepsilon$ because

$\int_{K}W_{\tilde{f}}\,d\lambda=\lim_{j\to+\infty}\int_{K}W_{\tilde{f}}\,d(\mu_{n_{k_{j}}}(x_{k_{j}})\times\mu_{n_{k_{j}}}(y_{k_{j}}))=\lim_{j\to+\infty}\int_{\mathbb{D}\times\mathbb{D}}W_{\tilde{f}}\,d(\mu_{n_{k_{j}}}(x_{k_{j}})\times\mu_{n_{k_{j}}}(y_{k_{j}}))$
$=\lim_{j\to+\infty}\frac{\sum_{i=0}^{n_{k_{j}}-1}\sum_{j=0}^{n_{k_{j}}-1}W_{\tilde{f}}(f^{i}(x_{{k_{j}}}),f^{j}(y_{{k_{j}}}))}{n_{k_{j}}^{2}}$

where each element of the sequence in the last limit satisfies (4.7).

In the case where $\lambda(\partial K)=0$, we can write

$\rho^{\prime}=\int_{K}W_{\tilde{f}}\leavevmode\nobreak\
d\lambda=\int_{K\setminus\partial K}W_{\tilde{f}}\leavevmode\nobreak\
d\lambda=\int_{(\mathbb{D}\times\mathbb{D})\setminus\Delta}W_{\tilde{f}}\leavevmode\nobreak\
d(\mu\times\nu)=\int_{\mathbb{D}\times\mathbb{D}}W_{\tilde{f}}\leavevmode\nobreak\
d(\mu\times\nu).$

By Theorem 3.9, we have the following:

$\int_{\mathbb{D}\times\mathbb{D}}W_{\tilde{f}}\leavevmode\nobreak\
d(\mu\times\nu)=\rho(\tilde{f}).$

This leads to a contradiction.

If $\lambda(\partial K)\neq 0$, then $\mu\times\nu(\Delta)\not=0$, indicating
that $\mu$ and $\nu$ share a common atom. However, the only invariant atomic
measure is the Dirac measure $\delta_{0}$ because $f$ is an irrational pseudo-
rotation. So, we can express $\mu=t\mu^{\prime}+(1-t)\delta_{0}$ and
$\nu=s\nu^{\prime}+(1-s)\delta_{0}$, where $\nu^{\prime}$ and $\mu^{\prime}$
do not have atoms and

$\lambda=st\,\mu^{\prime}\times\nu^{\prime}+t(1-s)\,\mu^{\prime}\times\delta_{0}+(1-t)\,s\delta_{0}\times\nu^{\prime}+(1-t)(1-s)\lambda^{\prime},$

where $\mu^{\prime}(\\{0\\})=\nu^{\prime}(\\{0\\})=0$ and
$\lambda^{\prime}(T^{1}_{0}(\mathbb{D}))=1$ (here $T^{1}_{0}(\mathbb{D})$
denotes the unit tangent plane of $\mathbb{D}$ at $0$).

For the same reasons as before, it follows that

$\displaystyle\int_{K}W_{\tilde{f}}\leavevmode\nobreak\ d\lambda$
$\displaystyle=st\int_{\mathbb{D}\times\mathbb{D}}W_{\tilde{f}}\leavevmode\nobreak\
d(\mu^{\prime}\times\nu^{\prime})+t(1-s)\int_{\mathbb{D}\times\mathbb{D}}W_{\tilde{f}}\leavevmode\nobreak\
d(\mu^{\prime}\times\delta_{0})$
$\displaystyle\quad+(1-t)s\int_{\mathbb{D}\times\mathbb{D}}W_{\tilde{f}}\leavevmode\nobreak\
d(\delta_{0}\times\nu^{\prime})+(1-t)(1-s)\int_{\partial
K}W_{\tilde{f}}\leavevmode\nobreak\ d\lambda^{\prime}.$

Here, the first three integrals are equal to $\rho(\tilde{f})$. To complete
the contradiction, we need to show that the last integral is also equal to
$\rho(\tilde{f})$. This is evident because the rotation number induced on the
unit tangent vector by the differential is $\rho(\tilde{f})$. This concludes
the proof of Theorem 1.8. ∎

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