Pathological center foliation with dimension greater than one

TL;DR

This paper investigates the properties of partially hyperbolic diffeomorphisms on the torus with higher-dimensional center foliations, establishing bounds on Lyapunov exponents and constructing examples with pathological center foliations.

Contribution

It proves bounds on Lyapunov exponents for such diffeomorphisms and constructs new examples with non-absolutely continuous and pathological center foliations.

Findings

Bound on sum of center Lyapunov exponents for certain diffeomorphisms.

Existence of open class of volume-preserving diffeomorphisms with non-absolute continuous center foliation.

Construction of examples with disintegration of volume neither Lebesgue nor atomic.

Abstract

In this paper we are considering partially hyperbolic diffeomorphims of the torus, with $dim(E^c) > 1.$ We prove, under some conditions, that if the all center Lyapunov exponents of the linearization $A,$ of a \mbox{DA-diffeomorphism} $f,$ are positive and the center foliation of $f$ is absolutely continuous, then the sum of the center Lyapunov exponents of $f$ is bounded by the sum of the center Lyapunov exponents of $A.$ After, we construct a $C^1-$open class of volume preserving \mbox{DA-diffeomorphisms}, far from Anosov diffeomorphisms, with non compact pathological two dimensional center foliation. Indeed, each $f$ in this open set satisfies the previously established hypothesis, but the sum of the center Lyapunov exponents of $f$ is greater than the corresponding sum with respect to its linearization. It allows to conclude that the center foliation of $f$ is non absolutely continuous. We still build an example of a DA-diffeomorphism, such that the disintegration of volume along the two dimensional, non compact center foliation is neither Lebesgue nor atomic.

Equations

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

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

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

$$\underset{n\to +\infty }{\lim }\frac{1}{n}\log ||D{f}^n(x)\cdot v||$$

$$d_{\tilde{W}}(x,y)\le Q{d}_{\tilde{M}}(x,y)+Q$$

$$||x-y||>M\Rightarrow \frac{1}{C}<\frac{||{\tilde{f}}^n(x)-{\tilde{f}}^n(y)||}{||{\tilde{A}}^n(x)-{\tilde{A}}^n(y)||}<C.$$

$$B_R(X)=\{y\in {\mathbb{R}}^d;||x-y||<R\text{mbox}forsomex\in X\}.$$

$$||{\pi }_A^{c\perp }(x-y)||<\epsilon ||{\pi }_A^c(x-y)||.$$

$$\mathrm{Vol}({\tilde{f}}^n{\pi }^{-1}(R))\le C_0(1+\epsilon {)}^{n{d}_c}{e}^{n\sum {\lambda }_i^c(A)}\mathrm{Vol}(R)$$

$$||D_{x}f(v)||=\underset{t\to 0}{\lim }\Vert \frac{f(x+tv)-f(x)}{t}\Vert \le \underset{t\to 0}{\lim }\frac{L||x+tv-x||}{|t|}=L||v||$$

$$|\mathrm{Jac}f(x)|=|\det D_{x}f|\le K.$$

$$||d\pi (y)(v)||\ge \beta ||v||$$

$$d_{{\tilde{W}}^c}({\pi }^{-1}(x),{\pi }^{-1}(y))\le length(\gamma )={\int }_{[x,y]}|d{\pi }^{-1}(t)|dt\le L||x-y||.$$

$$(1-\epsilon )e^{n{\lambda }_i^c(A)}||{\pi }^{-1}(x)-{\pi }^{-1}(y)||\le ||{\tilde{A}}^n{\pi }^{-1}(x)-{\tilde{A}}^n{\pi }^{-1}(y)||\le (1+\epsilon )e^{n{\lambda }_i^c(A)}||{\pi }^{-1}(x)-{\pi }^{-1}(y)||$$

$$\frac{{\pi }^{-1}(x)-{\pi }^{-1}(y)}{||{\pi }^{-1}(x)-{\pi }^{-1}(y)||}=v_i^c+e_M$$

$${\tilde{A}}^n\left(\frac{{\pi }^{-1}(x)-{\pi }^{-1}(y)}{||{\pi }^{-1}(x)-{\pi }^{-1}(y)||}\right)={\mu }^n{v}_i^c+{\tilde{A}}^n{e}_M={\mu }^n\left(\frac{{\pi }^{-1}(x)-{\pi }^{-1}(y)}{||{\pi }^{-1}(x)-{\pi }^{-1}(y)||}\right)-{\mu }^n{e}_M+{\tilde{A}}^n{e}_M$$

$$\frac{||{\tilde{A}}^n({\pi }^{-1}(x)-{\pi }^{-1}(y))||}{||{\pi }^{-1}(x)-{\pi }^{-1}(y)||}=\Vert {\mu }^n\left(\frac{{\pi }^{-1}(x)-{\pi }^{-1}(y)}{||{\pi }^{-1}(x)-{\pi }^{-1}(y)||}\right)-{\mu }^n{e}_M+{\tilde{A}}^n{e}_M\Vert$$

$$||{\pi }^{-1}(x)-{\pi }^{-1}(y)||$$

$$\le ||{\pi }^{-1}(x)-{\pi }^{-1}(y)||({\mu }^n+{\mu }^n||e_M||+||{\tilde{A}}^n||||e_M||)$$

$${\mu }^n||e_M||+||{\tilde{A}}^n||||e_M||\le \epsilon {\mu }^n$$

$$(1-\epsilon )e^{n{\lambda }_i^c(A)}||{\pi }^{-1}(x)-{\pi }^{-1}(y)||\le ||{\tilde{A}}^n{\pi }^{-1}(x)-{\tilde{A}}^n{\pi }^{-1}(y)||\le (1+\epsilon )e^{n{\lambda }_i^c(A)}||{\pi }^{-1}(x)-{\pi }^{-1}(y)||$$

$$\tilde{A}({\pi }^{-1}(R))\subset B_K(\tilde{f}({\pi }^{-1}(R)).$$

$$\lambda M(1+\epsilon )>\lambda M+2K,$$

$$\tilde{f}({\pi }^{-1}(R))\subset {\pi }^{-1}((1+\epsilon )\tilde{A}(R)),$$

$$||\tilde{f}{\pi }^{-1}(x)-\tilde{f}{\pi }^{-1}(y)||$$

$$\ge (1+\epsilon {)}^{-1}(1-\epsilon )e^{{\lambda }_i^c(A)}||{\pi }^{-1}(x)-{\pi }^{-1}(y)||$$

$$\ge ||x-y||\ge M,$$

$$\tilde{f}({\pi }^{-1}(R_1))\subset {\pi }^{-1}((1+\epsilon )\tilde{A}(R_1)),$$

$${\tilde{f}}^2({\pi }^{-1}(R))\subset {\pi }^{-1}((1+\epsilon {)}^2{\tilde{A}}^2(R)).$$

$${\tilde{f}}^n({\pi }^{-1}(R))\subset {\pi }^{-1}((1+\epsilon {)}^n{\tilde{A}}^n(R)),$$

$$\mathrm{Vol}({\tilde{f}}^n{\pi }^{-1}(R))$$

$$Z_q=\left\{x\in \tilde{Z};\sum _i{\lambda }_i^c(\tilde{f},x)>\sum _i{\lambda }_i^c(A)+\log \left(1+\frac{1}{q}\right)\right\}.$$

Full text

1. Introduction

A diffeomorphism $f:M\rightarrow M$ of a compact closed smooth manifold is partially hyperbolic if the tangent bundle splits into three invariant sub bundles $TM=E^{s}\oplus E^{c}\oplus E^{u}$ such that $E^{s}$ is contracting, $E^{u}$ is expanding and $E^{c}$ has an intermediate behavior, that is, not as contracting as $E^{s}$ and nor as expanding as $E^{u}$. If $E^{c}=\{0\}$, then $f$ is called uniformly hyperbolic or Anosov diffeomorphism. We denote by $PH^{r}_{\omega}(M),$ the set of all $C^{r}-$partially hyperbolic diffeomorphism preserving the $\omega-$form. Here we study partially hyperbolic diffeomorphisms on torus ${\mathbb{T}}^{d}$ homotopic to a linear Anosov which are known as Derived from Anosov (DA). By [16] and [7] for partially hyperbolic diffeomorphisms, there is foliations $W^{s}$ and $W^{u}$ tangent to $E^{s}$ and $E^{u}$, respectively, but the distribution $E^{c}$ may not be integrable, for instance see section 6.1 of [25]. If $E^{c}$ is one dimensional, then it is integrable, but not necessarily uniquely integrable (see [14]). In [6] shows that for (absolute) partially hyperbolic diffeomorphisms if the foliations $W^{s}$ and $W^{u}$ has a geometrical condition (quasi-isometric), then $E^{c}$ is uniquely integrable, that is, there is a foliation $W^{c}$ tangent to $E^{c}$. Our results relate absolute continuity of the center foliation $W^{c}$ and the Lyapunov exponents.

The Lyapunov exponents play an important role in the ergodic theory and dynamical systems. They are useful tool of the Pesin theory, in the study of entropy, equilibrium states among others. The existence of these exponents is guaranteed by celebrated Osceledec’s Theorem [24]. In general the Lyapunov exponents not vary continuously with $x\in M$ or with the dynamics in the ambient $\operatorname{Diff}^{1}(M)$. We show, under some conditions, that the partially hyperbolic diffeomorphsms in the same homotopic class has its center Lyapunov exponents bounded by center exponents of the its linearization, more precisely:

Theorem A**.**

Let $f:{\mathbb{T}}^{d}\rightarrow{\mathbb{T}}^{d}$ be a volume preserving DA diffeomorphism and consider $A$ its linearization such that:

1. (1)

$\mathrm{dim}E^{c}_{f}=\mathrm{dim}E^{c}_{A}=d_{c}.$** 2. (2)

$E^{c}_{A}=E^{c}_{1}\oplus E^{c}_{2}\oplus\cdots\oplus E^{c}_{d_{c}},$* such that each $E^{c}_{i},i=1,\ldots,d_{c}$ is a unidimensional eigenspace of $A$ with the corresponding Lyapunov exponent, $\lambda_{i}^{c}(A)>0,$ for any $i=1,2,\ldots,d_{c}.$* 3. (3)

$f$* is dynamically coherent and in the universal cover $\angle(T_{x}\widetilde{W}^{c}_{f},(\widetilde{E}^{c}_{A})^{\bot})>\alpha>0$ for any $x\in{\mathbb{R}}^{d}.$*

If the center foliation $W^{c}_{f}$ is absolutely continuous, then $\displaystyle\sum_{i=1}^{d_{c}}\lambda_{i}^{c}(f,x)\leq\displaystyle\sum_{i=1}^{d_{c}}\lambda_{i}^{c}(A)$ for $m-$ almost everywhere $x\in{\mathbb{T}}^{d}$.

Absolutely continuous of foliations is a very related property with Lyapunov exponents. We understand it by absolutely continuous the behavior of foliation relative to disintegration of the volume in the foliation. By Fubini’s Theorem we know that $C^{1}$-foliations are absolutely continuous, but this property is not valid for $C^{0}$-foliation. The non absolutely continuous foliation have been referred to as ”Fubini’s Nightmare” or ”Fubini Foiled”, Katok construct examples of foliation with this property (see [22]), for more definitions and discussions about absolutely continuous see [28]. In [1], [2] it is shown that the stable and unstable foliations of $C^{2}-$Anosov diffeomorphisms are absolutely continuous and this property is fundamental in the proof of the ergodicity of the volume preserving $C^{2}-$Anosov diffeomorphisms, [1]. We also know that for partially hyperbolic $C^{2}$ diffeomorphisms the stable and unstable foliations are absolutely continuous [7], but in general, we do not know if the center foliation (when there is) is absolutely continuous. One of the first to study the behavior of the center foliation was R. Mañé, in a letter (unpublished) to M. Shub, they relate the absolutely continuous of compact center foliations in which the Lyapunov exponents are non zero. These ideas were very useful in the study on absolutely continuous of compact center foliations (see [29] and [15]). The non absolutely continuous of the non compact center foliations is also very common, in [10] show there are open sets in $PH^{r}_{m}({\mathbb{T}}^{3}),r\geq 2,$ of diffeomorphisms with one dimensional non compact center foliation and non absolutely continuous. We build an open set $U\subset PH^{r}_{m}({\mathbb{T}}^{4}),$ $r\geq 2$, of diffeomorphisms with non compact two dimensional center foliation and non absolutely continuous.

Denote by $DA^{r}_{m}(\mathbb{T}^{d})$ the set of all $C^{r}$ and $m$ preserving DA diffeomorphism of $\mathbb{T}^{d}$ and $\mathcal{A}(\mathbb{T}^{d})$ the set of all Anosov diffeomorphism of $\mathbb{T}^{d}.$

Theorem B**.**

There is a $C^{1}-$open set $U\subset DA^{r}_{m}(\mathbb{T}^{4})\setminus\overline{\mathcal{A}(\mathbb{T}^{4})},$ with $r\geq 2,$ such that each $f\in U$ has the same diagonalizable linearization $A,$ $\dim E^{c}_{f}=2,$ satisfying the hypothesis $1),2)$ and $3)$ of Theorem A, but $\sum_{i=1}^{2}\lambda^{c}_{i}(f,x)>\sum_{i=1}^{2}\lambda^{c}_{i}(A),$ for $m$ almost everywhere $x\in\mathbb{T}^{4}.$ Particularly $W^{c}_{f}$ is non absolutely continuous for every $f\in U.$

This Theorem is related to a result of [30] which shows the existence of open sets in $PH^{r}_{m}({\mathbb{T}}^{d})$ of diffeomorphisms near a linear automorphism with non absolutely continuous one dimensional center foliation. In [20], the second autor display a open set in $PH^{r}_{m}({\mathbb{T}}^{3})$ of diffeomorphisms far from linear Anosov with non absolutely continuous one dimensional center foliation. Our results are a generalized version of results of [20].

A natural question is how is disintegration of the volume form along the foliations in these cases? This is, how are conditional measures of non absolutely continuous foliations? In [26] and [27] built examples of partially hyperbolic diffeomorphisms of ${\mathbb{T}}^{3}$ wherein the disintegration of the center foliation is atomic (totally contrary to the Lebesgue measure). In [3] it is shown for perturbation of time-one of geodesic flow, the disintegration along the center foliation is or Lebesgue or atomic. In [31] and [32] it is shown which the disintegration of the one dimensional center foliation of partially hyperbolic diffeomorphisms of ${\mathbb{T}}^{3}$ can be neither Lebesgue nor atomic (in contrast to the dichotomy of [3]). We construct an example of the partially hyperbolic diffeomorphism with two dimensional center foliation whose disintegration is neither Lebesgue nor atomic.

Theorem C**.**

There is a DA diffeomorphism $f:{\mathbb{T}}^{4}\rightarrow{\mathbb{T}}^{4}$ volume preserving with non compact two dimensional center foliation such that the disintegration of volume along the center leaves is neither Lebesgue nor atomic.

2. Preliminaries

Let $M$ be a $C^{\infty}$ Riemannian closed (compact, connected and boundaryless) manifold. A $C^{1}-$diffeomorphism $f:M\rightarrow M$ is called a 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 $v^{\sigma}\in E^{\sigma}_{x},\sigma\in\{s,c,u\}$ for every $x\in M$ satisfy:

[TABLE]

moreover

[TABLE]

We say that a partially hyperbolic diffeomorphism $f$ is an absolute partially hyperbolic diffeomorphism if

[TABLE]

for every $x,y,z\in M$ and $v^{s},v^{c},v^{u}$ are unitary vectors in $E^{s}_{x},E^{c}_{y},E^{u}_{z}$ respectively.

From now on, in this paper, when we require partial hyperbolicity, we mean absolute partially hyperbolicity and all diffeomorphisms considered are at least $C^{1}.$

Definition 2.1**.**

A partially hyperbolic diffeomorphism $f:M\rightarrow M$ is called dynamically coherent if $E^{cs}:=E^{c}\oplus E^{s}$ and $E^{cu}:=E^{c}\oplus E^{u}$ are uniquely integrable to invariant foliations $W^{cs}$ and $W^{cu},$ respectively the center stable and center unstable foliations. Particularly $E^{c}$ is uniquely integrable to the center foliation $W^{c},$ which is obtained by the intersection $W^{cs}\cap W^{cu}.$

Every diffeomorphism of the torus $\mathbb{T}^{d}$ induces an automorphism of the fundamental group and there exists a unique linear diffeomorphism $f_{\ast}$ which induces the same automorphism on $\pi_{1}(\mathbb{T}^{d}).$ The diffeomorphism $f_{\ast}$ is called linearization of $f.$

Definition 2.2**.**

Let $f:\mathbb{T}^{d}\rightarrow\mathbb{T}^{d}$ be a partially hyperbolic diffeomorphism, $f$ is called a derived from Anosov (DA) diffeomorphism if its linearization $f_{\ast}:\mathbb{T}^{d}\rightarrow\mathbb{T}^{d}$ is a linear Anosov automorphism.

Theorem 2.3** ([franks1969anosov], [19]).**

Let $f:\mathbb{T}^{d}\rightarrow\mathbb{T}^{d}$ be a derived from Anosov diffeomorphism, then $f$ is semi-conjugate to $A$, that is, there exists a continuous surjective function $h:{\mathbb{T}}^{d}\rightarrow{\mathbb{T}}^{d}$ such that $h\circ f=A\circ h$.

2.1. Lyapunov Exponents

Lyapunov exponents are important constants and measure the asymptotic behavior of dynamics in tangent space level. Let $f:M\rightarrow M$ be a measure preserving diffeomorphism. Then by Oseledec’s Theorem, for almost every $x\in M$ and any $v\in T_{x}M$ the following limit exists:

[TABLE]

and it is equal to one of the Lyapunov exponents of $f.$

2.2. Absolute Continuity

Roughly speaking a foliation $W$ of $M$ is absolutely continuous if satisfies: Given a set $Z\subset M,$ such that $Z$ intersects the leaf $W(x)$ on a zero measure set of the leaf, with $x$ along a full Lebesgue set of $M,$ then $Z$ is a zero measure set of $M.$ More precisely we write:

Definition 2.4**.**

We say that a foliation $W$ of $M$ is absolutely continuous if given any $W-$foliated box $B$ and a Lebesgue measurable set $Z,$ such that $Leb_{W(x)\cap B}(W(x)\cap Z)=0,$ for $m_{B}-$ almost everywhere $x\in B,$ then $m_{B}(Z)=0.$ Here $m_{B}$ denotes the Lebesgue measure on $B$ and $Leb_{W(x)\cap B}$ is the Lebesgue measure of the submanifold $W(x)$ restricted to $B.$

It means that if $P$ is such that $m_{B}(P)>0,$ then there are a measurable subset $B^{\prime}\subset B,$ such that $m_{B}(B^{\prime})>0$ and $Leb_{W(x)\cap B}(W(x)\cap Z)>0$ for every $x\in B^{\prime}.$

The study of absolute continuity of the center foliation started with Mañé, that noted a interesting relation between absolute continuity and the center Lyapunov exponent. The Mañé’s argument can be explained as the following theorem:

Theorem 2.5**.**

Let $f:M\rightarrow M$ be a partially hyperbolic, dynamically coherent such that $\dim(E^{c})=1$ and $W^{c}$ is a compact foliation. Suppose $f$ preserves a volume form $m$ on $M,$ and the set of $x\in M$ such that $\lambda^{c}_{f}(x)>0$ has positive volume. Then $W^{c}$ is non absolutely continuous.

Proof.

Denote by $P$ the set of $x\in M$ such that $\lambda^{c}_{f}(x)>0.$ Consider the set $\Lambda_{k,l,n}=\{x\in P|\;||Df^{j}(x)|E^{c}||\geq e^{\frac{j}{k}},\;\mbox{for every}\;j\geq l,\;\mbox{and}\;|W^{c}(x)|<n\},$ here $|W^{c}(x)|$ denotes the size of the center leaf $W^{c}(x)$ through $x.$

We have $P=\displaystyle\bigcup_{k,l,n\in\mathbb{N}}\Lambda_{k,l,n},$ in particular there are $k_{0},l_{0},n_{0}$ such that $m(\Lambda_{k_{0},l_{0},n_{0}})>0.$ Supposing that $W^{c}$ is an absolutely continuous foliation, there is a center leaf $W^{c}(x),$ such that it intersects $\Lambda_{k_{0},l_{0},n_{0}}$ on a positive Lebesgue measure set of the leaf. By Poincaré-recurrence Theorem, the point $x$ can be chosen a recurrent point, particularly there is a subsequence $n_{k}$ such that $f^{n_{k}}(x)\in\Lambda_{k_{0},l_{0},n_{0}},$ and it implies that the size $|W^{c}({f^{n_{k}}(x)})|<n_{0}.$

On the other hand, we denote $\alpha=Leb_{W^{c}(x)}(W^{c}(x)\cap\Lambda_{k_{0},l_{0},n_{0}}),$ so if $j\geq k_{0}$ we have $|W^{c}(f^{j}(x))|\geq\alpha\cdot e^{\frac{j}{k_{0}}}\rightarrow+\infty$ when $j\rightarrow+\infty,$ and it contradicts $|W^{c}({f^{n_{k}}(x)})|<n_{0}$ for a subsequence $n_{k}.$ ∎

Consequently all one dimensional compact and absolutely continuous center foliation implies that $\lambda^{c}_{f}(x)=0,$ for $m-$ almost everywhere $x\in M.$

We make a generalized version of the Mañé’s argument, comparing volumes, in the proof of the Theorem A.

Remark 2.6*.*

Katok exhibits an example of a volume preserving partially hyperbolic diffeomorphism $f:\mathbb{T}^{3}\rightarrow\mathbb{T}^{3}$ such that $W^{c}$ is compact, non absolutely continuous and $\lambda^{c}_{f}(x)=0$ for $m-$ a.e. $x\in\mathbb{T}^{3}.$ See [12] and citations therein.

2.3. Quasi-isometry

An important tool that we need is quasi-isometry of foliations.

Definition 2.7**.**

A foliation $W$ of a closed manifold $M$ is called quasi-isometric if there is a constant $Q>0$, such that in the universal cover $\widetilde{M}$ we have:

[TABLE]

for every $x,y$ points in the same lifted leaf $\widetilde{W}$, where $\widetilde{W}$ denotes the lift of $W$ on $\widetilde{M}$.

Here $d_{\widetilde{W}}$ denotes the Riemannian metric on $\widetilde{W}$ and $d_{\widetilde{M}}$ is a Riemannian metric of the ambient $\widetilde{M}$.

Every diffeomorphism of the torus ${\mathbb{T}}^{d}$ induces an automorphism of the fundamental group and there exists a unique linear diffeomorphism $f_{\ast}$ which induces the same automorphism on $\pi_{1}({\mathbb{T}}^{d})$. The diffeomorphism $f_{\ast}$ is called linearization of $f$. When $f$ is DA diffeomorphism, then its linearization is the linear Anosov automorphism. Below we have a well-known result.

Proposition 2.8**.**

Let $f:{\mathbb{T}}^{d}\rightarrow{\mathbb{T}}^{d}$ be a partially hyperbolic diffeomorphism with linearization $A$, then for each $n\in{\mathbb{Z}}$ and $C>1$ there is an $M>0$ such that for all $x,y\in{\mathbb{R}}^{d}$ and

[TABLE]

*. *

For a subset $X\subset{\mathbb{R}}^{d}$ and $R>0$, led $B_{R}(X)$ denote the neighbourhood

[TABLE]

Proposition 2.9** ([11]).**

Let $f:{\mathbb{T}}^{d}\rightarrow{\mathbb{T}}^{d}$ be a partially hyperbolic diffeomorphism dynamically coherent with linearization $A$, then there is a constant $R_{c}$ such that for all $x\in{\mathbb{R}}^{d}$,

• •

$\widetilde{W}^{cs}_{f}(x)\subset B_{R_{c}}(\widetilde{W}^{cs}_{A}(x))$,

• •

$\widetilde{W}^{cu}_{f}(x)\subset B_{R_{c}}(\widetilde{W}^{cu}_{A}(x))$,

• •

$\widetilde{W}^{c}_{f}(x)\subset B_{R_{c}}(\widetilde{W}^{c}_{A}(x))$.

Corollary 2.10** ([11]).**

If $||x-y||\rightarrow\infty$ where $y\in\widetilde{W}^{c}_{f}(x)$ then $\frac{x-y}{||x-y||}\rightarrow\widetilde{E}^{c}_{A}(x)$ uniformly. More precisely, for $\varepsilon>0$ there exists $M>0$ such that if $x\in{\mathbb{R}}^{d}$, $y\in\widetilde{W}^{c}_{f}(x)$ and $||x-y||>M$, then

[TABLE]

where $\pi_{A}^{c}$ is the orthogonal projection in the subspace $E^{c}_{A}$ and $\pi_{A}^{c\perp}$ is the projection in the orthogonal subspace $E^{c\perp}_{A}.$

From Brin, [6] we have.

Proposition 2.11** ([6]).**

Let $W$ be a $k-$dimensional foliation of $\mathbb{R}^{d}.$ Suppose that there is an $(d-k)$ dimensional plane $P$ such that $T_{x}W(x)\cap P=\{0\}$ and $\angle(T_{x}W(x),P)>\beta>0$ for every $x\in\mathbb{R}^{d}.$ Then $W$ is quasi-isometric.

Theorem 2.12** ([6]).**

Let $f:M\rightarrow M$ a partially hyperbolic diffeomorphism. If $W^{s}_{f}$ and $W^{u}_{f}$ are quasi-isometric foliations, then $f$ is dynamically coherent.

3. Proof of Theorem A

Lemma 3.1**.**

Let $f$ be as in the Theorem A, then the foliation $\widetilde{W}^{c}_{f}$ is quasi-isometric.

Proof.

We just need to apply the Proposition 2.11, doing $P=(\widetilde{E}^{c}_{A})^{\bot}.$ By the third item of the assumptions of Theorem A, we have $\angle(T_{x}\widetilde{W}^{c}_{f},P)>\alpha>0$, thus by the Proposition 2.11, $\widetilde{W}^{c}_{f}$ is quasi-isometric. ∎

Proposition 3.2**.**

Let $f:{\mathbb{T}}^{d}\rightarrow{\mathbb{T}}^{d}$ and $A$ be as in the Theorem A,

then given $\varepsilon>0$, there is $M>0$ such that

[TABLE]

for any $n,$ where $\pi$ is the orthogonal projection from $\widetilde{W}^{c}_{f}(z)$ to $\widetilde{E}^{c}_{A}(z)$ (parallel to $(\widetilde{E}^{c}_{A})^{\bot}$) and $R$ is a hypercube in $\widetilde{E}^{c}_{A}(z)$ contained $z,$ with dimension equal to $d$ and the length of each edge is bigger than $M.$

To prove the above proposition we need the following auxiliar results:

Lemma 3.3**.**

Let $f:M\rightarrow M$ be a Lipschitzian map, then there is $K>0$ such that $|\mathrm{Jac}f(x)|\leq K,$ for any $x\in M$ such that $f$ is differentiable at $x.$

Proof of Lemma.

Let $L$ be the the Lipschitz constant of $f$, give $x\in M$ such that $f$ is differentiable and $v\in T_{x}M$, then

[TABLE]

It implies that $||D_{x}f||=\displaystyle\sup_{v\in T_{x}M}\frac{||D_{x}f(v)||}{||v||}\leq L.$

Since $\mathbb{R}^{n^{2}}$ is isomorphic to $M_{n}(\mathbb{R}),$ the space of all $n\times n-$matrixes with real coefficients, by equivalence between norms of $\mathbb{R}^{n^{2}}$ we have $\{A\in M_{n}(\mathbb{R})|\;||A||\leq L\}$ is compact. Since $\det:M_{n}(\mathbb{R})\rightarrow\mathbb{R}$ is continuous, then there is $K\geq 0$ such that

[TABLE]

It concludes the proof. ∎

Proposition 3.4**.**

Let $f:X\rightarrow Y$ be a covering map such that $X$ is connected by path and $Y$ is simply connected, then $f$ is a homeomorphism.

Proof of the Proposition 3.2.

Claim 1: For $z\in{\mathbb{R}}^{d}$, the orthogonal projection $\pi:\widetilde{W}^{c}_{f}(z)\rightarrow\widetilde{E}^{c}_{A}$ (parallel to $(\widetilde{E}^{c}_{A})^{\bot}$) is a uniform bi-Lipschitz diffeomorphism.

Proof of Claim 1.

The argument used here is similar to that used in [6], Proposition 4. By item 3 of the Theorem A the plane $(\widetilde{E}_{A}^{c})^{\bot}$ is transversal to the foliation $\widetilde{W}^{c}_{f}$ then there is $\beta>0$ such that

[TABLE]

for any $x\in\widetilde{W}^{c}_{f}(z)$ and $v\in T_{x}\widetilde{W}^{c}_{f}(z)$, it implies that $\mathrm{Jac}\pi(x)\neq 0$, by Inverse Function Theorem for each $x\in\widetilde{W}^{c}_{f}(z)$ there is a ball $B(\delta,x)\subset\widetilde{W}^{c}_{f}(z)$ such that $\pi|_{B(\delta,x)}$ is a diffeomorphism. From proof of Inverse Function Theorem and Equation 3.1, $\delta$ can be taken independent of $x$. Again, by Equation 3.1 there is $\varepsilon>0$, independent of $x$ such that $B(\varepsilon,\pi(x))\subset\pi(B(\delta,x))$. For to prove which $\pi$ is surjective, we will show that $\pi(\widetilde{W}^{c}_{f}(z))$ is open and closed set. As $\pi$ is a local homeomorphism, then $\pi(\widetilde{W}^{c}_{f}(z))$ is open. To verify that it is also closed let $y_{n}\in\pi(\widetilde{W}^{c}_{f}(z))$ be a sequence converging to $y$, hence there is a $n_{0}$ enough large such that $y\in B(\varepsilon,y_{n_{0}})$ and therefore $y\in\pi(\widetilde{W}^{c}_{f}(z))$, then $\pi$ is surjective. Moreover $\pi$ is a covering map, in fact for any $y\in\widetilde{E}^{c}_{A}$ there is a neighborhood $B(\varepsilon,y)$ with $\pi^{-1}(B(\varepsilon,y))=\bigcup U_{i}$, where $\pi:U_{i}\rightarrow B(\varepsilon,y)$ is a diffeomorphism. The injectivity follows of the Proposition 3.4.

The map $\pi$ is Lipschitz. In fact $||\pi(x)-\pi(y)||\leq||x-y||\leq d_{\widetilde{W}_{f}^{c}}(x,y)$. Let us show that $\pi^{-1}$ is Lipschitz. From the Equation 3.1, consider $L=\frac{1}{\beta}$ such that $||d\pi^{-1}(y)(v)||\leq L||v||$ for any $y\in W^{c}_{f}(z)$ e $v\in T_{y}W^{c}_{f}(z)$. Let $[x,y]$ be the rect segment in $\widetilde{E}^{c}_{A}$ connecting $x$ to $y,$ so the set $\pi^{-1}([x,y])=\gamma$ is a smooth curve connecting the points $\pi^{-1}(x)$ and $\pi^{-1}(y)$ in $\widetilde{W}^{c}_{f}(z)$. So,

[TABLE]

∎

As $E^{c}(A)=E^{c}_{1}\oplus E^{c}_{2}\oplus\cdots\oplus E^{c}_{d_{c}}$, suppose that the edges of the $d_{c}$-cube $R$ lies in hypercubes parallel to $E^{c}_{i}.$

As in (Lemma 3.6, [21]), we can prove:

Claim 2: Let $x=(x_{1},\ldots,x_{i},\ldots,x_{d_{c}})$ and $y=(x_{1},\ldots,y_{i},\ldots,x_{d_{c}})$ with $x,y\in\partial R$, $x-y\in E_{i}^{c},$ then given $n$ and $\varepsilon>0$, there is $M>0$ such that, if $||x-y||>M,$ then

[TABLE]

for all $i=1,2,\ldots,d_{c}$

Proof of Claim 2.

Let $\widetilde{E}^{c}_{i}(A)$ be the eigenspace corresponding to $\lambda_{i}^{c}(A)$ and denote by $\mu:=e^{\lambda_{i}^{c}(A)},$ the absolute value of the an eigenvalue $\tilde{A}$ corresponding $\widetilde{E}^{c}_{i}(A)$. By Corollary 2.10 we have

[TABLE]

where $v_{i}^{c}$ is an unit eigenvector of $\tilde{A}$ in $\widetilde{E}^{c}_{i}(A)$ direction and $e_{M}$ is a vector correction, converges uniformly to zero when $M\rightarrow+\infty.$ It follows that

[TABLE]

it implies that

[TABLE]

thus

[TABLE]

Since $n$ is fixed and $||e_{M}||\rightarrow 0$ when $M\rightarrow\infty$, choose a large $M$ such that

[TABLE]

we conclude that

[TABLE]

∎

Claim 3: There is $M>0,$ such that $\tilde{f}^{n}(\pi^{-1}(R))\subset\pi^{-1}((1+\varepsilon)^{n}\tilde{A}^{n}(R)),$ for every $n\geq 1,$ up to translation of $(1+\varepsilon)^{n}\tilde{A}^{n}(R).$ We are considering $R$ a $d_{c}$ hypercube with edges parallel to each direction $E^{c}_{i},i=1,\ldots,d_{c},$ (see Figure 1), and the length of each edge of $R$ is at least $M.$

Proof.

Since $||\tilde{f}-\tilde{A}||<K,$ for some $K>0,$ We have

[TABLE]

Moreover, $\tilde{A}(\pi^{-1}(R))$ projects on $\tilde{A}(R),$ via $\pi.$ We can choose $M,$ very large, such that $\tilde{f}(\pi^{-1}(R))\subset\pi^{-1}((1+\varepsilon)\tilde{A}(R)),$ where $(1+\varepsilon)\tilde{A}(R)$ denotes a dilatation of $\tilde{A}(R),$ by a factor $1+\varepsilon.$ For this, is suffices that

[TABLE]

for every $\lambda$ a center eigenvalue of $A.$ So it is sufficient that $\frac{2K}{\lambda M}<\varepsilon.$

Thus, for $n=1,$ we have

[TABLE]

up to a possible translation of $\tilde{A}(R).$

Now, denote $R_{1}=(1+\varepsilon)\tilde{A}(R).$ It is important to note that $||\tilde{f}\pi^{-1}(x)-\tilde{f}\pi^{-1}(y)||>M.$ In fact, we can take $M$ satisfying also the Proposition 2.8 and Claim 2, so

[TABLE]

if $\varepsilon$ is enough small.

So we can do the same process before, for $R_{1},$ we obtain:

[TABLE]

Since $\tilde{f}^{2}(\pi^{-1}(R))\subset\tilde{f}(\pi^{-1}(R_{1})),$ it implies

[TABLE]

Following this inductive argument, we conclude that

[TABLE]

for every $n\geq 1.$ ∎

By Lemma 3.3 follows that

[TABLE]

The constant $C_{0},$ comes from Lemma 3.3. It concludes the proof of the Proposition 3.2. ∎

Proof of Theorem A.

Suppose by contradiction, $W^{c}_{f}$ is absolutely continuous and there exists a positive volume set $Z\in{\mathbb{T}}^{d}$ such that $\sum_{i}\lambda_{i}^{c}(f,x)>\sum_{i}\lambda_{i}^{c}(A)$ for any $x\in Z$.

Let $P:{\mathbb{R}}^{d}\rightarrow{\mathbb{T}}^{d}$ be the covering map, $D\subset{\mathbb{R}}^{d}$ a fundamental domain and $\widetilde{Z}=P^{-1}(Z)\cap D$. We have $\mathrm{Vol}(\widetilde{Z})>0$. Since $D\tilde{f}^{n}$ and $Df^{n}$ are conjugate matrices, $D\tilde{f}^{n}=DP^{-1}\circ Df^{n}\circ DP$, it follows that $\sum_{i}\lambda_{i}^{c}(\tilde{f},x)>\sum_{i}\lambda_{i}^{c}(A)$ for any $x\in\widetilde{Z}$.

For each $q\in{\mathbb{N}}-\{0\}$ we define the set

[TABLE]

We have $\bigcup_{q=1}^{\infty}Z_{q}=\widetilde{Z}$, thus there is $q$ such that $m(Z_{q})>0$. For each $x\in Z_{q}$ follows that

[TABLE]

So there is $n_{0}$ such that for $n\geq n_{0}$ we have

[TABLE]

it implies that

[TABLE]

By this fact, for every $n>0$ we define

[TABLE]

There is $N>0$ with $\mathrm{Vol}(Z_{q,N})>0,$ for some integer $q>0.$

For each $x\in D$, consider $B_{x}\subset{\mathbb{R}}^{d}$ a foliated box of $\widetilde{W}^{c}_{f}.$ By compactness there is finite cover $\{B_{x_{i}}\}_{i=1}^{j}$ covering $\overline{D}.$ Since $W^{c}_{f}$ is absolutely continuous and the covering map is smooth, then $\widetilde{W}^{c}_{f}$ is absolutely continuous, thus there is some $i$ and $p\in B_{x_{i}}$ such that $\mathrm{Vol}(B_{x_{i}}\cap\widetilde{W}^{c}_{f}(p)\cap Z_{q,N})>0$.

There is a set $\pi^{-1}(R)\subset\widetilde{W}^{c}_{f}(p)$, where $\pi^{-1}(R)$ is as in the Proposition 3.2 containing $p$ and the edges of the cube $R$ are enough large such that $\mathrm{Vol}(\pi^{-1}(R)\cap Z_{q,N})>0.$ Choose a number $\varepsilon>0$ as in Proposition 3.2, such that $(1+\varepsilon)^{d_{c}}<\left(1+\frac{1}{q}\right).$ By Proposition 3.2 we have

[TABLE]

for any $n\geq 1.$

On the other hand, let $\alpha>0$ be such that $\mathrm{Vol}(\pi^{-1}(R)\cap Z_{q,N})=\alpha\mathrm{Vol}(\pi^{-1}(R)),$ for $n>N,$ we have

[TABLE]

The equations $(\ref{eq4})$ and $(\ref{eq5})$ given us a contradiction when $n$ is enough large, thus proving the Theorem A.

∎

4. Proof of Theorem C

For the construction we need the following results.

Proposition 4.1** ([4]).**

Let $(M,m)$ be a compact manifold endowed with a $C^{r}$ volume form, $r\geq 2.$ Let $f$ be a $C^{1}$ and $m-$preserving diffeomorphisms of $M,$ admitting a dominated partially hyperbolic splitting $TM=E^{s}\oplus E^{c}\oplus E^{u}.$ Then there are arbitrarily $C^{1}-$close and $m-$preserving perturbation $g$ of $f,$ such that

[TABLE]

where $J^{c}_{f}(x)$ is the absolute value of the determinant of $Df$ restricted to $E^{c}_{f}(x).$

Remark 4.2*.*

When $f$ is $C^{r},r\geq 1,$ in Proposition 4.1 above the perturbation $g$ also can be taken $C^{r}.$

Theorem 4.3** ([8]).**

Let $f:{\mathbb{T}}^{d}\rightarrow{\mathbb{T}}^{d}$ be a partially hyperbolic diffeomorphism that is isotopic to a linear Anosov automorphism along a path of partially hyperbolic diffeomorphisms, then $f$ is dynamically coherent. Moreover, for each $x\in{\mathbb{T}}^{d}$, $W^{c}_{f}(x)=h^{-1}(W^{c}_{A}(h(x)))$, where $h$ is the semiconjugacy between $f$ and $A$.

Proof of Theorem C.

Consider a linear Anosov diffeomorphism of the ${\mathbb{T}}^{4}$ induced by the matrix:

[TABLE]

The characteristic polynomial of $A_{n}$ is $p(t)=t^{4}-8t^{3}+19t^{2}-14t+1$ and its eigenvalues are approximately $\beta^{s}=0,08;\,\,\,\beta^{c}_{1}=1,2;\,\,\,\beta^{c}_{2}=2,3$  and   $\beta^{u}=4,3$.

We use the Proposition 4.1 to get a $C^{2}-$ volume preserving Anosov diffeomorphism $f=A\circ\phi^{cu},$ where $\phi^{cu}$ is a volume preserving perturbation that preserves $E^{s}_{A_{n}}$ such that

[TABLE]

By Theorem A, the foliation $W^{c}_{f}$ is non absolutely continuous, thus the disintegration of the volume along center box foliated is not Lebesgue. By conjugacy between $f$ and $A,$ the stable index of $f$ is equal to one of $A.$ Using the Pesin’s formula, we get

[TABLE]

then the volume $m$ is the maximal measure entropy of $f.$ By uniqueness of the maximal entropy measure of $f$ it follows that $h_{\ast}m=m,$ it means that $h$ preserves $m.$

By Theorem 4.3, $W^{c}_{f}(x)=h^{-1}(W^{c}_{A}(h(x)))$, hence the disintegration of the volume along $W^{c}_{f}$ induces, by $h$, a disintegration of the volume along $W^{c}_{A}$, hence the disintegration along $W^{c}_{f}$ is not atomic, because the disintegration along $W^{c}_{A}$ is Lebesgue. ∎

5. Proof of Theorem B

For to begin the construction, for each $n\geq 1,$ we consider $A_{n}:\mathbb{T}^{4}\rightarrow\mathbb{T}^{4}$ the Anosov automorphism the $\mathbb{T}^{4}$ induced by the matrices

[TABLE]

Denote by $p_{n}(t)$ a characteristic polynomial of $A_{n},$ we have

[TABLE]

Proposition 5.1**.**

For large $n,$ the matrix $A_{n}$ have four eigenvalues $0<\beta^{s}_{n}<1<\beta^{c}_{1,n}<\beta^{c}_{2,n}<\beta^{u}_{n},$ such that

[TABLE]

Proof.

We have $p_{n}(n)=1-2n\;\mbox{and}\;p_{n}(n+1)=n^{3}-n^{2}-4n-1,$ for $n$ enough large $p_{n}$ has a root $\beta^{u}_{n}\in(n,n+1),$ so $\displaystyle\lim_{n\rightarrow+\infty}\frac{\beta^{u}_{n}}{n}=1.$

Fix $\varepsilon>0$, we have $p_{n}(1)=-1$ and $p_{n}(1+\varepsilon)=(2+\varepsilon-\varepsilon^{2})\varepsilon n+\varepsilon^{4}-3\varepsilon^{2}-4\varepsilon-1,$ if $\varepsilon$ is very small, when $n$ is large we have $p_{n}(1+\varepsilon)>0,$ so $p_{n}$ has a root $\beta^{c}_{1,n}\in(1,1+\varepsilon),$ for large values of $n.$

Again, fix $\varepsilon>0$ a small arbitrary number, we have $p_{n}(3)=-5$ and

$p_{n}(3-\varepsilon)=(6-5\varepsilon+\varepsilon^{2})\varepsilon n-16\varepsilon+21\varepsilon^{2}-8\varepsilon^{3}+\varepsilon^{4}-5,$ if $\varepsilon$ is very small, when $n$ is large we have $p_{n}(3-\varepsilon)>0,$ so $p_{n}$ has a root $\beta^{c}_{2,n}\in(3-\varepsilon,3),$ for large values of $n.$

Since $p_{n}$ has four real roots, $\beta^{u}_{n}\cdot\beta^{c}_{1,n}\cdot\beta^{c}_{2,n}\cdot\beta^{s}_{n}=1,$ and

[TABLE]

then as $\beta^{s}_{n}=\frac{1}{\beta^{u}_{n}\cdot\beta^{c}_{1,n}\cdot\beta^{c}_{2,n}},$ finally we have $\displaystyle\lim_{n\rightarrow+\infty}3n\beta^{s}_{n}=1.$ ∎

We go to denote $\beta^{c}_{1,n}=1+\alpha_{n},$ such that $\alpha_{n}\rightarrow 0^{+}.$

Proposition 5.2**.**

Let $v^{u}_{n},v^{c}_{2,n},v^{c}_{1,n},v^{s}_{n}$ be corresponding unit eigenvectors of $\beta^{u}_{n}$, $\beta^{c}_{2,n}$, $\beta^{c}_{1,n}$ and $\beta^{s}_{n}$ respectively. There are unit vectors $v^{u},v^{c}_{2},v^{c}_{1},v^{s},$ such that

[TABLE]

moreover $\mathcal{B}=\{v^{u},v^{c}_{2},v^{c}_{1},v^{s}\}$ is a basis of $\mathbb{R}^{4}.$

Proof.

The vectores

w^{s}_{n}=\left(\begin{array}[]{c}1\\ \beta^{s}_{n}\\ (\beta^{s}_{n})^{2}\\ (\beta^{s}_{n})^{3}\\ \end{array}\right), w^{c}_{1,n}=\left(\begin{array}[]{c}1\\ \beta^{c}_{1,n}\\ (\beta^{c}_{1,n})^{2}\\ (\beta^{c}_{1,n})^{3}\\ \end{array}\right), w^{c}_{2,n}=\left(\begin{array}[]{c}1\\ \beta^{c}_{2,n}\\ (\beta^{c}_{2,n})^{2}\\ (\beta^{c}_{2,n})^{3}\\ \end{array}\right) and w^{u}_{n}=\left(\begin{array}[]{c}1\\ \beta^{u}_{n}\\ (\beta^{u}_{n})^{2}\\ (\beta^{u}_{n})^{3}\\ \end{array}\right) are eigenvectors of $A_{n}$ associated with the respective eigenvalues $\beta^{s}_{n}$, $\beta^{c}_{1,n}$, $\beta^{c}_{2,n}$ and $\beta^{u}_{n}$. When $n\rightarrow\infty$, we have

[TABLE]

which are LI vectors. ∎

Denote by H^{u}_{n}=\langle v^{u}_{n}\rangle,$$H^{c}_{1,n}=\langle v^{c}_{1,n}\rangle, $H^{c}_{2,n}=\langle v^{c}_{2,n}\rangle$ and $H^{s}_{n}=\langle v^{s}_{n}\rangle.$

We can see $A_{n}:\mathbb{T}^{4}\rightarrow\mathbb{T}^{4}$ with two partially hyperbolic decomposition.

[TABLE]

where $F^{u}_{n}=H^{u}_{n}\oplus H^{c}_{2,n},$ $F^{c}_{n}=H^{c}_{1,n}$ and $F^{s}_{n}=H^{s}_{n}$ and

[TABLE]

where $E^{u}_{n}=H^{u}_{n},$ $E^{c}_{n}=H^{c}_{1,n}\oplus H^{c}_{2,n}$ and $E^{s}_{n}=H^{s}_{n}.$

Remark 5.3*.*

From now on, for simplicity we can consider $v^{u}=(1,0,0,0)$, $v^{c}_{2}=(0,1,0,0)$, $v^{c}_{1}=(0,0,1,0)$ and $v^{s}=(0,0,0,1).$

5.1. Perturbing $A_{n}$

By Proposition 4.1, consider $g_{n}$ a small perturbation of $A_{n},$ an Anosov diffeomorphism such that $A_{n}$ and $g_{n}$ are close in the $C^{1}$ topology and

[TABLE]

Also we go to require that

[TABLE]

It is possible by Proposition 4.1, since $g_{n}=A_{n}\circ H_{n}$ and $g_{n}$ can be found $C^{1}-$arbitrarely close to $A_{n}.$

By continuity of the partially hyperbolic decomposition, if $g_{n}$ is $C^{1}$ close to $A_{n},$ then there is an invariant partially hyperbolic splitting $T\mathbb{T}^{4}=E^{u}_{g_{n}}\oplus E^{c}_{g_{n}}\oplus E^{s}_{g_{n}},$ where $\dim(E^{\sigma}_{g_{n}})=\dim(E^{\sigma}_{n}),\sigma\in\{s,c,u\}$ and $E^{\sigma}_{g_{n}}$ and $E^{\sigma}_{n},\sigma\in\{s,c,u\}$ are close. Again, by continuity of partially hyperbolic splitting, each $g_{n}$ admits a splitting $T\mathbb{T}^{4}=F^{u}_{g_{n}}\oplus F^{c}_{g_{n}}\oplus F^{s}_{g_{n}},$ close to $F^{u}_{n}\oplus F^{c}_{n}\oplus F^{s}_{n}.$

Now we go to modify the stable index of a fixed point of $g_{n}$. For this we use the arguments of a well-known lemma of [9, Lema 1.1], which allows us to make a change in the differential of $g_{n}$ in a finite number of points. In [5, Proposition 7.4] shows that if the initial diffeomorphism is conservative then we can take the perturbation also conservative. There is a more general version of this result in [18, Lema 2.4]. For our proposes we only need the following lemma.

Lemma 5.4** ([5]).**

For any $N\in{\mathbb{N}}$ e $\rho>0$, there is a $\delta>0$ and a neighborhood of the identity $B(\delta,Id)\subset SL(N,{\mathbb{R}})$ such that for any $A\in B(\delta,Id)$ there is a $h\in\operatorname{Diff}^{r}_{m}({\mathbb{R}}^{N})$ satisfying the properties:

• •

$h=Id$* out of a unitary ball centered on [math],*

• •

$h(0)=0$* and $Dh(0)=A$,*

• •

$||Dh-Id||<\rho$.

Let $p_{n}$ be a fixed point for $g_{n}.$ For each fixed $n,$ consider a system $\{V_{n,j}\}_{j=0}^{+\infty}$ of small open balls centered in $p_{n}.$ The neighborhoods $V_{n,j}$ are constructed after.

Fixed $n>0,$ enough large, consider matrixes $D_{n}$ satisfying:

1. (1)

$D_{n}$ and $Dg_{n}(p_{n})$ have the same eigenspaces and the corresponding eigenvalues have the same sign. 2. (2)

$D_{n}|F^{u}_{g_{n}}(p_{n})=Dg_{n}(p_{n})|F^{u}_{g_{n}}(p_{n}),$ 3. (3)

$||D_{n}|F^{c}_{g_{n}}(p_{n})||=1-\alpha_{n},\;\;$ 4. (4)

$D_{n}|F^{s}_{g_{n}}(p_{n})$ is taken coherently with $\det(D_{n})=\det(Dg_{n}(p_{n})).$

Since $g_{n}$ can be taken $C^{1}-$arbitrarily close to $A_{n},$ we can to choose $g_{n}$ such that $\angle(F^{\sigma}_{g_{n}}(\cdot),F^{\sigma}_{n}(\cdot))\rightarrow 0$ uniformly in $\mathbb{T}^{4},$ for each $\sigma\in\{s,c,u\}.$ For each $n>0$ and $x\in\mathbb{T}^{4}$ it is possible to find a basis $\{w^{u}_{n}(x),w^{c}_{2,n}(x),w^{c}_{1,n}(x)w^{s}_{n}(x)\}$ composed by unit vectors of $T_{x}\mathbb{T}^{4}$ uniformly close to $\{v^{u}_{n},v^{c}_{2,n},v^{c}_{1,n}v^{s}_{n}\}.$ By uniform convergence, the change matrix $C_{n}(x),$ from the basis $\mathcal{B}_{n}^{\prime}=\{w^{u}_{n}(x),w^{c}_{2,n}(x),w^{c}_{1,n}(x),w^{s}_{n}(x)\}$ to the canonical basis, is uniform close to the identity matrix (see Remark 5.3).

The matrixes $D_{n}$ will be diagonal matrixes, in the basis $\mathcal{B}_{n}^{\prime}.$ Denote by $\theta^{u}_{n},$ $\theta^{c}_{1,n},$ $\theta^{c}_{2,n}$ and $\theta^{s}_{n}=\frac{1}{\theta^{u}_{n}\cdot\theta^{c}_{1,n}\cdot\theta^{c}_{2,n}},$ the four eigenvalues of $Dg_{n}(p_{n})$ respectively in the directions of the basis the basis $\mathcal{B}_{n}^{\prime}.$ Then, using the items above, in the basis $\mathcal{B}_{n}^{\prime}$ we have:

[TABLE]

From $\ref{convergeId1},$ we have $1<\theta^{c}_{1,n}<1+2\alpha_{n}$ it leads that $\max\left\{|1-\frac{\theta^{c}_{1,n}}{1-\alpha_{n}}|,|1-\frac{1-\alpha_{n}}{\theta^{c}_{1,n}}|\right\}<3,5\alpha_{n},$ for $n$ large. Since $C_{n}(x)$ is uniform close to the identity, then in the canonical basis we have

[TABLE]

Fix $\delta>0,$ such that every $h\in\operatorname{Diff}^{1}_{m}(\mathbb{T}^{4})$ that is $\delta-C^{1}-$close to $Id$ is homotopical to $Id.$

Since by 5.5 the norm $||[Dg_{n}(p_{n})]^{-1}\cdot D_{n}-Id||\rightarrow 0$ when $n\rightarrow+\infty.$

Now as the Lemma 5.4, fix $\sigma>0$, which depends only on $\delta$, such that for all $h\in\operatorname{Diff}^{r}_{m}({\mathbb{R}}^{4})$ with $||h-Id||_{C^{1}}\leq\sigma$ is homotopic to identity. For Equation 5.5 take $n$ so that $4\cdot\alpha_{n}$ is enough small such that by Lemma 5.4 there is a $h_{n}\in\operatorname{Diff}^{r}_{m}({\mathbb{R}}^{4})$ satisfying:

• •

$||h_{n}-Id||_{C^{1}}<\sigma,$

• •

$h_{n}=Id$ in $p_{n}$ and out of a unitary ball centered on $p_{n}$.

• •

$Dh_{n}(p_{n})=[Dg_{n}(p_{n})]^{-1}\cdot D_{n}.$

For each $j$ suppose that $V_{n,j}=B(p_{n},\varepsilon_{j})$ ($\varepsilon_{j}$ It will be defined later). We define $h_{n,j}\in\operatorname{Diff}^{r}_{m}({\mathbb{R}}^{4})$ by

[TABLE]

• •

$||h_{n,j}-Id||_{C^{1}}<\sigma,$

• •

$h_{n,j}(p_{n})=p_{n},\;h_{n,j}=Id\;\mbox{out of}\,\,V_{n,j},$

• •

$Dh_{n,j}(p_{n})=Dh_{n}(p_{n})=[Dg_{n}(p_{n})]^{-1}\cdot D_{n}.$

Let $g_{n,j}=g_{n}\circ h_{n,j}$. Then

[TABLE]

Remark 5.5*.*

We observe that if $g_{n}$ is $C^{r},r\geq 1,$ then each $g_{n,j}$ can be taken $C^{r}.$ That is because the perturbation of $g_{n}$ involves exponential function and some bump functions which are $C^{\infty}.$

5.2. The choice of the open system $\{V_{n,j}\}_{j=0}^{+\infty}$

Fix $n>0$ and $\varepsilon>0$ a small number. Let $V_{n,0}=B(p_{n},\varepsilon)$ be the open ball centered in $p_{n}$ with radius $\varepsilon.$ Since $p_{n}$ is a fixed point for the diffeomorphism $g_{n},$ then the distance between $p_{n}$ and $g_{n}^{j}(\mathbb{T}^{4}\setminus V_{n,0})$ is positive for any $j$ a integer number. Define recursively $\varepsilon_{j}=(20n)^{-j}\varepsilon_{j-1},j=1,2,\ldots$ and $V_{n,j}=B(p_{n},\varepsilon_{j}),j=1,2,\ldots.$ Since the Lipschitz constants of $g_{n,j}$ and its inverse is bounded by $20n,$ we have $g_{n,j}^{k}(x)\notin V_{n,j}$ for every $x\notin V_{n,j-1},$ $|k|\leq j$ and $j=1,2,\ldots.$

5.3. Properties of the diffeomorphisms $g_{n,j}$

Lemma 5.6**.**

The diffeomorphisms $g_{n,j}$ are partially hyperbolic diffeomorphisms homotopic to $A_{n}$.

Proof.

It is because $g_{n,j}=g_{n}\circ h_{n,j}=A_{n}\circ H_{n}\circ h_{n,j}$ and $H_{n}\circ h_{n,j}$ is arbitrarily $C^{1}-$close of the identity, for $n$ enough large. The partial hyperbolicity follows from Lemma 6.2. ∎

Lemma 5.7**.**

For each large $n$ the diffeomorphisms $g_{n,j}$ are (absolute) partially hyperbolic and there are two decompositions $F_{g_{n,j}}^{u}\oplus F_{g_{n,j}}^{c}\oplus F_{g_{n,j}}^{s}$ and $E_{g_{n,j}}^{u}\oplus E_{g_{n,j}}^{c}\oplus E_{g_{n,j}}^{s}$ such that

1. (1)

$E^{\sigma}_{g_{n,j}}(x)\rightarrow E^{\sigma}_{g_{n}}(x),\sigma\in\{s,c,u\},\;\mbox{where}\;j\rightarrow+\infty,$** 2. (2)

$F^{\sigma}_{g_{n,j}}(x)\rightarrow F^{\sigma}_{g_{n}}(x),\sigma\in\{s,c,u\},\;\mbox{where}\;j\rightarrow+\infty,$**

Proof.

The existence of two partially hyperbolic decompositions follow, as above, from the Lemma 6.2. In fact, for both decompositions we can take $\varepsilon$ (depends only on $\Theta$) constant, because $\Theta>0$ for $n$ great enough, moreover $H_{n}\circ h_{n,j}$ is arbitrarily $C^{1}-$close of the identity.

Now fix the decomposition $E_{g_{n,j}}^{u}\oplus E_{g_{n,j}}^{c}\oplus E_{g_{n,j}}^{s},$ for this decompositions the are the cone fields (constructed in the Lemma 6.2).

By construction of the family $\{V_{n,j}\}$, we have $\operatorname{diam}(V_{n,j})\rightarrow 0$ where $j\rightarrow\infty$ and for any $x\neq p_{n}$, there is $j_{x}\geq 0$ such that

• (1)

$x\notin\overline{V_{n,j}}$ for all $j\geq j_{x}$

• (2)

$g^{k}_{n,j}(x)=g^{k}_{n}(x)$ for all $-j<k<j$ and $j>j_{x}$

• (3)

$g_{n,j}=g_{n}$ out of $V_{n,j}$ for all $j>0$

For $\sigma=cu$, suppose that $C^{cu}(x,\beta)$ is the cone field in the direction $E^{cu}$, then

[TABLE]

and

[TABLE]

By cone construction, the sequence of cones $Dg^{k}_{n,j}(g^{-k}_{n,j}(x))(C^{cu}(g^{-k}_{n,j}(x),\beta))$ is decreasing nested for $k=1,2,3,\ldots$. If $j>j_{x}$ we have, by definition of $j_{x}$, that

[TABLE]

Since $Dg^{j}_{n}\left(g^{-j}_{n}(x)\right)\left(C^{cu}\left(g^{-j}_{n}(x),\beta\right)\right)\rightarrow E^{cu}_{g_{n}}$ where $j\rightarrow\infty$, then

[TABLE]

Moreover by construction $E^{cu}_{g_{n,j}}(p_{n})=E^{cu}_{g_{n}}(p_{n})$.

Analogously, we proof for the other directions $E_{g_{n,j}}^{cs},E_{g_{n,j}}^{u},E_{g_{n,j}}^{s}$ and also for the other decomposition $F_{g_{n,j}}^{u}\oplus F_{g_{n,j}}^{c}\oplus F_{g_{n,j}}^{s}$. ∎

Lemma 5.8**.**

For $n$ and $j$ enough large and for both partially hyperbolic decompositions of $g_{n,j}$ holds:

• (1)

There is $\alpha>0$ such that $\angle(E^{\sigma}_{g_{n,j}},E^{\sigma}_{A_{n}})>\alpha$ and $\angle(F^{\sigma}_{g_{n,j}},F^{\sigma}_{A_{n}})>\alpha,$

• (2)

The foliations $W^{s}(g_{n,j})$ and $W^{u}(g_{n,j})$ are quasi-isometric in the universal cover,

• (3)

$g_{n,j}$* is dynamically coherent and the foliation $W^{c}(g_{n,j})$ is quasi-isometric in the universal cover.*

• (4)

If $h$ is the semiconjugacy between $A_{n}$ and $g_{n,j}$, then $W^{c}_{g_{n,j}}(x)=h^{-1}(W^{c}_{A_{n}}(h(x)))$. In particular the two dimensional foliation $W^{c}_{g_{n,j}}$ is not compact.

Proof.

The item (1) follows of the Lemma 5.7. The item (2) follows of the item (1) and Proposition 2.11. The item (2) together with the Theorem 2.12 implies that $g_{n,j}$ is dynamically coherent and again by item (1) and Proposition 2.11 implies that $W^{c}(g_{n,j})$ is quasi-isometric in the universal cover. The item (4) follows of the Theorem 4.3. In fact, since $H_{n}\circ h_{n,j}$ is arbitrarily close to $Id$ in the $C^{1}-$topololy, its combined with the Lemma 6.2 of the appendix, leads us to each $A_{n}\circ I_{t},t\in[0,1],$ is partially hyperpolic, where $I_{t}=(1-t)Id+t(H_{n}\circ h_{n,j}).$ It is clear, because if $H_{n}\circ h_{n,j}$ is $C^{1}-$close to $Id,$ the $I_{t},$ so are. The path $t\mapsto A_{n}\circ I_{t}$ is fully contained in the partially hyperbolic path connect component of $A_{n},$ we are able to apply Theorem 4.3. ∎

5.4. Conclusion of the proof of Theorem C

Fix a large $n$ and $x\neq p_{n}.$ Since $E^{c}_{g_{n,j}}(x)\rightarrow E^{c}_{g_{n}}(x)$ and $g_{n,j}=g_{n}$ out of $V_{n,j},$ thus for every $j>j_{x}$ we have $Dg_{n,j}(x)=Dg_{n}(x),$ then by Lemma 5.7

[TABLE]

Since $H_{n}$ and $(H_{n}\circ h_{n,j})$ are $C^{1}$ close to $Id$ and $g_{n,j}=A_{n}\circ H_{n}\circ h_{n,j},$ we have $||Dg_{n,j}||<20n$ for large $n,$ then by dominated convergence:

[TABLE]

The DA-diffeomorphisms $g_{n,j}$ are not Anosov. In fact, if $g_{n,j}$ was Anosov, then they would be conjugated to $A_{n},$ in particular the stable index of fixed points of $A_{n}$ and $g_{n,j}$ would be the same.

Take a volume preserving partially hyperbolic diffeomorphism $g_{n,j}$ satisfying

[TABLE]

it is possible by the expression $(\ref{last}).$

Since $g_{n,j}$ is not Anosov, we have $g_{n,j}\in\partial(\overline{\mathcal{A}(\mathbb{T}^{4})})\cup DA^{r}_{m}(\mathbb{T}^{4})\setminus\overline{\mathcal{A}(\mathbb{T}^{4})},$ in both cases it is possible to perturb $g_{n,j}$ to an $f_{n}\in DA^{r}_{m}(\mathbb{T}^{4})\setminus\overline{\mathcal{A}(\mathbb{T}^{4})},r\geq 2,$ an stably ergodic partially hyperbolic diffeomorphism, such that

[TABLE]

The existence of such $f_{n}$ is due to the continuity of the structure $F^{u}\oplus F^{c}\oplus F^{s},$ $\dim F^{c}=1$ and Theorems A and B in [13]. Now consider $U\subset DA^{r}_{m}(\mathbb{T}^{4})\setminus\overline{\mathcal{A}(\mathbb{T}^{4})}$, $r\geq 2,$ an small $C^{1}-$neighborhood of $f_{n}.$ If we take $U$ a suitable neighborhood, $f_{n},$ such that any $f\in U$ is ergodic, isotopic to $A_{n},$ and

[TABLE]

Equation $(\ref{finalmente2}),$ joint with Theorem B implies that the bi-dimensional center foliation tangent to $E^{c}_{f}$ is not absolutely continuous.

6. Appendix: Cone Conditions

In this section we adapt a result of appendix of [26] from ${\mathbb{T}}^{3}$ to ${\mathbb{T}}^{d}$ using all the notation. As in [26] we remember the equivalent definition of absolutely partially hyperbolic diffeomorphism by discretion of the cone.

Definition 6.1**.**

Given an orthogonal splitting of the tangent bundle of $M$, $TM=E\oplus F$, and a constat $\beta>0$, for any $x\in M$ we define the cone centered in $E(x)$ with angle $\beta$ as

[TABLE]

Given a partially hyperbolic diffeomorphism $f:{\mathbb{T}}^{d}\rightarrow{\mathbb{T}}^{d}$ with invariant splitting $TM=E^{s}\oplus E^{c}\oplus E^{u}$, there is an adapted inner product (and then an adapted norm) with respect to which the splitting is orthogonal (see [25]). Thus, given $\beta>0$ we can define standard families of cones centered on the fiber bundles $E^{\sigma}(x)$ with angle $\beta>0$, $C^{\sigma}(x,\beta)$, $\sigma=s,c,u,cs,cu$.

Consider $f:M\rightarrow M$ an absolutely partially hyperbolic diffeomorphism. Using an adapted norm $||\cdot||$, we can consider the invariant splitting $TM=E^{s}\oplus E^{c}\oplus E^{u}$ As being an orthogonal splitting, and there are numbers

[TABLE]

for which

[TABLE]

Partial hyperbolicity can be described in terms of invariant cone families (see [25], pg. 15). More specifically, let $f:{\mathbb{T}}^{d}\rightarrow{\mathbb{T}}^{d}$ be a partially hyperbolic diffeomorphism and let

[TABLE]

a orthogonal splitting of $T{\mathbb{T}}^{d}$. Give $\beta>0$ define the families of cones

[TABLE]

where $E^{cs}(x)=E^{c}\oplus E^{s}(x),\mbox{e}E^{cu}(x)=E^{c}\oplus E^{u}(x)$. Then $f$ is absolutely partially hyperbolic if, and only if, there is $0<\beta<1$ and constants $0<\mu_{1}<\lambda_{2}\leq\mu_{2}<\lambda_{3}$ with $\mu_{1}<1$ and $\lambda_{3}>1$ such that

[TABLE]

and

[TABLE]

For the linear case we find an explicit relation between the angle of the invariant cones families and the ratio of domination between unstable, stable and central bundles.

Consider $f:{\mathbb{T}}^{d}\rightarrow{\mathbb{T}}^{d}$ a linear partially hyperbolic diffeomorphism and denote by $\lambda_{1}^{s},\lambda_{2}^{s},\ldots,\lambda_{i}^{s},\lambda_{1}^{c}\ldots,\lambda_{j}^{c},\lambda_{1}^{u},\ldots,\lambda_{k-1}^{u},\lambda_{k}^{u},$ its eigenvalues, where

[TABLE]

Consider

[TABLE]

We can choose a constant $\beta>0$ such that

[TABLE]

By definition of $\beta$ we have

[TABLE]

We can find constants $\mu_{1},\lambda_{2},\mu_{2},\lambda_{3}$ such that

[TABLE]

where ${\mu_{1}<1<\lambda_{3}}$

Now, we go to check that with the constants defined by 6.3 and 6.4, the families of stable, unstable, center-stable and center-unstable cones satisfies 6 and 6.

$\bullet$ Let $v=v_{s}+v_{cu}\in C^{cu}(x,\beta)$, then

[TABLE]

This is $Df(x)(C^{cu}(x,\beta))\subset C^{cu}(f(x),\beta)$.

Furthermore

[TABLE]

By 6.4 we have that $|\lambda_{1}^{c}|>(1+\beta)\lambda_{2}$. Hence,

[TABLE]

Then $||Df(x)v||>\lambda_{2}||v||$.

Similarly we do to $C^{cs}(x,\beta)$, $C^{u}(x,\beta)$ and $C^{s}(x,\beta)$.

The next lemma is completely analogous to (Lemma 6.1, [26]), as well as its demonstration. This lemma show the size of the $C^{1}$-neighborhood of the linear partially hyperbolic diffeomorphism such that all diffeomorphism in this neighborhood are absolutely partially hyperbolic diffeomorphism.

Lemma 6.2**.**

Let $f:{\mathbb{T}}^{d}\rightarrow{\mathbb{T}}^{d}$ be a linear partially hyperbolic diffeomorphism with eigenvalues $\lambda_{1}^{s},\lambda_{2}^{s},\ldots,\lambda_{i}^{s},\lambda_{1}^{c}\ldots,\lambda_{j}^{c},\lambda_{1}^{u},\ldots,\lambda_{k-1}^{u},\lambda_{k}^{u},$ where

[TABLE]

then there is a constant $\varepsilon>0$ such that, for any diffeomorphism $g:{\mathbb{T}}^{d}\rightarrow{\mathbb{T}}^{d}$ with $||g-Id||_{C^{1}}<\varepsilon$ (adapted norm of $f$), the composition $f\circ g$ is an absolutely partially hyperbolic diffeomorphism. The constant $\varepsilon$ depends only on $\Theta:=\min\left\{\frac{\lambda_{1}^{c}}{\lambda^{s}_{i}},\frac{\lambda^{u}_{1}}{\lambda^{c}_{j}}\right\}$.

Proof.

Let $f$ be as in the statement and denote the invariant splitting of $f$ by $T_{x}M=E^{s}(x)\oplus E^{c}(x)\oplus E^{u}(x)$. Consider the adapted norm $||\cdot||$ in a way that the invariant splitting is orthogonal. Now, since $f$ is a linear partially hyperbolic diffeomorphism we can choose constants

[TABLE]

and let $\beta$ as defined in the equations 6.3 e 6.4.

For $v\in T_{x}M$ we can write $v=v_{s}+v_{c}+v_{u}$ with $v_{\sigma}\in E^{\sigma}(x),\sigma=s,c,u.$

$\bullet$ If $v\in C^{u}(x,\beta)$, then $||v_{cs}||\leq\beta||v_{u}||$, where $v_{cs}=v_{s}+v_{c}$, thus

[TABLE]

this is, $Df(x)(C^{u}(x,\beta))\subset C^{u}(f(x),(\mu_{2}/\lambda_{3})\beta)$.

$\bullet$ If $v\in C^{cu}(x,\beta)$, then $||v_{s}||\leq\beta||v_{cu}||$, where $v_{cu}=v_{c}+v_{u}$, thus

[TABLE]

this is, $Df(x)(C^{cu}(x,\beta))\subset C^{u}(f(x),(\mu_{1}/\lambda_{2})\beta)$.

$\bullet$ If $v\in C^{s}(x,\beta)$, then $||v_{cu}||\leq\beta||v_{s}||$, where $v_{cu}=v_{c}+v_{u}$, thus

[TABLE]

this is, $Df^{-1}(x)(C^{s}(x,\beta))\subset C^{s}(f^{-1}(x),(\mu_{1}/\lambda_{2})\beta)$.

$\bullet$ If $v\in C^{cs}(x,\beta)$, then $||v_{u}||\leq\beta||v_{cs}||$, where $v_{cs}=v_{c}+v_{s}$, thus

[TABLE]

this is, $Df^{-1}(x)(C^{cs}(x,\beta))\subset C^{cs}(f^{-1}(x),(\mu_{2}/\lambda_{3})\beta)$.

Define $\gamma:=\max\left\{\frac{\mu_{2}}{\lambda_{3}},\frac{\mu_{1}}{\lambda_{2}}\right\}<1$. So we have

[TABLE]

Observe that by equations 6.3 and 6.4, $\beta$ and $\gamma$ depends only on the rations $\frac{\lambda_{1}^{c}}{\lambda^{s}_{i}}$ and $\frac{\lambda^{u}_{j}}{\lambda^{c}_{1}}$.

Since the invariant splitting is constant, we can take an $\varepsilon>0$ depending only on the ratios $\frac{\lambda_{1}^{c}}{\lambda^{s}_{i}}$ and $\frac{\lambda^{u}_{j}}{\lambda^{c}_{1}}$ such that if $||g-Id||_{0}<\varepsilon$, then

[TABLE]

let

[TABLE]

with

[TABLE]

Thus we have (see Figure 3)

[TABLE]

Now, we need to show uniform contraction and expansion on these families of cones.

$\bullet$ If $v\in C^{u}(x,\gamma\beta)$, then

[TABLE]

$\bullet$ If $v\in C^{cs}(x,\beta)$, then

[TABLE]

$\bullet$ If $v\in C^{cu}(x,\gamma\beta)$, then

[TABLE]

$\bullet$ If $v\in C^{s}(x,\beta)$, then

[TABLE]

Furthermore,

[TABLE]

so that $f\circ g$ is absolutely partially hyperbolic as we claimed. ∎