Constant periodic data and rigidity

TL;DR

This paper proves that constant periodic data for certain expanding maps and Anosov diffeomorphisms imply smooth conjugacy and rigidity, with results varying between global and local depending on the dimension.

Contribution

It establishes new rigidity results linking constant periodic data to smooth conjugacy for expanding maps and Anosov diffeomorphisms, extending known results to higher dimensions.

Findings

Constant periodic data implies smooth conjugacy for certain maps.

Global rigidity results for dimensions 2 and 3.

Local rigidity results for dimensions 4 and higher.

Abstract

In this work we lead with expanding maps of the circle and Anosov diffeomorphisms on $\mathbb{T}^d, d \geq 2.$ We prove that, for these maps, \textit{constant periodic data} imply \textit{same periodic data of these maps and their linearizations}, so in particular we have smooth conjugacy. For expanding maps of the circle and Anosov diffeomorphism on $\mathbb{T}^d, d= 2, 3,$ we have global rigidity. In higher dimensions, $d \geq 4,$ we can establish a result of local rigidity, in several cases. The main tools of this work are celebrated results of rigidity involving same periodic data with linearization and results involving topological entropy of a diffeomorphism along an expanding invariant foliation.

Equations

$$h\circ f=L\circ h.$$

$$\frac{1}{C_f}<\frac{|D{f}^n(x)|}{|D{f}^n(y)|}<C_f,$$

$$\frac{D{f}^n(x)}{D{f}^n(y)}=\frac{{\prod }_{i=0}^{n-1}Df(f^i(x))}{{\prod }_{i=0}^{n-1}Df(f^i(y))}=\prod _{i=0}^{n-1}\left(1+\frac{Df(x_i)-Df(y_i)}{Df(y_i)}\right).$$

$$\frac{D{f}^n(x)}{D{f}^n(y)}=\prod _{i=0}^{n-1}\left(1+\frac{(x_i-y_i)D^{2}f(z_i)}{Df(y_i)}\right)\le \prod _{i=0}^{n-1}(1+M({\lambda }^{-1}{)}^{n-i}|x_n-y_n|),$$

$$\log \left(\frac{|D{f}^n(x)|}{|D{f}^n(y)|}\right)\le M|x_n-y_n|\sum _{i=0}^{\infty }{\lambda }^{-i}\le M\sum _{i=0}^{\infty }{\lambda }^{-i}\le K=K_f,$$

$$\frac{|D{f}^n(x)|}{|D{f}^n(y)|}<C_f=\exp (K_f),\frac{|D{f}^n(y)|}{|D{f}^n(x)|}<C_f=\exp (K_f).$$

$${\int }_I{\omega }_f(x)dx={\int }_{h(I)}d(h_{\ast }({\mu }_f))={\int }_{h(I)}{\omega }_{g}dx={\int }_I{\omega }_g(h(x))|h^{\prime }(x)|dx,$$

$$z^{\prime }=\frac{{\omega }_f(t)}{{\omega }_g(z)}.$$

$$h(t)=h(0)+{\int }_0^t{h}^{\prime }(s)ds=h(0)+{\int }_0^t\frac{{\omega }_f(s)}{{\omega }_g(h(s))}ds,$$

$$|D{f}^n(p)|={\lambda }^n.$$

$$\frac{1}{C}|D{f}^n(p_{n,j})|\cdot |I_{n,j}|\le |S^1|=1\le C|D{f}^n(p_{n,j})|\cdot |I_{n,j}|,$$

$$\frac{1}{C}{\lambda }^n\cdot |I_{n,j}|\le 1\le C{\lambda }^n\cdot |I_{n,j}|.$$

$$\frac{1}{C{\lambda }^n}\le |I_{n,j}|\le \frac{C}{{\lambda }^n},$$

$$1\ge d^n\cdot \underset{1\le j\le d^n}{\min }\{|I_{n,j}|\}\ge d^n\cdot \frac{1}{C{\lambda }^n}\Rightarrow \frac{d^n}{{\lambda }^n}\le C$$

$$1\le d^n\cdot \underset{1\le j\le d^n}{\max }\{|I_{n,j}|\}\le d^n\cdot \frac{C}{{\lambda }^n}\Rightarrow \frac{d^n}{{\lambda }^n}\ge \frac{1}{C},$$

$$\frac{1}{C}\le \frac{d^n}{{\lambda }^n}\le C,$$

$$\frac{1}{C^2}\le \frac{|h(I_{n,j})|}{|I_{n,j}|}\le C^2,$$

$$\underset{k\to +\infty }{\lim }\frac{1}{n_k}\log ||D{f}^{n_k}(x)||=\lambda (p),$$

$$\underset{k\to +\infty }{\lim }\frac{1}{m_k}\log ||D{f}^{m_k}(x)||=\lambda (q),$$

$$||D_{x}f{v}^s||<||D_{y}f{v}^c||<||D_{z}f{v}^u||,$$

$$||D_{x}f{v}^s||<1\text{mbox}and||D_{z}f{v}^u||>1,$$

$$h_{\mu }(f)=\int \sum _{{\lambda }_i>0}{\lambda }_i\dim (E_i)d\mu .$$

$$h\circ f=g\circ h.$$

$$h\circ f=g\circ h,$$

$$E_g^s\oplus E_g^{wu}\oplus E_g^{su}=T{\mathbb{T}}^3=E_f^s\oplus E_f^{wu}\oplus E_f^{su}.$$

$$h_{\mu }(f)\le {\lambda }_f^u(p).$$

$$h_{\nu }(f)={\int }_{{\mathbb{T}}^2}{\lambda }_{\mu }^u(x,f)={\lambda }_f^u(p).$$

$${\lambda }_L^u=h_{top}(L)=h_{top}(f)=h_{\nu }(f)={\lambda }_f^u(p).$$

$${\lambda }_f^s(p)={\lambda }_L^s,$$

$${\Lambda }^{\ast }(x)={\Lambda }^{\ast },\ast \in \{s,u\}.$$

Full text

1. Introduction

Expanding endomorphisms of the circle are one of the most studied examples of dynamical systems. A classical class of expanding maps of the circle is the linear model $E_{d}:S^{1}\rightarrow S^{1},E_{d}(x)=dx(mod1),$ where $d\geq 2$ is a integer number. It is well known that if $f:S^{1}\rightarrow S^{1}$ is an orientation preserving expanding endomorphism of the circle with degree $d\geq 2,$ then $f$ is conjugated to $E_{d},$ meaning that there is a homeomorphism $h:S^{1}\rightarrow S^{1},$ such that $f\circ h=h\circ E_{d}.$ In particular two orientation preserving expanding endomorphism of the circle $f$ and $g$ with the same degree are conjugated. In the case that $f,g$ above are $C^{r},r\geq 2,$ two orientation preserving expanding endomorphism of $S^{1},$ it is known by [16], that $f$ and $g$ are absolutely continuous conjugated by a conjugacy $h$ if and only if $h$ is $C^{r},r\geq 2.$

Other important fact is that every $C^{r},r\geq 2$ expanding map $f$ of the circle admit a unique invariant measure $\mu_{f}$ that is absolute continuous with respect to Lebesgue measure $m$ of $S^{1},$ moreover $\mu_{f}$ is ergodic. By ergodicity of $\mu_{f}$ it is possible talk about the Lyapunov exponent with respect to $\mu_{f}$ for $f,$ that is $m-$almost everywhere constant and we will denote by $\lambda_{\mu_{f}}.$

Now consider $M$ a compact, connected, boundaryless $C^{\infty}$ manifold $M,$ we say that a diffeomorphism $f:M\rightarrow M$ is an Anosov diffeomorphism if $TM$ splits as $TM=E^{s}_{f}\oplus E^{u}_{f}$ a continuous and $Df$ invariant spliting, such that $Df$ is uniform contracting on $E^{s}_{f}$ and uniform expanding on $E^{u}_{f}.$

In this work we lead with Anosov diffeomorfisms $f:\mathbb{T}^{d}\rightarrow\mathbb{T}^{d},d\geq 2.$ Denote by $L$ the linearization of $f,$ the map induced on $\mathbb{T}^{d}$ by the matrix with integer coefficients given by the action of $f$ on $\Pi_{1}(\mathbb{T}^{d}).$ It is known by [7] that $L$ is an Anosov automorphism, and $f$ and $L$ are conjugated by a homeomorphism $h$ such that

[TABLE]

Before the results here, we define.

Definition 1.1**.**

Let $f:M\rightarrow M$ be a local diffeomorphism. We say that $f$ has constant periodic data if for any periodic points $p,q$ of $f,$ with period $k$ and $n$ respectively, then $Df^{\tau}(p)=Df^{\tau}(q)$ are conjugated, for every integer $\tau$ such that $f^{\tau}(p)=p$ and $f^{\tau}(q)=q.$ In particular the set of Lyapunov exponents of $p$ and $q,$ are equal and each common Lyapunov exponent has the same multiplicity for both.

Remark 1.2*.*

Constant periodic data is a more weaker condition than to suppose $f$ and its linearization $L$ have same periodic data at corresponding periodic points.

We are able to prove the following.

Theorem 1.3**.**

Consider $f:S^{1}\rightarrow S^{1}$ a $C^{r},r\geq 2$ orientation preserving expanding endomorphism with degree $d\geq 2.$ The map $f$ is $C^{r}$ conjugated to $E_{d}$ if and only if $\lambda_{\mu_{f}}$ is constant on $Per(f),$ where $Per(f)$ denotes the set of periodic points for $f.$

Note that, in the hypothesis of the previous Theorem we don’t have suppose $\lambda_{\mu_{f}}(p)=\log(d).$ In fact Theorem 1.3 generalizes a result by Arteaga in [1].

In dimension $d>1,$ we study regularity of conjugacy of Anosov diffeomorphism. For dimensions two and three we can state the following.

Theorem 1.4**.**

Consider $f:\mathbb{T}^{2}\rightarrow\mathbb{T}^{2}$ a $C^{r},r\geq 2,$ Anosov diffeomorphism. Suppose that for each $\ast\in\{s,u\},$ we have $\lambda^{\ast}_{f}(p)=\lambda^{\ast}_{f}(q),$ for any $p,q$ periodic points of $f,$ then $f$ is $C^{r-\varepsilon}$ conjugated with its linearization $L,$ for some $\varepsilon>0.$

Theorem 1.5**.**

Consider $f:\mathbb{T}^{3}\rightarrow\mathbb{T}^{3}$ a $C^{r},r\geq 2$ Anosov diffeomorphism. Suppose that $f$ admits a partially hyperbolic structure $T\mathbb{T}^{3}=E_{f}^{s}\oplus E_{f}^{wu}\oplus E_{f}^{su}$ and for each $\ast\in\{s,wu,su\},$ we have $\lambda^{\ast}_{f}(p)=\lambda^{\ast}_{f}(q),$ for any $p,q$ periodic points of $f,$ then $f$ is $C^{1+\varepsilon}$ conjugated with its linearization $L,$ for some $\varepsilon>0.$

The results above are general, it is sufficient the constant periodic data condition to ensure rigidity. For dimension $d\geq 4,$ we are able to prove a more restricted version of the previous Theorems in several cases. More precisely, we have.

Theorem 1.6**.**

Let $L:\mathbb{T}^{d}\rightarrow\mathbb{T}^{d},d\geq 4,$ be a linear Anosov automorphism, diaganalizable over $\mathbb{R},$ irreducible over $\mathbb{Q},$ with distinct eigenvalues. Suppose that $E^{s}_{L}=E^{s}_{1}\oplus E^{s}_{2}\oplus\ldots\oplus E^{s}_{k}$ and $E^{u}_{L}=E^{u}_{1}\oplus E^{u}_{2}\oplus\ldots\oplus E^{u}_{n}.$ If $f$ is a $C^{2}$ diffeomorphism of $\mathbb{T}^{d}$ sufficiently $C^{1}-$close to $L,$ such that $\lambda^{u}_{i}(p,f)=\lambda^{u}_{i}(q,f),$ for any $p,q\in Per(f),$ $i=1,\ldots,n$ and $\lambda^{s}_{i}(p,f)=\lambda^{s}_{i}(q,f),$ for any $p,q\in Per(f),$ $i=1,\ldots,k,$ then $f$ is $C^{1+\varepsilon}$ conjugated with its linearization $L,$ for some $\varepsilon>0.$

2. One dimensional case - Expanding Maps

In this section we present some important classical results about expanding endomorphism of the circle, which will be useful for our propose.

Lemma 2.1** (Bounded Distortion Lemma).**

Let $f$ be a $C^{1+\alpha}$ expanding endomorphism of $S^{1}.$ There is a constant $C_{f}\geq 1,$ such that if $I\subset S^{1}$ is an interval and $f^{n}$ is injective on $I,$ then

[TABLE]

for any $x,y\in I.$

Proof.

Since $f$ is expanding map, there is $\lambda>1,$ such that $|Df(z)|>\lambda,$ for every $z\in S^{1}.$

Consider $x,y\in I$ and denote $x_{i}=f^{i}(x),y_{i}=f^{i}(y).$ One has

[TABLE]

Using mean value theorem and since $f$ is $C^{2},$ we obtain

[TABLE]

once $f^{n-i}(x_{i})=x_{n},f^{n-i}(y_{i})=y_{n}.$ Passing modulus, taking $\log,$ and using the elementary fact $\log(1+x)<x,$ for any $x>0,$ we have

[TABLE]

finally

[TABLE]

∎

Theorem 2.2**.**

Any $C^{2}$ expanding map $f:S^{1}\rightarrow S^{1}$ has a unique absolutely continuous invariant measure, $\mu_{f}.$ Furthermore $\mu_{f}$ is ergodic.

For a proof, see [11], page 193.

Theorem 2.3** (Sackesteder, Krzyzewsky ).**

Let $f:M\rightarrow M$ be a $C^{r},r\geq 2$ expanding map. Then there is a normalized $C^{r-1}$ invariant measure for $f.$

Theorem 2.4** (Shub-Sullivan’s Theorem, [16]).**

Let $f,g$ be two $C^{r},r\geq 2$ orientation preserving endomorphism of the circle. Then $f$ and $g$ are absolutely continuous conjugated if and only if $f$ and $g$ are $C^{r}$ conjugated.

Remark 2.5*.*

The proof that we present here is different from the original one. Here we use an argument involving ordinary differential equations. A similar argument the reader can see in [12] and [14], for example.

Proof.

Let $\mu_{f}$ and $\mu_{g}$ be the unique invariant measures absolutely continuous with respect to Lebesgue measure on the circle, for $f$ and $g$ respectively. Let $\omega_{f}$ and $\omega_{g}$ be the positive densities of $\mu_{f}$ and $\mu_{g}$ respectively. Since $h$ is absolutely continuous, the measure $h_{\ast}(\mu_{f})$ is a probability invariant measure for $g,$ moreover $h_{\ast}(\mu_{f})$ is absolutely continuous. By the uniqueness established in the theorem 2.2, we have $h_{\ast}(\mu_{f})=\mu_{g}.$

Since $h$ is an absolutely continuous homeomorphism, there is $h^{\prime}$ for almost every point $x\in S^{1}.$ So using the chang of variable formula, we have:

[TABLE]

for any interval $I\subset S^{1}.$

So we have $\omega_{f}(x)=\omega_{g}(h(x))|h^{\prime}(x)|,$ for almost everywhere $x\in S^{1}.$

Since $f,g$ has the same orientation, then $h$ is not reverse wise, then $h^{\prime}>0.$ So for almost everywhere $t\in S^{1},$ the function $h$ satisfies the ordinary differential equation

[TABLE]

By Theorem $\ref{saks},$ the densities $\omega_{f}$ and $\omega_{g}$ are $C^{r-1}$ functions. So considering equation $(\ref{edo})$ for every $t\in S^{1},$ by classical theory of ordinary differential equations, the O.D.E above has a unique $C^{r}$ solution $H$ satisfying $H(0)=h(0).$

So, since $h$ has bounded variation, for every $t\in S^{1}$ we have

[TABLE]

on the other hand, changing $h$ by $H$ in the equation above, the analogous equations holds for $H.$ Moreover, by Picard’s Theorem $H$ is the unique continuous function satisfying the integral equation above with the initial condition $H(0)=h(0).$ Then, by a continuation argument, $H(t)=h(t),$ for every $t.$

∎

2.1. Proof of Theorem 1.3

Let us to prove Theorem 1.3.

Proof.

Since the degree of $f$ is $d\geq 2$ we know that $f$ is conjugated to $E_{d}(x)=dx(mod1).$ Fix the notation: $\log(\lambda),\lambda>1,$ denoting the Lyapunov exponent on periodic points. In particular if $p$ has period $n,$ then

[TABLE]

Fix $n>1$ and $\{{I_{n,j}}\}_{j=1}^{d^{n}}$ maximal intervals of injectivity of $f^{n},$ such that $f^{n}(I_{n,j})=S^{1},j=1,2\ldots,d^{n}.$ By conjugacy with $E_{d^{n}},$ in each $\overline{I}_{n,j},$ there is a periodic point $p_{n,j},f^{n}(p_{n,j})=p_{n,j}.$

Denote by $|I|$ the size of the oriented interval $I\subset S^{1}.$ By bounded distortion lemma we have that there is $C>1,$ such that

[TABLE]

rewriting we have:

[TABLE]

In particular, using the inequality above, we have

[TABLE]

also we have

[TABLE]

[TABLE]

we obtain

[TABLE]

for every $n\geq 1.$ Since the conjugacy $h$ between $f$ and $E_{d}$ carries intervals $I_{n,j}$ to intervals $X_{n,j}$ the maximal intervals of injectivity of $E_{d^{n}},$ which size is $\frac{1}{d^{n}}.$ So from equations $(\ref{lips1})$ and $(\ref{lips2})$ have

[TABLE]

for every $n\geq 1$ and $1\leq j\leq d^{n}.$

Since $n$ is arbitrary, by (2.3) we have $d^{n}=\lambda^{n}=|Df^{n}(p)|,$ for every $p\in Per(f),$ such that $f^{n}(p)=p,$ in particular $\lambda_{f}(p)=\log(d),$ for any $p\in Per(f).$ Also, by $(\ref{lips3}),$ and the fact $\displaystyle\max_{1\leq j\leq d^{n}}\{|I_{n,j}|\}\rightarrow 0,$ when $n\rightarrow+\infty,$ we have that $h$ is a bi-Lipschitz conjugacy, consequently an absolutely continuous conjugacy, which it is $C^{r}$ by Theorem 2.4.

The equality $\lambda_{f}(p)=\lambda_{f}(q)$ for any $p,q\in Per(f)$ is obvious when $f$ and $E_{d}$ are $C^{r}-$conjugated.

∎

Corollary 2.6**.**

Let $f:S^{1}\rightarrow S^{1}$ be a $C^{r},r\geq 2$ orientation preserving, with degree $d\geq 2.$ Denote by $\mathcal{R}_{f}$ the set of regular points of $f.$ Then $f$ is $C^{r}$ conjugated to $E_{d},$ if and only if $\mathcal{R}_{f}=S^{1}.$

Proof.

Of course if $f$ and $E_{d}$ are $C^{r}$ conjugated then $\mathcal{R}_{f}=S^{1}.$ On the other hand, suppose that $\mathcal{R}_{f}=S^{1}$ and the Lyapunov exponent is not constant on $Per(f).$ Let $\lambda(p)\neq\lambda(q)$ be two different Lyapunov exponents at periodic points $p$ and $q.$ Then using ’symbolic representation’ it is possible to obtain $x\in S^{1}$ and two subsequences $\{n_{k}\}_{k}$ and $\{m_{k}\}$ such that

[TABLE]

[TABLE]

so $x\notin\mathcal{R}_{f}.$ ∎

3. General Preliminaries on Anosov Diffeomorphism

First let us define basic concepts. Let $M$ be a $C^{\infty}$ riemannian closed (compact, connected and boundaryless) manifold. A $C^{1}-$diffeomorphism $f:M\rightarrow M$ is called an absolute partially hyperbolic diffeomorphism if the tangent bundle $TM$ admits a $Df$ invariant tangent decomposition $TM=E^{s}\oplus E^{c}\oplus E^{u}$ such that all unitary vectors $v^{s}\in E^{s}_{x},v^{c}\in E^{c}_{y},v^{u}\in E^{u}_{z},$ for every $x,y,z\in M$ satisfy:

[TABLE]

moreover,

[TABLE]

for a suitable norm.

When $TM=E^{s}\oplus E^{u},$ where $E^{s}$ and $E^{u}$ is as above, then $f$ is called an Anosov diffeomorphism.

Definition 3.1** (SRB measure).**

Let $f:M\rightarrow M$ be a $C^{2}$ diffeomorphism. An $f$ invariant Borel probability measure $\mu$ is called an SRB measure if $(f,\mu)$ has a positive Lyapunov exponent a.e. and $\mu$ has absolutely continuous conditional measures on unstable manifolds.

Theorem 3.2** (Ledrappier-Young, [13]).**

Let $f$ be a $C^{2}-$diffeomorphism and $\mu$ an $f-$invariant probability measure with a positive Lyapunov exponent a.e. Then $\mu$ is SRB measure of $f$ if and only if

[TABLE]

Theorem 3.3** (Bowen, [3]).**

Let $f:M\rightarrow M$ be a $C^{2}-$Anosov diffeomorphism. Suppose that for every periodic point $p,$ with period $n_{p}$ we have $|\det(Df^{n_{p}}(p))|=1,$ then $f$ preserves a unique borelian probability measure $\mu$ absolutely continuous with respect to Lebesgue measure $m,$ moreover $\frac{d\mu}{dm}=H$ is a positive Hölder function.

It is known by [3] that a $C^{2}-$axiom A diffeomorphism has a unique SRB measure. It includes transitive Anosov diffeomorphisms, particularly Anosov diffeomorphism on $\mathbb{T}^{d}.$

Lemma 3.4** (Anosov Closing Lemma).**

Let $f:M\rightarrow M$ be a $C^{1+\alpha}$ diffeomorphism preserving a hyperbolic Borel probability measure. For all $\delta>0$ and $\epsilon>0$ there exists $\beta=\beta(\delta,\epsilon)>0$ such that if $x,f^{n(x)}(x)\in\Delta_{\delta}$ (Pesin block) for some $n(x)>0$ and $d(x,f^{n(x)}(x))<\beta$ then there exists a hyperbolic periodic point of period $n(x)$, $z$ with $d(f^{k}(x),f^{k}(z))\leq\epsilon$ for all $0\leq k\leq n(x)-1.$

Note that, for the Anosov diffeomorphism $f,$ we may take (a.e) $\Lambda_{\delta}=\mathbb{T}^{d}$ for some $\delta>0.$

Definition 3.5** (Specification Property).**

Let $f:M\rightarrow M$ be a diffeomorphism. We say that $f$ has the specification property if given $\varepsilon>0$ there is a relaxation time $N\in\mathbb{N}$ such that every $N-$spaced collection of orbit segments is $\varepsilon-$shadowed by an actual orbit. More precisely, for points $x_{1},x_{2},\ldots,x_{n}$ and legths $k_{1},\ldots,k_{n}\in\mathbb{N}$ one can find times $a_{1},\ldots,a_{n}$ such that $a_{i+1}\leq a_{i}+N$ and a point $x$ such that $d(f^{a_{i}+j}(x),f^{j}(x_{i}))<\varepsilon$ whenever $0\leq j\leq k_{i}.$ Moreover, one can choose $x$ a periodic point with period no more than $a_{n}+k_{n}+N.$

Theorem 3.6** (Bowen, [2]).**

Every transitive Anosov diffeomorphism has the specification property.

Let us speak about rigidity of Anosov diffeomorphisms of the torus, that we will use in this work.

Theorem 3.7** (De La Llave, [12]).**

Let $f,g$ be two $C^{k},k\geq 2,$ Anosov diffeomorphism of $\mathbb{T}^{2}$ and $h$ a homeomorphism of $\mathbb{T}^{2},$ satisfying

[TABLE]

If the Lyapunov exponents at corresponding periodic orbits are the same, then $h\in C^{k-\varepsilon}.$

Rigidity in three dimensional torus was studied in [9]. We use indirectly the following result, that is strongly used in [14].

Theorem 3.8** (Gogolev-Guysinsky, [9]).**

Let f and g be Anosov diffeomorphisms of $\mathbb{T}^{3}$ and

[TABLE]

where $h$ is a homeomorphism homotopic to identity. Suppose that periodic data of $f$ and $g$ coincide, meaning Lyapunov exponents at corresponding periodic orbits are the same. Also assume that $f$ and $g$ can be viewed as partially hyperbolic diffeomorphisms:

[TABLE]

Then the conjugacy $h$ is $C^{1+\nu},$ for some $\nu>0.$

In higher dimensional torus $\mathbb{T}^{d},d\geq 4,$ same periodic data does not imply regularity of the conjugacy, De La Llave in [12] constructed two Anosov diffeomorphisms on $\mathbb{T}^{4},$ with the same periodic data which are only Hölder conjugated. Saghin and Yang in [17] proved.

Theorem 3.9**.**

Let $L$ be an irreducible Anosov automorphism of $\mathbb{T}^{d},d\geq 3,$ with simple real spectrum. If $f$ is a $C^{2}$ volume preserving diffeomorphism $C^{1}$ close to $L$ and has the same Lyapunov exponents of $L,$ at corresponding periodic points, then $f$ is $C^{1+\varepsilon}$ conjugated to $L$ for some $\varepsilon>0.$

4. Two dimensional case

First we remember the statement of Theorem 1.4.

Theorem 4.1**.**

Consider $f:\mathbb{T}^{2}\rightarrow\mathbb{T}^{2}$ a $C^{r},r\geq 2,$ Anosov diffeomorphism. Suppose that for each $\ast\in\{s,u\},$ we have $\lambda^{\ast}_{f}(p)=\lambda^{\ast}_{f}(q),$ for any $p,q$ periodic points of $f,$ then $f$ is $C^{1}$ conjugated with its linearization $L.$

Proof.

Given $\mu$ an $f-$invariant probability measure, by the Closing lemma for hyperbolic measures, we have $\lambda^{\ast}_{\mu}(x,f)=\lambda^{\ast}_{f}(p),\ast\in\{s,u\},$ for $\mu$ almost everywhere $x\in\mathbb{T}^{2},$ here $p$ is a given periodic point for $f.$ By Ruelle formula we have

[TABLE]

Since $f$ is $C^{r},r\geq 2,$ there is a unique SRB probability measure $\nu$ invariant for $f.$ In this case

[TABLE]

So $\nu$ is the maximal entropy measure, then by variational principal we have

[TABLE]

Taking $f^{-1},$ analogously we conclude that

[TABLE]

for any $p\in Per(f).$

By the De La Llave result in [12], we have $f$ and $L$ are $C^{r-\varepsilon},$ conjugated for some $\varepsilon>0.$ ∎

5. Three dimensional case

Let us first state the following lemma.

Lemma 5.1**.**

Let $f:M\rightarrow M$ be a $C^{2}-$Anosov diffeomorphism with constant periodic data. Then $f$ is conservative and the absolutely continuous measure is the maximal entropy measure.

Proof.

Let $x\in M$ be a regular point, define $\Lambda^{s}(x)$ be the sum of negative Lyapunov exponents of the point $x$ and $\Lambda^{u}(x),$ the sum of positive Lyapunov exponents of the point $x.$ Since we have constant periodic data, for any $p,q\in Per(f),$ holds $\Lambda^{\ast}(p)=\Lambda^{\ast}(q),\ast\in\{s,u\}.$ We call $\Lambda^{\ast}$ the value $\Lambda^{\ast}(p),\ast\in\{s,u\},$ where $p$ is a periodic point of $f.$

Let $\mu$ be an $f$-invariant probability measure. Obviously $\mu$ is a hyperbolic measure for $f.$ So, using Anosov Closing lemma, for $\mu$ a.e. $x\in M$ we have

[TABLE]

By Ruelle formula, we have:

[TABLE]

for any $\mu$ an $f-$invariant probability measure.

Let $\mu_{f}$ be a SRB measure of $f.$ So, for $\mu_{f}$ we also have $\Lambda^{\ast}(x)=\Lambda^{\ast},$ for $\mu_{f}$ a.e. $x\in M.$ Using the SRB property of $\mu_{f},$ we get

[TABLE]

by variational principle $h_{\mu_{f}}(f)=h_{top}(f)=\Lambda^{u}.$ So $\mu_{f}$ is the maximal entropy measure of $f.$ Analogously, taking $f^{-1},$ we conclude that $\mu_{f^{-1}}$ is also maximal entropy measure of $f,$ so $\mu_{f}=\mu_{f^{-1}}$ and

[TABLE]

then

[TABLE]

by Bowen result in [3], we conclude that $f$ is conservative and $\mu_{f}$ is the absolutely continuous measure for $f.$ ∎

The conservative case of Theorem 1.5 is proved in [14].

Lemma 5.2** (Corollary 3.3 of [14]).**

Let $f:\mathbb{T}^{3}\rightarrow\mathbb{T}^{3}$ be a $C^{r},r\geq 2,$ volume preserving Anosov diffeomorphism such that $T\mathbb{T}^{3}=E^{s}_{f}\oplus E^{wu}_{f}\oplus E^{su}_{f}.$ Suppose that there are constants $\Lambda^{\sigma}_{f},\sigma\in\{s,wu,su\},$ such that for any $p\in Per(f)$ we have $\lambda^{\sigma}_{f}(p)=\Lambda^{\sigma}_{f},\sigma\in\{s,wu,su\}.$ Then $f$ is $C^{1}$ conjugated with its linearization $L.$

It is important to note that Lemma 5.2 holds in conservative context and the proof is the same. In fact the conjugacy above is $C^{1+\varepsilon},$ for some $\varepsilon>0,$ because [9].

Then Theorem 1.5, follows from the above previous lemmas. In fact, since in Theorem 1.5 we are supposing constant periodic data, by Lemma 5.1 we have $f$ is conservative. So we can apply Lemma 5.2 to conclude that the conjugacy is $C^{1+\varepsilon},$ for some $\varepsilon>0.$

Corollary 5.3**.**

Let $f$ be as in Theorem 1.4( or Theorem 1.5). If every $x$ is regular, then $f$ is $C^{1+\varepsilon}$ conjugated with its linearization $L,$ for some $\varepsilon>0.$

Proof.

It is as consequence of Specification Property of Anosov diffeomorphisms. In fact, if every point is regular, then the periodic data of $f$ is constant. Suppose that $p,q\in Per(f)$ and for some $\ast$ we have $\lambda_{f}^{\ast}(p)<\lambda_{f}^{\ast}(q),$ using specification it is possible to find a point $z$ and subsequences $\{n_{k}\}_{k=1}^{+\infty}$ and $\{m_{k}\}_{k=1}^{+\infty},$ such that for a given small $\varepsilon$ we have

[TABLE]

for all $k\geq 1.$ So $z$ can not be a regular point.

∎

6. Higher dimensions

In this section we will use results about entropy a diffeomorphism $f$ along an expanding and $f-$invariant foliation from [10] and the tools presented in [5] and citation therin.

Proposition 6.1**.**

Let $L:\mathbb{T}^{d}\rightarrow\mathbb{T}^{d}$ be an Anosov linear automorphism, diagonalizable and irreducible over $\mathbb{Q}.$ If $f$ is a $C^{2}-$Anosov diffeomorphism sufficiently $C^{1}-$close to $L$ with constant periodic data, then every point $x\in\mathbb{T}^{d}$ is regular and $\lambda^{\ast}_{f}(x)=\lambda^{\ast}_{f}(p),$ where $p$ is a given periodic point of $f.$ Moreover the convergence as in the definition of Lyapunov exponent is uniform, for all possible invariant direction.

We prove the proposition after some lemmas, in the same lines as in [5].

Lemma 6.2**.**

Let $\mathcal{M}$ be the space of $f-$invariant measures, $\phi$ be a continuous function on $M.$ If $\int\phi d\mu<\lambda,\;\forall\mu\in\mathcal{M},$ then for every $x\in M,$ there exists $n(x)$ such that

[TABLE]

Proof.

See [5]. ∎

Lemma 6.3**.**

Let $\mathcal{M}$ be the space of $f-$invariant measures, $\phi$ be a continuous function on $M.$ If $\int\phi d\mu<\lambda,\;\forall\mu\in\mathcal{M},$ then there exists $N$ such that for all $n\geq N,$ we have

[TABLE]

for all $x\in M.$

Proof.

See [5]. ∎

In the previous lemmas if we replace $\int\phi d\mu<\lambda$ by $\int\phi d\mu>\lambda,$ we can get analogous statements. Let us to prove Proposition 6.1

of the Proposition 6.1.

We go to use $\phi=\log|Df|E^{\ast}_{f}(\cdot)|.$ Since we have ergodic decomposition, we also go to consider $\mu$ an ergodic and $f-$invariant measure. The diffeomorphism $f$ is Anosov, then every $\nu\in\mathcal{M}$ is a hyperbolic measure. Fix $p$ a periodic point of $f.$ Since $f$ has constant periodic data, by Anosov Closing Lemma, we have $\lambda^{\ast}_{f}(x)=\lambda^{\ast}_{f}(p),$ for $\mu$ a.e. $x\in\mathbb{T}^{d}.$ Now, by Ergodic Birkhoff Theorem, we have

[TABLE]

for any $\mu\in\mathcal{M},$ ergodic measure. Consider $\varepsilon>0,$ we can apply the previous lemmas with number $\lambda^{\ast}_{f}(p)+\varepsilon.$ So we obtain that there is a integer $N_{1}>0,$ such that

[TABLE]

for any $x\in\mathbb{T}^{d}$ and $n\geq N_{1}.$ So we have

[TABLE]

Using the previous lemmas in their versions $`<`,$ we can find $N_{2}>0,$ integer such that

[TABLE]

Taking $N=N_{1}+N_{2}$ and using the equations $(\ref{menor})$ and $(\ref{maior}),$ we conclude that $\frac{1}{n}\log(Df|E^{\ast}_{f}(x))$ converges uniformly to $\lambda^{\ast}_{f}(p).$

∎

In Theorem 1.6 we can suppose that the eigenvalues of $L$ satisfying $0<|\beta_{1}^{s}|<\ldots<|\beta_{k}^{s}|<1<|\beta_{1}^{u}|<\ldots<|\beta_{n}^{u}|.$ The Lyapunov exponents of $L,$ are $\lambda^{s}_{i}(L)=\log(|\beta_{i}^{s}|),i=1,\ldots,k$ and $\lambda^{u}_{i}(L)=\log(|\beta_{i}^{u}|),i=1,\ldots,n.$ For $f$ we denote by $\lambda^{u}_{i}(x,f)$ the Lyapunov exponent of $f$ at $x$ in the direction $E^{u,f}_{i},i=1,\ldots,n$ and by $\lambda^{s}_{i}(x,f)$ the Lyapunov exponent of $f$ at $x$ in the direction $E^{s,f}_{i},i=1,\ldots,k,$ in the cases that Lyapunov exponents are defined.

Let us introduce a notation $E^{s,L}_{(1,i)}=E^{s}_{1}\oplus\ldots\oplus E^{s}_{i},i=1,\ldots,k$ and $E^{u,L}_{(1,i)}=E^{u}_{1}\oplus\ldots\oplus E^{u}_{i},i=1,\ldots,n.$ If $j>i,$ we denote $E^{s,L}_{(i,j)}=E^{s}_{i}\oplus\ldots\oplus E^{s}_{j}$ and $E^{u,L}_{(i,j)}=E^{u}_{i}\oplus\ldots\oplus E^{u}_{j}.$

In the setting of Theorem 1.6, it is known by Pesin [15], that if $f$ is $C^{1}-$close to $L,$ then $T\mathbb{T}^{d}$ admits a similar splitting $E^{s}_{f}=E^{s,f}_{1}\oplus E^{s,f}_{2}\oplus\ldots\oplus E^{s,f}_{k}$ and $E^{u}_{L}=E^{u,f}_{1}\oplus E^{u,f}_{2}\oplus\ldots\oplus E^{u,f}_{n}.$ As before, define $E^{u,f}_{(1,i)}=E^{u,f}_{1}\oplus\ldots\oplus E^{u,f}_{i}$ and $E^{s,f}_{(1,i)}=E^{s,f}_{1}\oplus\ldots\oplus E^{s,f}_{i},$ analogously, for $i>j,$ we define $E^{s,f}_{(i,j)}$ and $E^{u,f}_{(i,j)}.$

By continuity of each subbundle, we can take the decomposition $E^{s}_{f}\oplus E^{u,f}_{(1,i)}\oplus E^{u,f}_{(i+1,n)}$ such that it is a uniform partially hyperbolic splitting.

Moreover, by [4], each $E^{u,f}_{(1,i)}=E^{u,f}_{1}\oplus\ldots\oplus E^{u,f}_{i},$ is integrable to an invariant foliation $W^{u,f}_{(1,i)},$ with $i=1,\ldots,n.$ An analogous construction holds for stable directions. By [6], since $f$ is $C^{1}-$close to $L,$ the conjugacy $h$ between $L$ and $f$ is such that $h(W^{u,L}_{(1,i)})=W^{u,f}_{(1,i)},$ the same for stable foliations.

In [10] the authors lead with a notion of topological entropy $h_{top}(f,\mathcal{W})$ of an invariant expanding foliation $\mathcal{W}$ of a diffeomorphism $f.$ They establish variational principle in this sense and relation between $h_{top}(f,\mathcal{W})$ and volume growth of $\mathcal{W}.$

Here $W(x)$ will denote the leaf of $\mathcal{W}$ by $x.$ Given $\delta>0,$ we denote by $W(x,\delta)$ the $\delta-$ball centered in $x$ on $W(x),$ with the induced riemannian distance, that we will denote by $d_{W}.$

Given $x\in M,$ $\varepsilon>0,$ $\delta>0$ and $n\geq 1$ a integer number, let $N_{W}(f,\varepsilon,n,x,\delta)$ be the maximal number of points in $\overline{W(x,\delta)}$ such that $\displaystyle\max_{j=0\ldots,n-1}d_{W}(f^{j}(x),f^{j}(y))\leq\varepsilon.$

Definition 6.4**.**

The unstable entropy of $f$ on $M,$ with respect to the expanding foliation $\mathcal{W}$ is given by

[TABLE]

where

[TABLE]

Define $\mathcal{W}-$volume growth by

[TABLE]

where

[TABLE]

Note that, since we are supposing $\mathcal{W}$ a expanding foliation, the above definition is independent of $\delta$ and the riemannian metric.

Theorem 6.5** (Theorem C of [10]).**

With the notations above we have

[TABLE]

As a consequence of Proposition 6.1 and Theorem 6.5, we have.

Corollary 6.6**.**

If $f$ as in Theorem 1.6 and $W^{u,f}_{(1,i)}$ is the foliation tangent to $E^{u,f}_{(1,i)},$ then $h_{top}(f,W^{u}_{(1,i)})=\displaystyle\sum_{j=1}^{i}\lambda^{u}_{j}(p,f).$

Proof.

Fix $p$ a periodic point of $f.$ By Proposition 6.1we have

[TABLE]

Using Proposition 6.1, the right side of the above expression converges uniformly to $\sum_{j=1}^{i}\lambda^{u}_{j}(p,f).$ So, by Theorem C of [10], we have $h_{top}(f,W^{u}_{(1,i)})=\displaystyle\sum_{j=1}^{i}\lambda^{u}_{j}(p,f),$ as required. ∎

We are ready to prove Theorem 1.6.

Proof.

Since $h(W^{u,L}_{(1,i)})=W^{u,f}_{(1,i)},$ we have $h_{top}(f,W^{u,f}_{(1,i)})=h_{top}(L,W^{u,L}_{(1,i)}).$ Now, consider $\beta^{s}_{i},i=1,\ldots,k,$ the eigenvalues of $L,$ we have

[TABLE]

and $\beta^{u}_{i},i=1,\ldots,n,$ such that

[TABLE]

Let $p$ be a periodic point of $f.$ Since we have constant periodic data, so for any $i=1,\ldots,n$ we have

[TABLE]

for any $i=1,\ldots,n.$ So, for $i=1,$ we have

[TABLE]

for $i=2,$ we have $\lambda^{u}_{1}(p,f)+\lambda^{u}_{2}(p,f)=\lambda^{u}_{1}(L)+\lambda^{u}_{2}(L),$ since $\lambda^{u}_{1}(p,f)=\lambda^{u}_{1}(L),$ we get

[TABLE]

Analogously $\lambda^{u}_{i}(p,f)=\lambda^{u}_{i}(L),i=1,\ldots,n.$

Taking the inverses, we obtain

[TABLE]

so, $f$ and $L$ has the same periodic data, by [8] and [17], the maps $f$ and $L$ are $C^{1+\varepsilon}$ conjugated for some $\varepsilon>0,$ if $f$ is enough $C^{1}-$close to $L.$

∎

As in Corollary 5.3 we have.

Corollary 6.7**.**

Let $L:\mathbb{T}^{d}\rightarrow\mathbb{T}^{d}$ be as in Theorem 1.6. If $f$ is a $C^{r},r\geq 2,$ diffeomorphism, $C^{1}-$close to $L$ and every $x\in\mathbb{T}^{d}$ is regular point of $f,$ then $f$ and $L$ are $C^{1+\varepsilon}-$conjugated, for some $\varepsilon>0.$

The proof is similar to Corollary 5.3.