# Physical measures and absolute continuity
for one-dimensional center direction

Marcelo Viana1 and Jiagang Yang2,1<EMAIL_ADDRESS>www.impa.br/ viana/
<EMAIL_ADDRESS>
###### Abstract.

For a class of partially hyperbolic $C^{k}$, $k>1$ diffeomorphisms with circle
center leaves we prove existence and finiteness of physical (or Sinai-Ruelle-
Bowen) measures, whose basins cover a full Lebesgue measure subset of the
ambient manifold. Our conditions contain an open and dense subset of all
$C^{k}$ partially hyperbolic skew-products on compact circle bundles.

Our arguments blend ideas from the theory of Gibbs states for diffeomorphisms
with mostly contracting center direction together with recent progress in the
theory of cocycles over hyperbolic systems that call into play geometric
properties of invariant foliations such as absolute continuity. Recent results
show that absolute continuity of the center foliation is often a rigid
property among volume preserving systems. We prove that this is not at all the
case in the dissipative setting, where absolute continuity can even be robust.

Partially supported by CNPq and PRONEX-Dynamical Systems. J.Y. was supported
by scholarships from TWAS-CNPq and FAPERJ.

1IMPA, Est. D. Castorina 110, 22460-320 Rio de Janeiro, Brazil

2Instituto de Matemática, Universidade Federal Fluminense, Niterói, Brazil

###### Contents

  1. 1 Introduction
  2. 2 Statement of results
  3. 3 Gibbs $u$-states
  4. 4 Mostly contracting center
  5. 5 Finiteness and stability of physical measures
  6. 6 Absolute continuity for mostly contracting center
  7. 7 Robust absolute continuity

##  1\. Introduction

Let $f:N\to N$ be a diffeomorphism on some compact Riemannian manifold $N$. An
invariant probability $\mu$ is a _physical (Sinai, Ruelle, Bowen) measure_ for
$f$ if the set of points $z\in N$ for which

(1) $\frac{1}{n}\sum_{j=0}^{n-1}\delta_{f^{i}(z)}\to\mu\quad\text{(in the
weak${}^{*}$ sense)}$

has positive volume. This set is denoted $B(\mu)$ and called the _basin_ of
$\mu$. A program for investigating the physical measures of partially
hyperbolic diffeomorphisms was initiated by Alves, Bonatti, Viana in [5, 19],
who proved existence and finiteness when $f$ is either “mostly expanding”
(asymptotic forward expansion) or “mostly contracting” (asymptotic forward
contraction) along the center direction.

In this paper we analyze the existence and finiteness problem without a priori
conditions on the behavior along the center direction, in the case when the
center bundle has dimension $1$. Our results are illustrated by the following
example.

Suppose $N=M\times S^{1}$, for some compact manifold $M$, and $f_{0}:N\to N$
is a partially hyperbolic _skew-product_

(2) $f_{0}:M\times S^{1}\to M\times S^{1},\quad
f_{0}(x,\theta)=(g_{0}(x),h_{0}(x,\theta))$

with center bundle $E^{c}$ coinciding with the vertical direction
$\\{0\\}\times TS^{1}$ at every point. This implies $g_{0}$ is an Anosov
diffeomorphism, and we also take it to be transitive (all known Anosov
diffeomorphisms being transitive). Assume $f_{0}$ is of class $C^{k}$ for some
$k>1$, not necessarily an integer.

######  Theorem A.

There exists a $C^{k}$ neighborhood ${\mathcal{U}}_{0}$ of $f_{0}$ such that
for every $f\in{\mathcal{U}}_{0}$ which is accessible and whose center stable
foliation is absolutely continuous there exists a finite number of physical
measures. These measures are ergodic, the union of their basins has full
volume in $N$, and the center Lyapunov exponents are either negative or zero.
In the latter (zero) case the physical measure is unique.

The subset of accessible diffeomorphisms is $C^{1}$ open and $C^{k}$ dense in
the neighborhood of $f_{0}$ (Theorem 1.5 of Niţicǎ, Török [32]). Absolute
continuity is also quite common in this context as we are going to see. That
is surprising, since Avila, Viana, Wilkinson [14] have recently shown that
absolute continuity of the center foliation is a rigid property for _volume
preserving_ perturbations of skew-products. In contrast, here we prove

######  Theorem B.

Suppose $f_{0}$ exhibits some periodic vertical leaf $\ell$ such that
$f_{0}^{per(\ell)}\mid\ell$ is Morse-Smale with a unique periodic attractor
and repeller. Then $f_{0}$ is in the closure of an open set ${\mathcal{V}}$ of
$C^{k}$ diffeomorphisms such that for every $f\in{\mathcal{V}}$,

  * •

the center stable, the center unstable, and the center foliation are
absolutely continuous

  * •

both $f$ and its inverse have a unique physical measure, whose basin has full
Lebesgue measure in $N$.

Then the same is true for $f_{0}=g_{0}\times\operatorname{id}$, since it is
$C^{k}$ approximated by diffeomorphisms as in the hypothesis of the theorem.

Although we are primarily interested in general (dissipative) diffeomorphisms,
our methods also shed some light on the issue of absolute continuity in the
volume preserving context. Let $\lambda^{c}(f)$ denote the integrated center
Lyapunov exponent of $f$ relative to the Lebesgue measure.

######  Theorem C.

For any small $C^{1}$ neighborhood ${\mathcal{W}}$ of
$f_{0}=g_{0}\times\operatorname{id}$ in the space of volume preserving
diffeomorphisms of $N$,

  1. (1)

the subset ${\mathcal{W}}_{0}$ of diffeomorphisms $f\in{\mathcal{W}}$ such
that $\lambda^{c}(f)\neq 0$ is $C^{1}$ open and dense in ${\mathcal{W}}$;

  2. (2)

if $f\in{\mathcal{W}}_{0}$ and $\lambda^{c}(f)>0$ then the center foliation
and the center stable foliation are not (even upper leafwise) absolutely
continuous;

  3. (3)

there exists a non-empty $C^{1}$ open set
${\mathcal{W}}_{1}\subset\\{f\in{\mathcal{W}}_{0}:\lambda^{c}(f)>0\\}$ such
that the center unstable foliation of every $g\in{\mathcal{W}}_{1}$ is
absolutely continuous.

Claims (2) and (3) remain true when $\lambda^{c}(f)<0$, if one exchanges
center stable with center unstable. Every $C^{k}$, $k>1$ diffeomorphism
$f\in{\mathcal{W}}_{1}$ has a $C^{k}$ neighborhood ${\mathcal{W}}_{f}$ in the
space of all (possibly dissipative) diffeomorphisms where the center unstable
foliation remains absolutely continuous.

Theorems A and B follow from more detailed statements that we present in the
next section, where we also recall the main notions involved. A discussion of
the volume preserving case is given in Section 7.3, including the proof of
Theorem C and a (partly conjectural) scenario.

##  2\. Statement of results

Let ${\mathcal{P}}_{*}^{k}(N)$ be the space of partially hyperbolic,
dynamically coherent, $C^{k}$ diffeomorphisms whose center leaves are compact,
with any dimension, and form a fiber bundle. Unless otherwise stated, we
always assume $k>1$. Most of our results concern the subspace
${\mathcal{P}}_{1}^{k}(N)$ of diffeomorphisms with $1$-dimensional center
dimension. Let us begin by recalling the notions involved in these
definitions.

###  2.1. Basic concepts

A diffeomorphism $f:N\to N$ is _partially hyperbolic_ if there exists a
continuous $Df$-invariant splitting $TN=E^{u}\oplus E^{c}\oplus E^{s}$ and
there exist constants $C>0$ and $\lambda<1$ such that

  * (a)

$\|Df_{x}^{-n}(v^{u})\|\leq C\lambda^{n}$ and $\|Df_{x}^{n}(v^{s})\|\leq
C\lambda^{n}$

  * (b)

$\|Df_{x}^{-n}(v^{u})\|\leq C\lambda^{n}\|Df_{x}^{-n}(v_{c})\|$ and
$\|Df_{x}^{n}(v^{s})\|\leq C\lambda^{n}\|Df_{x}^{n}(v_{c})\|$

for all unit vectors $v^{u}\in E_{x}^{u}$, $v^{c}\in E_{x}^{c}$, $v^{s}\in
E_{x}^{s}$, and all $x\in N$ and $n\geq 0$. Condition (a) means that the
derivative $Df$ is uniformly expanding along $E^{u}$ and uniformly contracting
along $E^{s}$. Condition (b) means that the behavior of $Df$ along the _center
bundle_ $E^{c}$ is dominated by the behavior along the other two factors. Here
all three bundles are assumed to have positive dimension.

The bundles $E^{u}$ and $E^{s}$ are always integrable: there exist foliations
${\mathcal{W}}^{u}$ and ${\mathcal{W}}^{s}$ of $N$ tangent to $E^{u}$ and
$E^{s}$, respectively, at every point. In fact these foliations are unique.
Moreover, they are _absolutely continuous_ , meaning that the projections
along the leaves between any two cross-sections preserve the class of sets
with zero volume inside the cross-section. See [20, 28, 42]. A diffeomorphism
$f:N\to N$ is _dynamically coherent_ if the bundles $E^{cu}=E^{c}\oplus E^{u}$
and $E^{cs}=E^{c}\oplus E^{s}$ also admit integral foliations,
${\mathcal{W}}^{cu}$ and ${\mathcal{W}}^{cs}$. Then, intersecting their leaves
one obtains a _center foliation_ ${\mathcal{W}}^{c}$ tangent at every point to
the center bundle $E^{c}$.

We say that the center leaves _form a fiber bundle_ over the leaf space
$N/{\mathcal{W}}^{c}$ if for any ${\mathcal{W}}^{c}(x)\in N/{\mathcal{W}}^{c}$
there is a neighborhood $V\subset N/{\mathcal{W}}^{c}$ of
${\mathcal{W}}^{c}(x)$ and a homeomorphism

$h_{x}:V\times{\mathcal{W}}^{c}(x)\to\pi_{c}^{-1}(V)$

smooth along the verticals $\\{\ell\\}\times{\mathcal{W}}^{c}(x)$ and mapping
each vertical onto the corresponding center leaf $\ell$.

######  Remark 2.1.

The fiber bundle condition is probably not necessary. Indeed, when the
diffeomorphisms are volume preserving, Avila, Viana, Wilkinson [14] prove that
if $\dim E^{c}=1$ and the generic center leaves are circles then the center
leaves form a fiber bundle _up to a finite cover_. In particular, all leaves
are circles. Our arguments extend easily to this situation.

A partially hyperbolic diffeomorphism $f:N\to N$ is _accessible_ if any points
$z$, $w\in N$ can be joined by a piecewise smooth curve $\gamma$ such that
every smooth leg of $\gamma$ is tangent to either $E^{u}$ or $E^{s}$ at every
point. Equivalently, every smooth leg of the curve $\gamma$ is contained in a
leaf of either ${\mathcal{W}}^{u}$ or ${\mathcal{W}}^{s}$.

The _center Lyapunov exponent_ $\lambda^{c}(\mu)$ of an $f$-invariant
probability measure $\mu$ is defined by

(3) $\lambda^{c}(\mu)=\int\lambda^{c}(z)\,d\mu(z)\quad\text{where
$\lambda^{c}(z)=\lim_{n\to\infty}\frac{1}{n}\log|Df^{n}\mid E^{c}_{z}|$.}$

By the ergodic theorem, this may be rewritten

(4) $\lambda^{c}(\mu)=\int\log|Df\mid E^{c}_{z}|\,d\mu(z).$

If $\mu$ is ergodic then $\lambda^{c}(\mu)=\lambda^{c}(z)$ for $\mu$-almost
every $z$.

Finally, the center direction is _mostly contracting_ (Bonatti, Viana [19]) if

(5) $\limsup_{n\to+\infty}\frac{1}{n}\log\|Df^{n}\mid E^{c}_{x}\|<0.$

for a positive volume measure subset of any disk inside a strong unstable
leaf. It was shown by Andersson [9] that this is a $C^{k}$, $k>1$ open
property.

###  2.2. The leaf space

Let $d$ be the Riemannian distance on $N$. We endow the leaf space
$N/{\mathcal{W}}^{c}$ with the distance defined by

$d_{c}(\xi,\eta)=\sup_{x\in\xi}\inf_{y\in\eta}d(x,y)+\sup_{y\in\eta}\inf_{x\in\xi}d(x,y)\quad\text{for
each }\xi,\eta\in N/{\mathcal{W}}^{c}.$

The quotient map $\pi_{c}:(N,d)\to(N/{\mathcal{W}}^{c},d_{c})$ is continuous
and onto. In particular, the metric space $(N/{\mathcal{W}}^{c},d_{c})$ is
compact.

Let $f_{c}:N/{\mathcal{W}}^{c}\to N/{\mathcal{W}}^{c}$ be the map induced by
$f$ on the quotient space $N/{\mathcal{W}}^{c}$. The stable set of a point
$\xi\in N/{\mathcal{W}}^{c}$ for $f_{c}$ is defined by

$W^{s}(\xi)=\\{\eta\in
N/{\mathcal{W}}^{c}:d_{c}(f_{c}^{n}(\xi),f_{c}^{n}(\eta))\to 0\text{ when
}n\to+\infty\\}$

and the local stable set of size $\varepsilon>0$ is defined by

$W^{s}_{\varepsilon}(\xi)=\\{\eta\in
N/{\mathcal{W}}^{c}:d_{c}(f_{c}^{n}(\xi),f_{c}^{n}(\eta))\leq\varepsilon\text{
for all }n\geq 0\\}.$

The unstable set and local unstable set of size $\varepsilon>0$ are defined in
the same way, for backward iterates. It follows from the definitions that
there exist constants $K$, $\tau$, $\varepsilon$, $\delta>0$ such that

  1. (1)

$d_{c}(f_{c}^{n}(\eta_{1}),f_{c}^{n}(\eta_{2}))\leq Ke^{-\tau
n}d_{c}(\eta_{1},\eta_{2})$ for all $\eta_{1},\eta_{2}\in
W^{s}_{\varepsilon}(\xi)$, $n\geq 0$;

  2. (2)

$d_{c}(f_{c}^{-n}(\zeta_{1}),f_{c}^{-n}(\zeta_{2}))\leq Ke^{-\tau
n}d_{c}(\zeta_{1},\zeta_{2})$ for all $\zeta_{1},\zeta_{2}\in
W^{u}_{\varepsilon}(\xi)$, $n\geq 0$;

  3. (3)

if $d_{c}(\xi_{1},\xi_{2})\leq\delta$ then $W^{s}_{\varepsilon}(\xi_{1})$ and
$W^{u}_{\varepsilon}(\xi_{2})$ intersect at exactly one point, denoted
$[\xi_{1}\,\xi_{2}]$ and this point depends continuously on
$(\xi_{1},\xi_{2})$.

This means that $f_{c}$ is a hyperbolic homeomorphism (in the sense of Viana
[48]). We denote ${\mathcal{W}}^{c}(\Lambda)=\pi_{c}^{-1}(\Lambda)$, for any
subset $\Lambda$ of $N/{\mathcal{W}}^{c}$.

By Anosov’s closing lemma [10], periodic points are dense in the non-wandering
set of $f_{c}$. Smale’s spectral decomposition theorem [45], the non-wandering
set splits into a finite number of compact, invariant, transitive, pairwise
disjoint subsets. Among these basic pieces of the non-wandering set, the
_attractors_ $\Lambda_{i}$, $i=1,\dots,k$ of $f_{c}$ are characterized by the
fact that

$\Lambda_{i}=\bigcap_{n=0}^{\infty}f_{c}^{n}(U_{i})$

for some neighborhood $U_{i}$ of $\Lambda_{i}$ and it is transitive. The union
of the stable sets $W^{s}(\Lambda_{i})$, $i=1,\dots,k$ is an open dense subset
of $N/{\mathcal{W}}^{c}$. Every attractor $\Lambda_{i}$ consists of entire
unstable sets, and so ${\mathcal{W}}^{c}(\Lambda_{i})$ is
${\mathcal{W}}^{u}$-saturated, that is, it consists of entire strong unstable
leaves of $f$. Additionally, every $\Lambda_{i}$ has finitely many connected
components $\Lambda_{i,j}$, $j=1,\dots,n_{i}$ that are mapped to one another
cyclically. The unstable set $W^{u}(x)$ of every $x\in\Lambda_{i,j}$ is
contained and dense in $\Lambda_{i,j}$. In particular,
${\mathcal{W}}^{c}(\Lambda_{i,j})$ is also ${\mathcal{W}}^{u}$-saturated. If
$f_{c}$ is transitive, there is a unique attractor
$\Lambda_{1}=N/{\mathcal{W}}^{c}$.

We say $f$ is _accessible on $\Lambda_{i}$_ if, for every $j$, any points $z$,
$w\in{\mathcal{W}}^{c}(\Lambda_{i,j})$ can be joined by a piecewise smooth
curve $\gamma$ such that every smooth leg of $\gamma$ is tangent to either
$E^{u}$ or $E^{s}$ at every point and the corner points belong to the same
${\mathcal{W}}^{c}(\Lambda_{i,j})$. The center direction of
$f\mid{{\mathcal{W}}^{c}(\Lambda_{i})}$ is _mostly contracting_ if (5) holds
for a positive volume measure subset of any disk inside a strong unstable leaf
contained in ${\mathcal{W}}^{c}(\Lambda_{i})$.

###  2.3. Physical measures

We are ready to state our main result on existence and finiteness of physical
measures:

######  Theorem D.

If $f\in{\mathcal{P}}_{1}^{k}(N)$, $k>1$ is accessible on every attractor and
the center stable foliation is absolutely continuous then, for each attractor
$\Lambda_{i}$, either

  * (a)

there is a Lipschitz metric on each leaf of ${\mathcal{W}}^{c}(\Lambda_{i})$,
depending continuously on the leaf and invariant under $f$; then $f$ admits a
unique physical measure, which is ergodic, whose basin has full volume in the
stable set of ${\mathcal{W}}^{c}(\Lambda_{i})$, and whose center Lyapunov
exponent vanishes;

  * (b)

or the center direction of $f\mid{{\mathcal{W}}^{c}(\Lambda_{i})}$ is mostly
contracting; then $f\mid{{\mathcal{W}}^{c}(\Lambda_{i})}$ has finitely many
physical measures, they are ergodic for $f$ and Bernoulli for some iterate,
the union of their basins is a full volume subset of the stable set of
${\mathcal{W}}^{c}(\Lambda_{i})$, and their center Lyapunov exponents are
negative.

The union of the basins of these physical measures has full volume in $N$.

To see that Theorem A is contained in Theorem D let us to note that, for every
$k\geq 1$, any $C^{k}$ partially hyperbolic skew-product $f_{0}$ is in the
interior of ${\mathcal{P}}_{1}^{k}(N)$. Indeed, partial hyperbolicity is well
known to be a $C^{1}$ open property and the stability theorem for normally
hyperbolic foliations (Hirsch, Pugh, Shub [28]) gives that every $f$ in a
$C^{1}$ neighborhood of $f_{0}$ admits an invariant ${\mathcal{W}}^{*}_{f}$
foliation, for each $*\in\\{cu,cs,c\\}$, and there exists a homeomorphism
mapping the leaves of ${\mathcal{W}}^{*}_{f}$ diffeomorphically to the leaves
of ${\mathcal{W}}^{*}_{f_{0}}$. In particular, the center leaves of $f$ form a
circle fiber bundle.

######  Remark 2.2.

When the center fiber bundle is trivial, as happens near skew-products, part
(a) of the Theorem D gives that $f\mid{{\mathcal{W}}^{c}(\Lambda_{i})}$ is
topologically conjugate to a rotation extension

$\Lambda_{i}\times{\mathbb{R}}/{\mathbb{Z}}\to\Lambda_{i}\times{\mathbb{R}}/{\mathbb{Z}},\quad(x,\theta)\mapsto(f_{c}(x),\theta+\omega(x)).$

To see this, fix some consistent orientation of the center leaves and any
continuous section $\sigma:N/{\mathcal{W}}^{c}\to N$ of the center foliation,
that is, any continuous map such that $\sigma(\ell)\in\ell$ for every $\ell\in
N/{\mathcal{W}}^{c}$. Then define

$h:{\mathcal{W}}^{c}(\Lambda_{i})\to\Lambda_{i}\times{\mathbb{R}}/{\mathbb{Z}},\quad
h(z)=(\pi_{c}(z),|\sigma(\pi_{c}(z)),z|)$

where $|\sigma(\pi_{c}(z)),z|$ denotes the length, with respect to the
$f$-invariant Lipschitz metric, of the (oriented) curve segment from
$\sigma(\pi_{c}(z))$ to $z$ inside the center leaf. This map sends the center
leaves of $f$ to verticals $\\{w\\}\times{\mathbb{R}}/{\mathbb{Z}}$, mapping
the $f$-invariant Lipschitz metric on the center leaves to the standard metric
on ${\mathbb{R}}/{\mathbb{Z}}$. Then $h\circ f\circ h^{-1}$ preserves the
standard metric measure on the verticals, and so it is a rotation extension,
as stated. Observe that, in addition, both $h$ and its inverse are Lipschitz
on every leaf.

Explicit bounds on the number of physical measures can be given in many cases.
For instance, we will see in Theorem 5.3 that if $f$ admits some periodic
center leaf $\ell$ restricted to which $f$ is Morse-Smale then the number of
physical measures over the attractor containing $\pi_{c}(\ell)$ is bounded by
the number of periodic orbits on $\ell$. Notice that we must have alternative
(b) of Theorem D in this case, since alternative (a) is incompatible with the
existence of hyperbolic periodic points.

We also want to analyze the dependence of the physical measures on the
dynamics. For this, we assume $N=M\times S^{1}$ and restrict ourselves to the
subset ${\mathcal{S}}^{k}(N)\subset{\mathcal{P}}^{k}_{1}(N)$ of skew-product
maps. We prove in Theorem 5.6 that there is an open and dense subset of
diffeomorphisms $f\in{\mathcal{S}}^{k}(N)$ with mostly contracting center
direction, such that the number of physical measures is locally constant and
the physical measures vary continuously with the diffeomorphism. This property
of _statistical stability_ has been studied in a number of recent works,
including Alves, Viana [8], Vásquez [47], Andersson [9].

As mentioned before, existence and finiteness of physical measures for
partially hyperbolic diffeomorphisms was proved by Alves, Bonatti, Viana [5,
19], under certain assumptions of weak hyperbolicity along the center
direction. Substantial improvements followed, by Alves, Luzzatto, Pinheiro [6,
7], Alves, Araujo [4], Vasquez [47], Pinheiro [36], and Andersson [9], among
others. Perturbations of certain skew-products over hyperbolic maps have been
studied by Alves [2, 3], Buzzi, Sestier, Tsujii [23], and Gouezel [25]. In a
remarkable recent paper, Tsujii [46] proved that generic (dense $G_{\delta}$)
partially hyperbolic surface _endomorphisms_ do admit finitely many physical
measures, such that the union of their basins has full Lebesgue measure. His
approach is very different from the one in the present paper and it is not
clear how it could be extended to diffeomorphisms in higher dimensions, even
in the case of one-dimensional center bundle.

###  2.4. Absolute continuity

It has been pointed out by Shub, Wilkinson [44] that foliations tangent to the
center subbundle $E^{c}$ are often _not_ absolutely continuous. In fact,
Ruelle, Wilkinson [41] showed that the disintegration of Lebesgue measure
along the leaves is often atomic. Moreover, Avila, Viana, Wilkinson [14]
observed recently that for certain classes of volume preserving
diffeomorphisms, including perturbations of skew-products (2), absolute
continuity of the center foliation is a rigid property: it implies that the
center foliation is actually smooth, and the map is smoothly conjugate to a
rigid model.

However, we prove that this is not at all the case in our dissipative setting:

######  Theorem E.

There is an open set ${\mathcal{U}}\subset{\mathcal{P}}_{1}^{k}(N)$, $k>1$,
such that the center stable, the center unstable, and the center foliation are
absolutely continuous for every $f\in{\mathcal{U}}$. Moreover, ${\mathcal{U}}$
may be chosen to accumulate on every skew-product map $f_{0}$ that admits a
periodic vertical fiber restricted to which the map is Morse-Smale with a
unique periodic attractor and repeller.

Two weaker forms of absolute continuity are considered by Avila, Viana,
Wilkinson [14]. Let $\operatorname{vol}$ denote Lebesgue measure in the
ambient manifold and $\operatorname{vol}_{L}$ be Lebesgue measure restricted
to some submanifold $L$. A foliation ${\mathcal{F}}$ on $N$ is (_lower_)
_leafwise absolutely continuous_ if for every zero
$\operatorname{vol}$-measure set $Y\subset N$ and $\operatorname{vol}$-almost
every $z\in M$, the leaf $L$ through $z$ meets $Y$ in a zero
$\operatorname{vol}_{L}$-measure set. Similarly, ${\mathcal{F}}$ is _upper
leafwise absolutely continuous_ if $\operatorname{vol}_{L}(Y)=0$ for every
leaf $L$ through a full measure subset of points $z\in M$ implies
$\operatorname{vol}(Y)=0$. Absolute continuity implies both lower and upper
leafwise absolute continuity (see [14, 21]); the converse is not true in
general. We will see in Proposition 6.2 that the center stable foliation of a
partially hyperbolic, dynamically coherent diffeomorphism with mostly
contracting center direction is always upper leafwise absolutely continuous.
This does not extend to lower leafwise absolutely continuity, in general:
robust counter-examples will appear in [49]; see also Example 6.1 for a
related construction. However, as stated before, full absolute continuity of
the center foliation does hold on some open subsets of diffeomorphisms with
mostly contracting center.

##  3\. Gibbs $u$-states

Let $f:N\to N$ be a partially hyperbolic diffeomorphism. In what follows we
denote $I_{r}=[-r,r]$ for $r>0$ and $d_{*}=\dim E^{*}$ for each
$*\in\\{u,cu,c,cs,s\\}$. We use $\operatorname{vol}^{*}$ to represent the
volume measure induced by the restriction of the Riemannian structure on the
leaves of the foliation ${\mathcal{W}}^{*}$ for each $*\in\\{u,cu,c,cs,s\\}$.

Following Pesin, Sinai [34] and Alves, Bonatti, Viana [5, 19] (see also [17,
Chapter 11]), we call _Gibbs $u$-state_ any invariant probability measure $m$
whose conditional probabilities (Rokhlin [40]) along strong unstable leaves
are absolutely continuous with respect to the volume measure
$\operatorname{vol}^{u}$ on the leaf. More precisely, let

$\Phi:I_{1}^{d_{u}}\times I_{1}^{d_{cs}}\to N$

be any _foliated box_ for the strong unstable foliation. By this we mean that
$\Phi$ is a homeomorphism and maps every horizontal plaque
$I_{1}^{d_{u}}\times\\{\eta\\}$ diffeomorphically to a disk inside some strong
unstable leaf. Pulling $m$ back under $\Phi$ one obtains a measure $m_{\Phi}$
on $I_{1}^{d_{u}}\times I_{1}^{d_{cs}}$. The definition of Gibbs $u$-state
means that there exists a measurable function
$\alpha_{\Phi}(\cdot\,,\cdot)\geq 0$ and a measure $m_{\Phi}^{cs}$ on
$I_{1}^{d_{cs}}$ such that

(6)
$m_{\Phi}(A)=\int_{A}\alpha_{\Phi}(\xi,\zeta)\,d\xi\,dm_{\Phi}^{cs}(\zeta)$

for every measurable set $A\subset I_{1}^{d_{u}}\times I_{1}^{d_{cs}}$.

Proofs for the following basic properties of Gibbs $u$-states can be found in
Section 11.2 of Bonatti, Díaz, Viana [17]:

######  Proposition 3.1.

Let $f:N\to N$ be a partially hyperbolic diffeomorphism.

  1. (1)

The densities of a Gibbs $u$-state with respect to Lebesgue measure along
strong unstable plaques are positive and bounded from zero and infinity.

  2. (2)

The support of every Gibbs $u$-state is ${\mathcal{W}}^{u}$-saturated, that
is, it consists of entire strong unstable leaves.

  3. (3)

The set of Gibbs $u$-states is non-empty, weak∗ compact, and convex. Ergodic
components of Gibbs $u$-states are Gibbs $u$-states.

  4. (4)

Every physical measure of $f$ is a Gibbs $u$-state and, conversely, every
ergodic $u$-state whose center Lyapunov exponents are negative is a physical
measure.

Now let $f\in{\mathcal{P}}^{k}_{*}(N)$. Recall that $\pi_{c}:N\to
N/{\mathcal{W}}^{c}$ denotes the natural quotient map and
$f_{c}:N/{\mathcal{W}}^{c}\to N/{\mathcal{W}}^{c}$ is the hyperbolic
homeomorphism induced by $f$ in the leaf space. Given small neighborhoods
$V^{s}_{\xi}\subset W_{\varepsilon}^{s}(\xi)$ and $V^{u}_{\xi}\subset
W_{\varepsilon}^{u}(\xi)$ inside the corresponding stable and unstable sets,
the map

(7) $(\eta,\zeta)\mapsto[\eta,\zeta]$

defines a homeomorphism between $V^{u}_{\xi}\times V^{s}_{\xi}$ and some
neighborhood $V_{\xi}$ of $\xi$. A probability measure $\mu$ on
$N/{\mathcal{W}}^{c}$ has _local product structure_ if for $\mu$-almost every
point $\xi$ and any such product neighborhood $V_{\xi}$ the restriction
$\mu\mid V_{\xi}$ is equivalent to a product $\nu^{u}\times\nu^{s}$, where
$\nu^{u}$ is a measure on $V_{\xi}^{u}$ and $\nu^{s}$ is a measure on
$V_{\xi}^{s}$.

In the sequel we prove three additional facts about Gibbs $u$-states that are
important for our arguments.

######  Proposition 3.2.

Take $f\in{\mathcal{P}}_{*}^{k}(N)$, $k>1$ such that the center stable
foliation is absolutely continuous. For every ergodic Gibbs $u$-state $m$ the
support of the projection $(\pi_{c})_{*}(m)$ coincides with some attractor of
$f_{c}$. In particular, periodic points are dense in the support of
$(\pi_{c})_{*}(m)$.

Moreover, any two such projections with the same support must coincide. In
particular, the set of projections of all ergodic Gibbs $u$-states of $f$ down
to $N/{\mathcal{W}}^{c}$ is finite.

######  Proposition 3.3.

Take $f\in{\mathcal{P}}_{*}^{k}(N)$, $k>1$ such that the center stable
foliation is absolutely continuous. If $m$ is a Gibbs $u$-state for $f$ then
$\mu=(\pi_{c})_{*}(m)$ has local product structure.

######  Remark 3.4.

Suppose $f$ is volume preserving. The Lebesgue measure $\operatorname{vol}$ is
both an $s$-state and a $u$-state, because the strong stable foliation and the
strong unstable foliation are both absolutely continuous. Thus, Proposition
3.3 implies that $(\pi_{c})_{*}(m)$ has local product structure if _either_
${\mathcal{W}}^{cu}$ _or_ ${\mathcal{W}}^{cs}$ is absolutely continuous.

######  Proposition 3.5.

Let $f\in{\mathcal{P}}_{*}^{k}(N)$, $k>1$ and $\Lambda$ be an attractor of
$f_{c}$. Suppose the center stable foliation of $f$ is absolutely continuous
and $f$ is accessible on $\Lambda$. Then every ergodic Gibbs $u$-state of $f$
supported in ${\mathcal{W}}^{c}(\Lambda)$ has some non-positive center
Lyapunov exponent.

As a special case, we get that if $f\in{\mathcal{P}}_{1}^{k}(N)$, $k>1$ is
accessible on an attractor $\Lambda$ of $f_{c}$ and the center stable
foliation is absolutely continuous, then the (unique) center Lyapunov exponent
of every ergodic Gibbs $u$-state supported in ${\mathcal{W}}^{c}(\Lambda)$ is
non-positive.

The proofs of these propositions are given in Sections 3.1 through 3.3.

###  3.1. Finiteness in leaf space

Here we prove Proposition 3.2. Let $m_{1}$ be any ergodic Gibbs $u$-state and
$\mu_{1}=(\pi_{c})_{*}(m_{1})$. Notice that $\mu_{1}$ is ergodic and so its
support is a transitive set for $f_{c}$. Moreover,
$\operatorname{supp}\mu_{1}=\pi_{c}(\operatorname{supp}m_{1})$ consists of
entire unstable sets, because the support of $m_{1}$ is
${\mathcal{W}}^{u}$-saturated (Proposition 3.1). Thus,
$\operatorname{supp}\mu_{1}$ is an attractor $\Lambda$ of $f_{c}$. As pointed
out before, periodic points are dense in each attractor of $f_{c}$.

Now we only have to show that if $\mu_{2}=(\pi_{c})_{*}m_{2}$ for another
ergodic Gibbs $u$-state $m_{2}$ and
$\operatorname{supp}\mu_{2}=\Lambda=\operatorname{supp}\mu_{1}$ then
$\mu_{1}=\mu_{2}$. For this, take $x_{c}\in\Lambda$, let $U_{c}$ be a
neighborhood of $x_{c}$ in the quotient space $N/{\mathcal{W}}^{c}$, and let
$U=\pi_{c}^{-1}(U_{c})$. Then $U$ has positive $m_{i}$-measure for $i=1,2$.
So, since the $m_{i}$ are ergodic Gibbs $u$-states, there are disks
$D_{i}\subset U$, $i=1,2$ contained in strong unstable leaves and such that
Lebesgue almost every point in $D_{i}$ is in the basin $B(m_{i})$ of $m_{i}$.
Moreover, these disks may be chosen such that the center stable foliation
induces a holonomy map $h^{cs}:D_{1}\to D_{2}$. Since the center stable
foliation is absolutely continuous, it follows that $h^{cs}$ maps some point
$x_{1}\in D_{1}\cap B(m_{1})$ to a point $x_{2}\in D_{2}\cap B(m_{2})$ in the
basin of $m_{2}$. Then $x_{1}$ and $x_{2}$ belong to the same center stable
leaf of $f$, and so their projections $\pi_{c}(x_{1})$ and $\pi_{c}(x_{2})$
belong to the same stable set of $f_{c}$. Notice that
$\pi_{c}(B(m_{i}))\subset B(\mu_{i})$ for $i=1,2$, and so each point
$\pi(x_{i})\in B(\mu_{i})$. Since either basin consists of entire stable sets,
this proves that $B(\mu_{1})$ and $B(\mu_{2})$ intersect each other, and so
$\mu_{1}=\mu_{2}$. This completes the proof of Proposition 3.2.

###  3.2. Local product structure

Here we prove Proposition 3.3. Let $m$ be any Gibbs $u$-state and $\ell_{0}$
be any center leaf. Since the center leaves form a fiber bundle, we may find a
neighborhood $V\subset N/{\mathcal{W}}^{c}$ and a homeomorphism

$\phi:V\times\ell_{0}\mapsto\pi_{c}^{-1}(V),\quad(\ell,\zeta)\mapsto\phi(\theta,\zeta)$

that maps each vertical $\\{\ell\\}\times\ell_{0}$ to the corresponding center
leaf $\ell$. Clearly, we may choose $V$ to be the image of the bracket (recall
Section 2.2)

$W^{u}_{\varepsilon}(\ell_{0})\times W^{s}_{\varepsilon}(\ell_{0})\to
V,\quad(\xi,\eta)\mapsto[\xi,\eta]$

for some small $\varepsilon>0$. Then, by dynamical coherence, the
homeomorphism

(8) $W^{u}_{\varepsilon}(\ell_{0})\times
W^{s}_{\varepsilon}(\ell_{0})\times\ell_{0}\to\pi_{c}^{-1}(V),\quad(\xi,\eta,\zeta)\mapsto\phi([\xi,\eta],\zeta)$

maps each $\\{\xi\\}\times W_{\varepsilon}^{s}(\ell_{0})\times\ell_{0}$ onto a
center stable leaf and each
$W_{\varepsilon}^{u}(\ell_{0})\times\\{\eta\\}\times\ell_{0}$ onto a center
unstable leaf. For each $x\in\pi_{c}^{-1}(V)$, let
${\mathcal{W}}^{u}_{loc}(x)$ denote the local strong unstable leaf over $V$,
that is, the connected component of ${\mathcal{W}}^{u}(x)\cap\pi_{c}^{-1}(V)$
that contains $x$. Each ${\mathcal{W}}^{u}_{loc}(x)$ is a graph over the
unstable set $W^{u}(\pi_{c}(x))$ and the center stable holonomy defines a
homeomorphism

$h_{x,y}^{cs}:{\mathcal{W}}^{u}_{loc}(x)\to{\mathcal{W}}^{u}_{loc}(y)$

between any two local strong unstable leaves. By assumption, all these
homeomorphisms are absolutely continuous. Now let

$m\mid\pi_{c}^{-1}(V)=\int m_{x}\,d\hat{m}$

be the disintegration of $m$ relative to the partition of $\pi^{-1}_{c}(V)$
into local strong unstable leaves. By definition of Gibbs $u$-states, each
$m_{x}$ is equivalent to the Lebesgue measure along
${\mathcal{W}}^{u}_{loc}(x)$. It follows that the center stable holonomies are
absolutely continuous relative to the conditional probabilities of $m$ along
local strong unstable leaves:

(9) $m_{x}(E)=0\text{ if and only if }m_{y}(h_{x.y}^{cs}(E))=0$

for $x$ and $y$ in some full $m$-measure subset of $\pi_{c}^{-1}(V)$ and for
any measurable set $E\subset{\mathcal{W}}^{u}_{loc}(x)$. By the construction
of (8), center stable holonomies preserve the coordinate $\xi$. Thus,
identifying $\pi_{c}^{-1}(V)$ with the space $W_{\varepsilon}(\ell_{0})\times
W^{s}_{\varepsilon}(\ell_{0})\times\ell_{0}$ through the homeomorphism (8),
property (9) becomes

(10) $m_{x}(A\times W_{\varepsilon}^{s}(\ell_{0})\times\ell_{0})=0\text{ if
and only if }m_{y}(A\times W^{s}_{\varepsilon}(\ell_{0})\times\ell_{0})=0$

for any measurable set $A\subset W^{u}_{\varepsilon}(\ell_{0})$ and for
$m$-almost every $x$ and $y$ in $\pi_{c}^{-1}(V)$. Let $\mu\mid
V=\int\mu^{u}_{\eta}\,d\mu^{s}(\eta)$ be the disintegration of $\mu$ relative
to the partition of $V$ into unstable slices
$W^{u}(\ell_{0})\times\\{\eta\\}$; notice that $\mu^{s}$ is just the
projection of $\mu\mid V$ to $W^{s}_{\varepsilon}(\ell_{0})$. Projecting
$m\mid\pi_{c}^{-1}(V)$ down to $V\approx W^{u}_{\varepsilon}(\ell_{0})\times
W^{s}_{\varepsilon}(\ell_{0})$, property (10) yields

(11) $\mu_{\eta}(A\times W_{\varepsilon}^{s}(\ell_{0}))=0\text{ if and only if
}\mu_{\eta^{\prime}}(A\times W^{s}_{\varepsilon}(\ell_{0}))=0$

for any measurable set $A\subset W^{u}_{\varepsilon}(\ell_{0})$ and for
$\mu$-almost every $\eta$ and $\eta^{\prime}$ in $V$. This means that the
conditional probabilities $\mu^{u}_{\eta}$ are (almost) all equivalent.
Consequently, there is $\rho:W^{u}_{\varepsilon}(\ell_{0})\times
W^{s}_{\varepsilon}(\ell_{0})\to(0,\infty)$ such that
$\mu^{u}_{\eta}=\rho(\cdot,\eta)\mu^{u}$ at $\mu$-almost every point, where
$\mu^{u}$ denotes the projection of $\mu\mid V$ to
$W^{u}_{\varepsilon}(\ell_{0})$. Replacing in the disintegration of $\mu\mid
V$, we get that $\mu\mid V=\rho\,\mu^{u}\times\mu^{s}$. This proves that $\mu$
has local product structure, as claimed.

###  3.3. Positive Gibbs $u$-states

Here we prove Proposition 3.5. We begin by proving the following fact, which
is interesting in itself:

######  Proposition 3.6.

For $f\in{\mathcal{P}}_{*}^{1}(N)$, given $c>0$ and $l\geq 1$ there is $n_{0}$
such that $\\#\big{(}S\cap\Gamma_{c,l}\big{)}<n_{0}$ for every center leaf
$S$, where

$\Gamma_{c,l}=\\{x\in
N:\liminf\frac{1}{n}\sum_{i=1}^{n}\log\|Df^{-l}\mid{E^{c}(f^{il}(x))}\|^{-1}\geq
c\\}.$

###### Proof.

Recall that $\operatorname{vol}^{c}$ denotes the Riemannian volume on center
leaves. The main ingredient is

######  Lemma 3.7.

Given $c>0$ and $l\geq 1$ there exists $\delta>0$ such that for any $x\in
S\cap\Gamma_{c,l}$ and any neighborhood $U$ of $x$ inside the center leaf $S$
that contains $x$, one has

$\liminf\frac{1}{n}\sum_{i=0}^{n-1}\operatorname{vol}^{c}(f^{il}(U))\geq\delta.$

###### Proof.

Let $x\in S\cap\Gamma_{c,l}$ be fixed. Fix $0<c_{1}<c_{2}<c$ and define
$H(c_{2})$ to be the set of $c_{2}$-hyperbolic times for $x$, that is, the set
of times $m\geq 1$ such that

(12) $\frac{1}{k}\sum_{i=m-k+1}^{m}\log\|Df^{-l}\mid
E^{c}_{f^{il}(x)}\|^{-1}\geq c_{2}\quad\text{for all $1\leq k\leq m$.}$

By the Pliss Lemma (see [2, 5]) , there exist $n_{1}\geq 1$ and $\delta_{1}>0$
such that

$\\#\big{(}H(c_{2})\cap[1,n)\big{)}\geq n\delta_{1}\quad\text{for all $n\geq
n_{1}$.}$

Notice that (12) implies $Df^{-kl}$ is an exponential contraction on
$E^{c}_{f^{ml}(x)}$:

$\|Df^{-kl}\mid E^{c}_{f^{ml}(x)}\|\leq\prod_{i=m-k+1}^{m}\|Df^{-l}\mid
E^{c}_{f^{il}(x)}\|\leq e^{-c_{2}k}\quad\text{for all $1\leq k\leq m$.}$

It also follows from [5] that the points $f^{ml}(x)$ with $m\in H(c_{2})$
admit backward-contracting center disks with size uniformly bounded from
below: there is $r>0$ depending only on $f$ and the constants $c_{1}$ and
$c_{2}$ such that

$f^{-kl}(B_{r}^{c}(f^{ml}(x)))\subset
B^{c}_{e^{-c_{1}k}r}(f^{(m-k)l}(x))\quad\text{for all $1\leq k\leq m$.}$

where $B^{c}_{\rho}(y)$ denotes the ball inside ${\mathcal{W}}^{c}_{y}$ of
radius $\rho$ around any point $y$. Let $a_{1}>0$ be a lower bound for
$m^{c}(B^{c}_{r}(y))$ over all $y\in N$. Fix $n_{2}$ such that the ball of
radius $e^{-c_{1}k}r$ around $x$ is contained in $U$ for every $k\geq n_{2}$.
Then, in particular,

$f^{ml}(U)\supset B^{c}_{r}(f^{ml}(x))\quad\text{and so}\quad
m^{c}(f^{ml}(U))\geq a_{1}$

for every $m\in H(c_{2})$ with $m\geq n_{2}$. So, for
$n\gg\max\\{n_{1},n_{2}\\}$,

$\frac{1}{n}\sum_{i=0}^{n-1}m^{c}(f^{il}(U))\geq\frac{1}{n}a_{1}\big{[}\\#(H(c_{2})\cap[1,n))-n_{2}\big{]}\geq\frac{1}{n}a_{1}\big{[}n\delta_{1}-n_{2}\big{]}\geq\frac{\delta_{1}}{2}a_{1}\,.$

To finish the proof of Lemma 3.7 it suffices to take
$\delta=a_{1}\delta_{2}/2$. ∎

To deduce Proposition 3.6 from Lemma 3.7, take any $n_{0}\geq V/\delta$ where
$V$ is an upper bound for the volume of center leaves. Suppose
$S\cap\Gamma_{c,l}$ contains $n_{0}$ distinct points $x_{j}$,
$j=1,\ldots,n_{0}$. Let $U_{j}$, $j=1,\ldots,n_{0}$ be pairwise disjoint
neighborhoods of the $x_{j}$ inside $S$. Take $n$ large enough that

$\frac{1}{n}\sum_{i=0}^{n-1}m^{c}(f^{i}(U_{j}))>\delta\quad\text{for $1\leq
j\leq n_{0}$.}$

Then

$V\geq\frac{1}{n}\sum_{i=0}^{n-1}m^{c}(f^{i}(S))\geq\sum_{j=1}^{n_{0}}\frac{1}{n}\sum_{i=0}^{n-1}m^{c}(f^{i}(U_{j}))>n_{0}\delta>V.$

This contradiction proves Proposition 3.6. ∎

###### Proof of Proposition 3.5.

We argue by contradiction. Suppose there exists some ergodic Gibbs $u$-state
$\nu$ supported in ${\mathcal{W}}^{c}(\Lambda)$ whose center Lyapunov
exponents are all positive.

######  Lemma 3.8.

There is $k_{0}\geq 1$ and some ergodic Gibbs $u$-state $\nu_{*}$ of
$f^{k_{0}}$ supported in ${\mathcal{W}}^{c}(\Lambda)$ such that

(13) $\int\log\|Df^{-k_{0}}\mid E^{c}_{x}\|^{-1}d\nu_{*}(x)>0.$

###### Proof.

Arguing as in [48, Section 2.1] one can find $k_{0}\geq 1$ such that

$\int\log\|Df^{-k_{0}}\mid E^{c}_{x}\|^{-1}d\nu(x)>0$

The measure $\nu$ needs not be ergodic for $f^{k_{0}}$ but, since it is
ergodic for $f$, it has a finite number $k$ of ergodic components $\nu_{i}$
($k$ divides $k_{0}$). Moreover,

$\int\log\|Df^{-k_{0}}\mid E^{c}_{x}\|^{-1}d\nu_{i}(x)>0$

for some ergodic component $\nu_{i}$. Since, by Proposition 3.1, each ergodic
component $\nu_{i}$ is a Gibbs $u$-state, this completes the proof of the
lemma. ∎

Let $k_{0}\geq 1$ be fixed from now on and $\lambda>0$ denote the expression
on the left hand side of (13). Let $g=f^{k_{0}}$ and

$\Gamma=\\{x\in N:\lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n}\log\|Dg^{-1}\mid
E^{c}_{g^{j}(x)}\|^{-1}=\lambda\\}$

be the set of regular points of $\log\|Dg^{-1}\mid E^{c}\|$ for the
transformation $g$. By ergodicity, $\nu_{*}(\Gamma)=1$. A statement similar to
the next corollary was proved by Ruelle, Wilkinson [41] when the
diffeomorphism is $C^{1+\varepsilon}$ and the center is $1$-dimensional.

######  Corollary 3.9.

There is $n_{0}\geq 1$ such that
$\\#\big{(}{\mathcal{W}}^{c}(w)\cap\Gamma\big{)}<n_{0}$ for every $w\in N$.

###### Proof.

Just use Proposition 3.6 with $c=\lambda/2$ and $l=k_{0}$. Clearly,
$\Gamma\subset\Gamma_{c,l}$. ∎

Let $\ell_{0}$ be any periodic center leaf intersecting
$\operatorname{supp}\nu_{*}$ (periodic center leaves are dense in the support,
by Proposition 3.2) and $\kappa\geq 1$ be minimal such that
$g^{\kappa}(\ell_{0})=\ell_{0}$. Since $\nu_{*}$ is a Gibbs $u$-state and
$\Gamma$ has full measure,
$\operatorname{vol}^{u}({\mathcal{W}}^{u}(x)\setminus\Gamma)=0$ for
$\nu_{*}$-almost every $x$, where $\operatorname{vol}^{u}$ denotes the
Riemannian volume along strong unstable manifolds. In particular, the stable
set ${\mathcal{W}}^{s}(\ell_{0})=\cup_{z\in\ell_{0}}{\mathcal{W}}^{s}(z)$ must
intersect some strong unstable disk $D^{u}$ such that
$\operatorname{vol}^{u}(D^{u}\setminus\Gamma)=0$. See Figure 1.

Figure 1.

######  Lemma 3.10.

Every point $x\in D^{u}\cap{\mathcal{W}}^{s}(\ell_{0})$ belongs to the strong
stable manifold of some periodic point $y\in\ell_{0}$ of $f$ with period
bounded by $k_{0}\kappa n_{0}$.

###### Proof.

Let $y\in\ell_{0}$ be such that $x\in{\mathcal{W}}^{s}(y)$ and let
$g_{0}=g^{\kappa}\mid\ell_{0}$. Suppose first that the orbit of $y$ under
$g_{0}$ is infinite. We refer the reader to Figure 2. Fix $y^{*}\in\omega(y)$
and let $(y_{j})_{j}$ be an injective sequence of iterates of $y$ converging
to $y^{*}$. Let $(x_{j})_{j}$ be a sequence of iterates of $x$ with
$x_{j}\in{\mathcal{W}}^{s}(y_{j})$ and $d(x_{j},y_{j})\to 0$. Choose disks
$D^{u}_{j}$ around the $x_{j}$ inside the forward iterates of $D^{u}$, small
but with _uniform size_. Since $\Gamma$ is an invariant set,
$m^{u}(D_{j}^{u}\setminus\Gamma)=0$ for every $j$. For every large $j$, the
center leaves ${\mathcal{W}}^{c}(x_{j})$ is close to $\ell_{0}$ and so one can
define a $cs$-holonomy map $\pi^{cs}$ from $D^{u}_{j}$ to the local strong
unstable leaf through $y^{*}$. Since ${\mathcal{W}}^{cs}$ is absolutely
continuous, the image of every $D_{j}^{u}\cap\Gamma$ is a full volume measure
subset of a neighborhood of $y^{*}$ inside ${\mathcal{W}}^{u}(y^{*})$, where
these neighborhoods also have uniform size for all large $j$. Let
$J=\\{j_{0},j_{0}+1,\ldots,j_{0}+n_{0}\\}$ where $j_{0}$ is some large integer
and $n_{0}$ is as in Corollary 3.9. On the one hand, it follows from the
previous considerations that

$\Gamma^{*}=\bigcap_{j\in J}\pi^{cs}(D_{j}^{u}\cap\Gamma)$

is a full volume measure subset of some neighborhood of $y^{*}$ inside
${\mathcal{W}}^{u}(y^{*})$. Fix some $w\in\Gamma^{*}$ close to $y^{*}$. For
each $j\in J$, let $w_{j}\in D_{j}^{u}\cap\Gamma$ be such that
$\pi^{cs}(w_{j})=w$. Moreover, let $z_{j}$ be the point where the local strong
stable manifold of $w_{j}$ intersects
${\mathcal{W}}^{cu}(y^{*})={\mathcal{W}}^{cu}(w)$. It is clear from the
definition that $w_{j}\in{\mathcal{W}}^{cs}(w)$ and so
$z_{j}\in{\mathcal{W}}^{c}(w)$ for all $j\in J$. Moreover, by choosing $w$
close enough to $y^{*}$ we can ensure that $w_{j}$ is close to $x_{j}$ for
every $j\in J$ and so $z_{j}$ is close to $y_{j}$ for all $j\in J$. The latter
implies that the $z_{j}$ are all distinct. Observe also that $z_{j}\in\Gamma$
for all $j\in J$, because $\Gamma$ is (clearly) saturated by strong stable
leaves. This proves that $\\#({\mathcal{W}}^{c}(w)\cap\Gamma)\geq\\#J>n_{0}$,
in contradiction with Corollary 3.9. This contradiction proves that the
$g_{0}$-orbit of $y$ can not be infinite.

Figure 2.

Similar arguments handle the case when $y$ is a periodic point for $g_{0}$.
Let $k\geq 1$ be the (minimal) period of $y$ for $g_{0}$. Forward iterates of
$D^{u}$ accumulate on the strong unstable manifolds of the iterates of $y$.
Using, in much the same way as before, that the center stable foliation is
absolutely continuous and $\Gamma$ is saturated by strong stable leaves, we
find $w\in{\mathcal{W}}^{cu}(y)$ arbitrarily close to $y$ whose center leaf
${\mathcal{W}}^{c}_{w}$ intersects $\Gamma$ at points close to each of the $k$
iterates of $y$. In view of Corollary 3.9 this implies that $k<n_{0}$. This
means that the period of $y$ for $f$ is less than $k_{0}\kappa n_{0}$ as
stated. The proof of Lemma 3.10 is complete. ∎

######  Lemma 3.11.

Every point $z\in\ell_{0}$ is periodic for $f$, with period bounded by
$k_{0}\kappa n_{0}$.

###### Proof.

Let $y\in\ell_{0}$ be a periodic point as in Lemma 3.10 and let $z\in\ell_{0}$
be arbitrary. Choose
$y^{\prime}\in{\mathcal{W}}^{u}(y)\cap{\mathcal{W}}^{c}(\Lambda_{i})\setminus\ell_{0}$
and
$z^{\prime}\in{\mathcal{W}}^{s}(z)\cap{\mathcal{W}}^{c}(\Lambda_{i})\setminus\ell_{0}$.
By accessibility, there exists some $su$-path connecting $y^{\prime}$ to
$z^{\prime}$ or, in other words, there exist points

$b_{0}=y,a_{1}=y^{\prime},b_{1},\ldots,a_{i},b_{i},\ldots,a_{s}=z^{\prime},b_{s}=z$

which belong to ${\mathcal{W}}^{c}(\Lambda_{i})$ such that $a_{j}$ and $b_{j}$
belong to the same strong stable manifold and $b_{j}$ and $a_{j+1}$ belong to
the same strong unstable manifold. We are going to find an (arbitrarily)
nearby $su$-path

(14)
$\tilde{b}_{0}=y,\tilde{a}_{1},\tilde{b}_{1},\ldots,a_{i},b_{i},\ldots,\tilde{a}_{s},\tilde{b}_{s}$

with $\tilde{b}_{s}\in\ell_{0}$ and such that every $\tilde{b}_{i}$ belongs to
some periodic center leaf in ${\mathcal{W}}^{c}(\Lambda_{i})$. The first step
is to observe that, since periodic leaves are dense, one may always find
periodic leaves $\ell_{1},\ldots,\ell_{s-1}$ arbitrarily close to
${\mathcal{W}}^{c}({b_{1}}),\ldots,{\mathcal{W}}^{c}({b_{s-1}})$,
respectively. Let $\ell_{s}=\ell_{0}$. Assume
$\tilde{b}_{0},\tilde{a}_{1},\ldots,\tilde{b}_{k}$ have been defined, for some
$0\leq k<s$. Since ${\mathcal{W}}^{u}(b_{k})$ intersects the stable set of
${\mathcal{W}}^{c}(b_{k+1})$ transversely at $a_{k+1}$, and stable and
unstable sets vary continuously with the base point, we can find
$\tilde{b}_{k+1}\in\ell_{k+1}$ close to $b_{k+1}$ such that
${\mathcal{W}}^{u}(\tilde{b}_{k})$ intersects
${\mathcal{W}}^{s}(\tilde{b}_{k+1})$ at some point $\tilde{a}_{k+1}$ close to
$a_{k}$. Repeating this procedure $s$ times, we obtain an $su$-path as in
(14).

The next step is to prove that the points $\tilde{b}_{i}$ themselves are
periodic. Recall that $\tilde{b}_{0}=y$ is taken to be periodic and $D^{u}$
intersects ${\mathcal{W}}^{s}(\tilde{b}_{0})$. So, the iterates accumulate on
${\mathcal{W}}^{u}(\tilde{b}_{0})$ and, in particular, on $\tilde{a}_{1}$.
This implies there exist points $w\in\ell_{1}$ arbitrarily close to
$\tilde{b}_{1}$ whose strong stable manifold intersects $f^{n}(D^{u})$ for
some $n$. Since $\Gamma$ has full volume inside every $f^{n}(D^{u})$, we may
use Lemma 3.10 to conclude that $w$ is periodic, with period uniformly
bounded. Consequently, $\tilde{b}_{1}$ itself is periodic. It also follows
that the iterates of $D^{u}$ accumulate on ${\mathcal{W}}^{u}(\tilde{b}_{1})$.
This means we may now repeat the construction with $\tilde{b}_{1}$ in the
place of $\tilde{b}_{0}$ and conclude that $\tilde{b}_{2}$ is periodic. After
$s$ steps we conclude that $\tilde{z}=\tilde{b}_{s}$ is periodic. Since
$\tilde{z}$ is arbitrarily close to $z$, and all the periods are bounded, we
get that $z$ itself is periodic. This completes the proof of the lemma. ∎

In particular, Lemma 3.11 implies that no periodic point on the support of
$\nu_{*}$ is hyperbolic. This is a contradiction since, by a classical result
of Katok [29], the support of any hyperbolic measure contains hyperbolic
periodic points. This completes the proof of Proposition 3.5. ∎

##  4\. Mostly contracting center

In this section we prove some useful facts about partially hyperbolic
diffeomorphisms with mostly contracting center direction. We call
_${\mathcal{W}}^{u}$ -disk_ any image of a ball in $E^{u}$ embedded inside
some strong unstable leaf.

######  Lemma 4.1.

The center direction of $f$ is mostly contracting if and only if the center
Lyapunov exponents of all ergodic Gibbs $u$-states are negative.

If $f\in{\mathcal{P}}_{1}^{k}(N)$, $k>1$ and $\Lambda$ is an attractor of
$f_{c}$, then the center direction of $f\mid{{\mathcal{W}}^{c}(\Lambda)}$ is
mostly contracting if and only if the center Lyapunov exponent is negative for
every ergodic Gibbs $u$-state supported in ${\mathcal{W}}^{c}(\Lambda)$.

###### Proof.

Bonatti, Viana [19] show that if the center direction is mostly contracting
then the center exponents of every ergodic Gibbs $u$-state are negative. To
prove the converse, let $D$ be any disk inside a strong unstable leaf. By [17,
Lemma 11.12] every Cesaro accumulation point of the iterates of Lebesgue
measure on $D$ is a Gibbs $u$-state. By [17, Lemma 11.13] every ergodic
component of a Gibbs $u$-state is again a Gibbs $u$-state. This implies that
the iterates $f^{n}(D)$ accumulate on the support of some ergodic Gibbs
$u$-state $\nu$. The hypothesis implies that $\nu$-almost every point has a
Pesin (local) stable manifold which is an embedded disk of dimension $d_{cs}$.
Using also the absolute continuity of the Pesin stable foliation (Pesin [35]),
we conclude that a positive Lebesgue measure subset of points in some
$f^{n}(D)$ belong to the union of these $d_{s}$-disks. This implies that (5)
holds on a positive Lebesgue measure subset of $D$, as we wanted to show.

The second part of the lemma follows from similar arguments. ∎

###  4.1. Supports of Gibbs $u$-states

######  Lemma 4.2.

If the center direction of $f$ is mostly contracting then the supports of the
ergodic Gibbs $u$-states of $f$ are pairwise disjoint.

###### Proof.

Let $m_{1}$ and $m_{2}$ be ergodic Gibbs $u$-states of $f$ and suppose
$\operatorname{supp}m_{1}\cap\operatorname{supp}m_{2}$ contains some point
$z$. Let $D$ be any ${\mathcal{W}}^{u}$-disk around $z$. Then
$D\subset\operatorname{supp}m_{1}\cap\operatorname{supp}m_{2}$, since the
supports are ${\mathcal{W}}^{u}$-saturated (Proposition 3.1). By Lemmas 11.12
and 11.13 in [17], every ergodic component $\nu$ of every Cesaro accumulation
point of the iterates of Lebesgue measure on $D$ is an ergodic Gibbs
$u$-state. Clearly, the support of $\nu$ is contained in
$\operatorname{supp}m_{1}\cap\operatorname{supp}m_{2}$. By Pesin theory (see
[19] for this particular setting) $\nu$-almost every point has a local stable
manifold which is an embedded $d_{cs}$-disk. Recall (Proposition 3.1) that the
density of Gibbs $u$-states along strong unstable leaves is positive and
finite. Thus, we may find a ${\mathcal{W}}^{u}$-disk
$D_{\nu}\subset\operatorname{supp}\nu$ such that every point $x$ in a full
Lebesgue measure subset $D_{\nu}^{*}$ has a Pesin stable manifold and belongs
to the basin of $\nu$. Moreover, $D_{\nu}$ is accumulated by
${\mathcal{W}}^{u}$-disks $D_{i}\subset\operatorname{supp}m_{1}$ such that
Lebesgue almost every point is in the basin of $m_{1}$. Assuming $D_{i}$ is
close enough to $D_{\nu}$, it must intersect the union of the local stable
manifolds through the points of $D_{\nu}^{*}$ on some positive Lebesgue
measure subset $D_{i}^{*}$ (because the Pesin local stable lamination is
absolutely continuous [35]). Then $D_{i}^{*}$ is contained in the basin of
$\nu$, and some full Lebesgue measure subset is contained in the basin of
$m_{1}$. That implies $m_{1}=\nu$. Analogously, $m_{2}=\nu$, and so the
ergodic Gibbs $u$-states $m_{1}$ and $m_{2}$ coincide. That completes the
proof of the lemma. ∎

######  Remark 4.3.

It follows from Proposition 3.1 and Lemma 4.2 that if $f$ has mostly
contracting center direction and minimal strong unstable foliation then it has
a unique Gibbs $u$-state. This was first observed in [19].

######  Proposition 4.4.

Suppose the center direction of $f$ is mostly contracting, and let $m$ be an
ergodic Gibbs $u$-state of $f$. Then the support of $m$ has a finite number of
connected components. Moreover, each connected component $S$ is
${\mathcal{W}}^{u}$-saturated and ${\mathcal{W}}^{u}(x)$ is dense in $S$ for
any $x\in S$.

###### Proof.

Let $p$ be any periodic point in the support of $m$ with stable index equal to
$d_{cs}$ (such periodic points do exist, by Katok [29]) and let $\kappa$ be
its period. By Proposition 3.1, the unstable manifold of every $f^{j}(p)$ is
contained in $\operatorname{supp}m$. We claim that
$\cup_{j=1}^{\kappa}{\mathcal{W}}^{u}(f^{j}(p))$ is dense in
$\operatorname{supp}m$. To see this, let $D$ be any disk inside
${\mathcal{W}}^{u}(p)$. Consider the forward iterates of Lebesgue measure on
$D$. Using Lemmas 11.12 and Lemma 11.13 in [17], one gets that any ergodic
component of any Cesaro accumulation point of these iterates is an ergodic
Gibbs $u$-state $\nu$ supported inside the closure of
$\cup_{j=1}^{\kappa}{\mathcal{W}}^{u}(f^{j}(p))$. By Lemma 4.2, the Gibbs
$u$-states $m$ and $\nu$ must coincide. In particular, $\operatorname{supp}m$
is contained in the closure of
$\cup_{j=1}^{\kappa}{\mathcal{W}}^{u}(f^{j}(p))$. That proves our claim.

Since $m$ is ergodic for $f$, its ergodic decomposition relative to
$f^{\kappa}$ has the form $m=l^{-1}\sum_{i=1}^{l}f^{i}_{*}\tilde{m}$ where $l$
divides $\kappa$ and $\tilde{m}$ is $f^{\kappa}$-invariant and ergodic. Then

$\operatorname{supp}m=\bigcup_{i=1}^{l}f^{i}(\operatorname{supp}\tilde{m}).$

We claim that the $f^{i}(\operatorname{supp}\tilde{m})$, $i=1,\dots,l$ are
precisely the connected components of $\operatorname{supp}m$. On the one hand,
the previous paragraph gives that $p\in f^{s}(\operatorname{supp}\tilde{m})$
for some $s$. Replacing either $p$ or $\tilde{m}$ by an iterate, we may
suppose $s=0$. Then, by the argument in the previous paragraph applied to
$f^{\kappa}$ (it is clear from the definition (5) that if $f$ has mostly
contracting then so does any positive iterate), $\operatorname{supp}\tilde{m}$
coincides with the closure of ${\mathcal{W}}^{u}(p)$ and, in particular, it is
connected. On the other hand, Lemma 4.2 gives that the
$f^{i}(\operatorname{supp}\tilde{m})$, $i=1,\dots,l$ are pairwise disjoint.
Since they are closed, it follows that they are also open in
$\operatorname{supp}m$. This proves our claim.

We are left to prove that the strong unstable foliation is minimal in each
connected component $S_{i}=f^{i}(\operatorname{supp}\tilde{m})$. This will
follow from an argument of Bonatti, Díaz, Ures [16]:

######  Lemma 4.5.

There is a neighborhood $U^{s}_{i}$ of $f^{i}(p)$ inside $W^{s}(f^{i}(p))$
such that every unstable leaf in $S_{i}$ has some transverse intersection with
$U^{s}_{i}$.

###### Proof.

For any $x\in S_{i}$, let $D_{x}$ be a small ${\mathcal{W}}^{u}$-disk around
$x$. Since $\tilde{m}_{j}$ is the unique ergodic $u$-state of $f^{\kappa}$
with support contained in $S_{j}$. It is also the unique Cesaro accumulation
point of the iterates of $\operatorname{vol}_{D_{x}}$ under $f^{\kappa}$. In
particular, there is $n_{x}\geq 1$ such that $f^{n_{x}\kappa}(D_{x})$
intersects the local stable manifold of $f^{i}(p)$ transversely. This implies
that $D_{x}$ intersects the global stable manifold of $f^{i}(p)$ transversely.
Then, by continuity of the strong unstable foliation, there is a neighborhood
$V_{x}$ of $x$ and a bounded open set $U_{x}\subset
W_{f^{\kappa}}^{s}(f^{i}(p))$ such that ${\mathcal{W}}^{u}(y)$ intersects
$U_{x}$ transversely for every $y\in V_{x}$. The family $\\{V_{x}:x\in
S_{i}\\}$ is an open cover of the compact set $S_{i}$. Let
$\\{V_{x_{1}},\cdots,V_{x_{m}}\\}$ be a finite subcover. Choose $U^{s}_{j}$ a
bounded neighborhood of $f^{i}(p)$ inside $W_{f^{\kappa}}^{s}(f^{i}(p))$
containing $U_{x_{j}}$ for all $j=1,\dots,m$. It follows from the construction
that every strong unstable leaf contained in $S_{i}$ intersects $U_{i}^{s}$
transversely. This finishes the proof of the lemma. ∎

Let us go back to proving Proposition 4.4. The lemma gives that
${\mathcal{W}}^{u}(f^{-n\kappa}(x))$ intersects $U_{i}^{s}$ transversely, and
so ${\mathcal{W}}^{u}(x)$ intersects $f^{n\kappa}(U_{i}^{s})$ transversely,
for every $x\in S_{i}$ and every $n\geq 0$. Since $f^{n\kappa}(U_{i}^{s})$
converges to $f^{i}(p)$ when $n\to\infty$, it follows that $W^{u}(f^{i}(p))$
is contained in the closure of $W^{u}(x)$. Hence, $W^{u}(x)$ is dense in
$S_{j}$, as claimed. The proof of the proposition is complete. ∎

###  4.2. Bernoulli property

An invariant ergodic measure $\eta$ of a transformation $g$ is called
_Bernoulli_ if $(g,\eta)$ is ergodically conjugate to a Bernoulli shift.

######  Theorem 4.6.

Suppose $f$ is a $C^{k}$, $k>1$ partially hyperbolic diffeomorphism with
mostly contracting center direction. Then there is $l\geq 1$ and a $C^{k}$
neighborhood $\mathcal{U}$ of $f$ such that for any $g\in\mathcal{U}$, every
ergodic $u$-state of $g^{l}$ is Bernoulli.

###### Proof.

Let $m_{1},\dots,m_{u}$ be the ergodic Gibbs $u$-states of $f$. Proposition
4.4 gives that for each $j=1,\dots,u$ there exists $l_{j}\geq 1$ such that the
support of $m_{j}$ has $l_{j}$ connected components $S_{j,i}$,
$i=1,\dots,l_{j}$. Moreover, each connected component $S_{j,i}$ carries an
ergodic component $m_{j,i}=f^{i}_{*}\tilde{m}_{j}$ of the Gibbs $u$-state
$m_{j}$ for the iterate $f^{l_{j}}$. Let $l$ be any common multiple of
$l_{1},\dots,l_{u}$. Then every $S_{j,i}$ is fixed under $f^{l}$. Moreover,
every Gibbs $u$-state $m_{j,i}$ is $f^{l}$-invariant and $f^{nl}$-ergodic for
every $n\geq 1$: otherwise $S_{i}$ would break into more than one connected
component (cf. the proof of Lemma 4.2). Then, by Ornstein, Weiss [33], every
$m_{i,j}$ is a Bernoulli measure for $f^{l}$. We claim that $\\{m_{j,i}:1\leq
j\leq u\text{ and }1\leq i\leq l_{j}\\}$ contains all the ergodic $u$-states
of $f^{nl}$ for every $n\geq 1$. Indeed, let $m_{*}$ be any ergodic $u$-state
for $f^{nl}$. Then

$m=\frac{1}{nl}\sum_{k=1}^{nl}f_{*}^{k}m_{*}$

is a $u$-state for $f$. Let $m=a_{1}m_{1}+\cdots+a_{u}m_{u}$ be its ergodic
decomposition for $f$ and let $s$ be such that $a_{s}>0$. Then
$\operatorname{supp}m_{s}\subset\operatorname{supp}m$. Since
$\operatorname{supp}m_{s}$ is $f$-invariant, it must intersect
$\operatorname{supp}m_{*}$. Using Lemma 4.2 for $f^{nl}$ we conclude that
$m_{*}$ must coincide with some ergodic component of $m_{s}$ for the iterate
$f^{nl}$. In other words, it must coincide with $m_{s,i}$ for some
$i=1,\dots,l_{s}$, and this proves our claim.

Now we extend these conclusions to any diffeomorphism $g$ in a $C^{k}$, $k>1$
neighborhood of $f$. By Andersson [9], any such $g$ has mostly contracting
center direction, and so the previous argument applies to it. However, we must
also prove that the integer $l$ can be taken uniform on a whole neighborhood
of $f$. Notice that the only constraint on $l$ was that it should be a
multiple of the periods $l_{j}$ of the ergodic components $m_{j}$. Observe
that [9] also gives that the number of ergodic Gibbs $u$-states does not
exceed the number of ergodic Gibbs $u$-states of $f$. So, we only need to
check that the periods $l_{j}$ remain uniformly bounded for any $g$ in a
neighborhood. We do this by arguing with periodic points, as follows. Let us
fix, once and for all, $f$-periodic points $p_{j}$ with stable index $d_{cs}$
in the support of each $m_{j}$, $j=1,\dots,u$. The period of each $p_{j}$ is a
(fixed) multiple of $l_{j}$. Let $p_{j}(g)$ be the continuation of these
periodic points for some nearby diffeomorphism $g$, and let
$\\{m_{1}(g),\dots,m_{s}(g)\\}$, with $s\leq u$ be the ergodic Gibbs
$u$-states of $g$. We claim that every $\operatorname{supp}m_{j}(g)$, $1\leq
j\leq s$ contains some $p_{i}(g)$, $1\leq i\leq u$. This can be seen as
follows. As observed before, any accumulation point of Gibbs $u$-states of $g$
when $g\to f$ is a Gibbs $u$-state for $f$. We fix some small $\varepsilon>0$
and consider the $\varepsilon$-neighborhoods $B(p_{j},\varepsilon)$ of the
periodic points $p_{j}$. Then, for any $g$ close enough to $f$ every ergodic
Gibbs $u$-state $m_{j}(g)$ must give positive weight to some
$B(p_{i},\varepsilon)$ and, consequently, also to $B(p_{i}(g),2\varepsilon)$.
By continuous dependence of stable manifolds of periodic points on the
dynamics, and the fact that the supports of Gibbs $u$-states are
$u$-saturated, it follows that $\operatorname{supp}m_{j}(g)$ contains some
${\mathcal{W}}^{u}$-disk that intersects $W^{s}(p_{i}(g))$ transversely. Then,
the support of $m_{j}(g)$ must contain $p_{i}(g)$. This proves our claim. It
follows that the period $l_{j}(g)$ of each ergodic Gibbs $u$-state of $g$
divides the period of some $p_{i}(g)$ which, of course, coincides with the
period of $p_{i}$. Since the latter have been fixed once and for all, this
proves that the $l_{j}(g)$ are indeed uniformly bounded on a neighborhood of
$f$. The proof of the theorem is complete. ∎

###  4.3. Abundance of mostly contracting center

We also give a family of new examples of diffeomorphisms with mostly
contracting center.

######  Theorem 4.7.

Suppose $\dim M=3$. The set of ergodic diffeomorphisms such that either $f$ or
$f^{-1}$ has mostly contracting center direction is $C^{1}$ open and dense in
the space of $C^{k}$, $k>1$ partially hyperbolic volume preserving
diffeomorphisms with 1-dimensional center and some fixed compact center leaf.

###### Proof.

Denote by ${\mathcal{V}}_{m}^{k}$ the set of $C^{k}$ volume preserving
partially hyperbolic diffeomorphisms with 1-dimensional center and some fixed
compact center leaf. This is a $C^{1}$ open set, cf. [28, Theorem 4.1].
Moreover, the diffeomorphisms such that both the strong stable foliation and
the strong unstable foliation is minimal fill an open and dense subset $U_{1}$
of ${\mathcal{V}}_{m}^{1}$. This follows from a conservative version of the
results of [16]: one only has to observe that blenders, that they use for the
proof in the dissipative context, can be constructed also in the conservative
setting, as shown by [26]. By [15], there is an open and dense subset $U_{2}$
for which the center Lyapunov exponent

$\int\log|Df\mid{E^{c}(x)}|dm(x)\neq 0.$

Furthermore, by [24], there is an open and dense subset $U_{3}$ of
${\mathcal{V}}^{1}_{m}$ consisting of accessible diffeomorphisms. Let
$U=U_{1}\cap U_{2}\cap U_{3}$. Before proceeding, let us recall that
$C^{\infty}$ are $C^{1}$ dense in the space of volume preserving
diffeomorphisms, by [11]. In particular, the $C^{1}$ open and dense subset $U$
has non-trivial intersection with the space of $C^{k}$ diffeomorphisms, for
any $k>1$.

We claim that for every $C^{k}$, $k>1$ diffeomorphism $f$ in $U$, either $f$
or its inverse has a unique ergodic Gibbs $u$-state and the corresponding
center Lyapunov exponent is negative. In particular, by Lemma 4.1, either $f$
or its inverse has mostly contracting center direction. The first step is to
note that $f$ is ergodic, since it is accessible (see [22, 27, 38]). Then the
Lebesgue measure $\operatorname{vol}$ is an ergodic Gibbs $u$-state for both
$f$ and $f^{-1}$. Since the strong stable and strong unstable foliations are
minimal, the Gibbs $u$-state is unique; see Remark 4.3. This completes the
proof of Theorem 4.7. ∎

##  5\. Finiteness and stability of physical measures

In this section we prove Theorem D. As remarked before, Theorem A is a
particular case. We begin by recalling certain ideas from Bonatti, Gomez-Mont,
Viana [18] and Avila, Viana [13] that we use for handling the case when the
center Lyapunov exponent vanishes.

###  5.1. Smooth cocycles

By assumption, the center leaves of $f$ define a fiber bundle $\pi_{c}:N\to
N/{\mathcal{W}}^{c}$ over the leaf space. Then $f$ may be seen as a smooth
cocycle (as defined in [13]) over $f_{c}$:

$\begin{array}[]{rrcl}f:&N&\rightarrow&N\\\ &\downarrow&&\downarrow\\\
f_{c}:&N/{\mathcal{W}}^{c}&\rightarrow&N/{\mathcal{W}}^{c}\end{array}$

It follows from the form of our maps that the strong stable manifold
${\mathcal{W}}^{s}(x)$ of every point $x\in M$ is a graph over the stable set
$W^{s}_{\pi_{c}(x)}$ of $\pi_{c}(x)\in N/{\mathcal{W}}^{c}$. For each $\eta\in
W^{s}(\xi)$, the strong stable holonomy defines a homeomorphism
$h^{s}_{\xi,\eta}:\xi\to\eta$ between the two center leaves. In fact (see [13,
Proposition 4.1]),

(15)
$h^{s}_{\xi,\eta}(\theta)=\lim_{n\to\infty}(f^{n}\mid\eta)^{-1}\circ(f^{n}\mid\xi)(\theta),$

(for large $n$ one can identify $f^{n}(\xi)\approx f^{n}(\eta)$ via the fiber
bundle structure) for each $\theta\in\xi$ and the limit is uniform on the set
of all $(\xi,\eta,\theta)$ with $\theta\in\xi$ and $\xi$ and $\eta$ in the
same local stable set. These _$s$ -holonomy_ maps satisfy

  * •

$h^{s}_{\eta,\zeta}\circ h^{s}_{\xi,\eta}=h^{s}_{\xi,\zeta}$ and
$h^{s}_{\xi,\xi}=\operatorname{id}$

  * •

$f\circ h^{s}_{\xi,\eta}=h^{s}_{f_{c}(\xi),f_{c}(\eta)}\circ f$

  * •

$(\xi,\eta,\theta)\mapsto h^{s}_{\xi,\eta}(\theta)$ is continuous on the set
of triples $(\xi,\eta,\theta)$ with $\xi$ and $\eta$ in the same local stable
set and $\theta\in{\mathcal{W}}^{c}(\xi)$.

Let $m$ be any $f$-invariant probability measure and $\mu=(\pi_{c})_{*}(m)$. A
_disintegration_ of $m$ into conditional probabilities along the center leaves
is a measurable family $\\{m_{\xi}:\xi\in\operatorname{supp}\mu\\}$ of
probability measures with $m_{\xi}(\xi)=1$ for $\mu$-almost every $\xi$ and

(16) $m(E)=\int m_{\xi}(E)\,d\mu(\xi)$

for every measurable set $E\subset M$. By Rokhlin [40], such a family exists
and is essentially unique. A disintegration is called _$s$ -invariant_ if

$(h^{s}_{\xi,\eta})_{*}m_{\xi}=m_{\eta}\quad\text{for every $\xi$,
$\eta\in\operatorname{supp}\mu$ in the same stable set.}$

In a dual way one defines _$u$ -holonomy_ maps and _$u$ -invariance_. We call
a disintegration _bi-invariant_ if it is both $s$-invariant and $u$-invariant,
and we call it _continuous_ if $m_{\xi}$ varies continuously with $\xi$ on the
support of $\mu$, relative to the weak∗ topology.

######  Proposition 5.1.

Let $f\in{\mathcal{P}}_{*}^{k}(N)$, $k>1$ be such that the center stable
foliation is absolutely continuous. Let $m$ be an ergodic Gibbs $u$-state with
vanishing center Lyapunov exponents. Then $m$ admits a disintegration
$\\{m_{\xi}:\xi\in\operatorname{supp}\mu\\}$ into conditional probabilities
along the center leaves which is continuous and bi-invariant.

###### Proof.

Proposition 3.3 gives that $(\pi_{c})_{*}m$ has local product structure. Thus,
we are in a position to use Theorem D of Avila, Viana [13] to obtain the
conclusion of the present proposition. ∎

###  5.2. Zero Lyapunov exponent case

The following result provides a characterization of the systems exhibiting
ergodic Gibbs $u$-states with vanishing central exponent.

######  Proposition 5.2.

Let $f\in{\mathcal{P}}_{1}^{k}(N)$, $k>1$ be such that the center stable
foliation is absolutely continuous. Let $\Lambda$ be an attractor of $f_{c}$
such that $f$ is accessible on $\Lambda$, and let $m$ be an ergodic Gibbs
$u$-state with vanishing center Lyapunov exponent. Then

  * (1)

the conditional probabilities $\\{m_{x}:x\in\Lambda\\}$ along the center
leaves are equivalent to the Lebesgue measure on the leaves, with densities
uniformly bounded from zero and infinity;

  * (2)

$\operatorname{supp}m=W^{c}(\Lambda)$ and this is the unique Gibbs $u$-state
supported in ${\mathcal{W}}^{c}(\Lambda)$;

  * (3)

the basin of $m$ covers a full Lebesgue measure subset of a neighborhood of
${\mathcal{W}}^{c}(\Lambda)$.

###### Proof.

By Proposition 5.1, there is a disintegration $\\{m_{x}:x\in\Lambda\\}$ of $m$
along the center foliation which is continuous, $s$-invariant, and
$u$-invariant. Let $\xi$ and $\eta$ be any two points in
${\mathcal{W}}^{c}(\Lambda)$. By accessibility on $\Lambda$, one can find an
$su$-path $b_{0}=\xi,b_{1},\ldots,b_{s-1},b_{s}=\eta$ connecting $\xi$ to
$\eta$. This $su$-path induces a holonomy map
$h:{\mathcal{W}}^{c}(\xi)\to{\mathcal{W}}^{c}(\eta)$, defined as the
composition of all strong stable/unstable holonomy maps
$h_{i}:{\mathcal{W}}^{c}(b_{i-1})\to{\mathcal{W}}^{c}(b_{i})$. The fact that
the disintegration is bi-invariant gives, in particular, that

(17) $m_{\eta}(h(B^{c}_{\varepsilon}(\xi))=m_{\xi}(B^{c}_{\varepsilon}(\xi))$

It is a classical fact that the strong stable and strong unstable foliations
are absolutely continuous in a strong sense: their holonomy maps have bounded
Jacobians. See [20, 31, 1]. Those arguments extend directly to their
restrictions to each center stable or center unstable leaf, respectively: the
restricted strong stable and strong unstable foliations are also absolutely
continuous with bounded Jacobians. By compactness, the $su$-path may be chosen
such that the number $s$ of legs and the length of each leg are uniformly
bounded, independent of $\xi$ and $\eta$ 111This may be deduced from [12] as
follows. By Proposition 8.3 in [12], given any $x_{0}\in M$ there exists $w\in
M$ such that $x_{0}$ is connected to every point in a neighborhood of $w$ by a
uniformly bounded $su$-path. Then the same is true if one replaces $w$ by an
arbitrary point $z\in M$: connect $w$ to $z$ by some $su$-path; the ”same”
$su$-path determines a bijection between neighborhoods of $w$ and $z$;
concatenating with $su$-paths from $x_{0}$ to the neighborhood of $w$ one
obtains uniformly bounded $su$-paths from $x_{0}$ to any point near $z$. The
claim now follows by compactness of the ambient manifold.. Then, we may fix a
uniform upper bound constant $K>1$ on the Jacobians of all associated strong
stable and strong unstable holonomies. Notice
$\operatorname{vol}^{c}(B_{r}(\zeta))=2r$, since the center leaves are one-
dimensional. Then

(18)
$K^{-1}\operatorname{vol}^{c}(B^{c}_{\varepsilon}(\xi))\leq\operatorname{vol}^{c}(h(B^{c}_{\varepsilon}(\xi)))\leq
K\operatorname{vol}^{c}(B^{c}_{\varepsilon}(\xi)).$

From (17) and (18) we obtain

$\frac{1}{K}\frac{m_{\xi}(B^{c}_{\varepsilon}(\xi))}{\operatorname{vol}^{c}(B^{c}_{\varepsilon}(\xi))}\leq\frac{m_{\eta}(h(B^{c}_{\varepsilon}(\xi)))}{\operatorname{vol}^{c}(h(B^{c}_{\varepsilon}(\xi)))}\leq
K\frac{m_{\xi}(B^{c}_{\varepsilon}(\xi))}{\operatorname{vol}^{c}(B^{c}_{\varepsilon}(\xi))}.$

and, taking the limit as $\varepsilon\to 0$,

$\frac{1}{K}\frac{dm_{\xi}}{d\operatorname{vol}^{c}}(\xi)\leq\frac{dm_{\eta}}{d\operatorname{vol}^{c}}(\eta)\leq
K\frac{dm_{\xi}}{d\operatorname{vol}^{c}}(\xi).$

Since we can always find $\eta$ where the density is less or equal than $1$
(respectively, greater or equal than $1$), this implies that

(19) $\frac{dm_{\xi}}{d\operatorname{vol}^{c}}(\xi)\in\big{[}K^{-1},K]$

for every $\xi$, and that proves claim (1).

Now let $m^{\prime}$ be any other ergodic Gibbs $u$-state supported in
${\mathcal{W}}^{c}(\Lambda)$. The center Lyapunov exponent of $m^{\prime}$
must vanish: otherwise, by [29], there would be some hyperbolic periodic point
in ${\mathcal{W}}^{c}(\Lambda)$, and that is incompatible with the conclusion
in part (1) that there exist invariant conditional probabilities equivalent to
Lebesgue measure along the center leaves. So, all the previous considerations
apply to $m^{\prime}$ as well. In particular, it has a continuous
disintegration $\\{m^{\prime}_{x}:x\in\Lambda\\}$ along the center foliation
such that each $m^{\prime}_{x}$ is equivalent with $\operatorname{vol}^{c}$.
Moreover, by Proposition 3.2, $(\pi_{c})_{*}(m)=(\pi_{c})_{*}(m^{\prime})$.
Then, $\operatorname{vol}^{c}$-almost every point in almost every center leaf,
relative to $(\pi_{c})_{*}(m)=(\pi_{c})_{*}(m^{\prime})$, belongs to the basin
of both $m$ and $m^{\prime}$. In particular, the two basins intersect, and
that implies $m=m^{\prime}$. That completes the proof of claim (2).

From conclusion (1) we get that there exists a full $m$-measure set $\Delta$
consisting of leaves such that $\operatorname{vol}^{c}$-almost every point in
the leaf belongs to the basin of $m$. Then, since $m$ is a Gibbs $u$-state, we
can find a ${\mathcal{W}}^{u}$-disk $D^{u}_{0}$ such that $\Delta\cap
D^{u}_{0}$ has full measure in $D^{u}_{0}$. Consider the $cu$-disk

$D_{0}^{cu}=\bigcup_{\xi\in D_{0}^{u}}{\mathcal{W}}^{c}(\xi)$

Observe that the center foliation ${\mathcal{W}}^{c}$ is absolutely continuous
on each center unstable leaf, because the corresponding holonomy maps between
unstable leaves coincide with the corresponding holonomy maps for the center
stable foliation, which we assume to be absolutely continuous. Using this
fact, and a Fubini argument, we get that Lebesgue almost every point in
$D_{0}^{cu}$ belongs to the basin of $m$. Next, consider the open set

$D_{0}=\bigcup_{\zeta\in D_{0}^{cu}}{\mathcal{W}}^{s}_{loc}(\zeta).$

Since the strong stable foliation is absolutely continuous and the basin is
${\mathcal{W}}^{s}$-saturated, it follows that Lebesgue almost every point in
$D_{0}$ belongs to the basin of $m$. It is clear that we can cover
${\mathcal{W}}^{c}(\Lambda)$ by finitely many such open sets $D_{0}$. This
proves (3), and so the proof of the lemma is complete. ∎

###  5.3. Construction of physical measures

We are nearly done with the proof of Theorem D. By Proposition 3.5, all
ergodic Gibbs $u$-states have non-negative center Lyapunov exponent. The case
when the exponent vanishes for some Gibbs $u$-state is handled by Proposition
5.2: we get alternative (a) of the theorem in this case. Finally, if the
center Lyapunov exponent is negative for all Gibbs $u$-states over some
attractor $\Lambda_{i}$ of $f_{c}$ then, by Lemma 4.1, the center direction of
$f$ is mostly contracting on that attractor $\Lambda_{i}$. Then, by Bonatti,
Viana [19], there are finitely many ergodic Gibbs $u$-states supported in
${\mathcal{W}}^{c}(\Lambda_{i})$, these $u$-states are the physical measures
of $f$, and the union of their basins covers a full volume measure subset of a
neighborhood of ${\mathcal{W}}^{c}(\Lambda_{i})$. By Theorem 4.6, all these
physical measures are Bernoulli for some iterate of $f$. Thus, we get
alternative (b) of the theorem in this case.

From now on, let $\\{m_{i,j}\\}_{j=1}^{J(i)}$ be the physical measures
supported on each attractor $\Lambda_{i}$. As we have just seen, their basins
cover a full Lebesgue measure subset of a neighborhood $U_{i}$ of
${\mathcal{W}}^{c}(\Lambda_{i})$. We want to prove that the union of all these
basins contains a full Lebesgue measure subset of the ambient manifold.
Suppose otherwise, that is, suppose the complement $C$ of this union has
positive Lebesgue measure. Let $C_{0}\subset C$ be the set of Lebesgue density
points of $C$. Notice that $C_{0}$ is $f$-invariant and
$\operatorname{vol}(C_{0})=\operatorname{vol}(C)$. Since the unstable
foliation is absolutely continuous, there is a ${\mathcal{W}}^{u}$-disk
$D^{u}$ such that $\operatorname{vol}_{D^{u}}(D^{u}\cap C_{0})>0$. Denote
$I^{u}=D^{u}\cap C_{0}$. Then every Cesaro accumulation point of the iterates
of Lebesgue measure on $I^{u}$ is a Gibbs $u$-state (see [17], section 11.2),
and so its ergodic components are ergodic Gibbs $u$-states. Let $m^{*}$ be any
such accumulation point and $m_{i,j}$ be an ergodic component of $m^{*}$. The
support of $m_{i,j}$ is contained in $U_{i}$, and so there is $n_{0}\geq 1$
such that $f^{n_{0}}(I^{u})$ intersects $U_{i}$. Recalling that $C_{0}$ is
invariant, we get that $\operatorname{vol}(C_{0}\cap U_{i})>0$. This
contradicts the definition of $C_{0}$, since Lebesgue almost every point in
$U_{i}$ belongs to the basin of $m_{i,l}$ for some $l=1,\dots,J(i)$. This
contradiction proves that the union of the basins does have full Lebesgue
measure in $N$. That completes the proof of Theorem D.

###  5.4. Number of physical measures

In this section, we give explicit upper bounds on the number of physical
measures for some diffeomorphisms with mostly contracting center direction:

######  Theorem 5.3.

Let $f\in{\mathcal{P}}_{1}^{k}(N)$, $k>1$ be accessible on some attractor
$\Lambda$ and have absolutely continuous center stable foliation. Assume there
exists some center leaf $\ell\subset{\mathcal{W}}^{c}(\Lambda)$ such that
$f^{\kappa}(\ell)=\ell$ for some $\kappa\geq 1$ and $f^{\kappa}\mid\ell$ is
Morse-Smale with periodic points $p_{1},\cdots,p_{s}$.

Then the center direction is mostly contracting over $\Lambda$ and $f$ has at
most $s$ physical measures supported in ${\mathcal{W}}^{c}(\Lambda)$. If
${\mathcal{W}}^{u}(p_{i})$ intersects
$W^{s}(\ell)\setminus\cup_{j=1}^{s}{\mathcal{W}}^{s}(p_{j})$ for every $i$
then $f$ has at most $s/2$ physical measures supported in
${\mathcal{W}}^{c}(\Lambda)$.

###### Proof.

Since $f$ has hyperbolic periodic point in ${\mathcal{W}}^{c}(\Lambda)$ the
restriction of $f$ to ${\mathcal{W}}^{c}(\Lambda)$ can not be conjugate to a
rotation extension over $\Lambda$. Thus, by Theorem D, $f$ has mostly
contracting center direction over $\Lambda$.

######  Lemma 5.4.

Suppose $f\in{\mathcal{P}}_{1}^{k}(N)$, $k>1$ has mostly contracting center
direction on an attractor $\Lambda$ and let $p$ be any periodic point in
${\mathcal{W}}^{c}(\Lambda)$. Then any disk $D^{u}$ in unstable manifold of
$p$ contains a positive measure subset $I^{u}$ such that any $\xi\in I^{u}$
belongs to the basin of some physical measure and has local stable manifold
$W_{loc}^{s}(\xi)$.

###### Proof.

As in the proof of Lemma 4.1, there is a positive measure subset $I^{u}$ of
$D^{u}$ belonging to the basin of some physical measure $m$, and for $\xi\in
I^{u}$, there is $n_{0}$ such that $f^{n_{0}}(\xi)$ belongs to the Pesin
stable manifold of some point $\zeta$. Iterating backward we obtain a local
stable manifold for $\xi$. ∎

Suppose $f$ has physical measures $\\{m_{j}\\}_{j=1}^{J}$ on
${\mathcal{W}}^{c}(\Lambda)$. Let $p_{t}$, $t=1,\dots,s$ be fixed as in
Theorem 5.3. Since the support of each physical measure is a $u$-saturated
compact set, the following fact is an immediate consequence of Lemma 4.2:

######  Corollary 5.5.

For each $1\leq t\leq s$ there is at most one physical measure whose support
intersects $W^{s}(p_{t})$.

As observed before, the unstable foliation is minimal in every attractor in
the quotient. So, the orbit of every strong unstable leaf intersects
$W^{s}(\ell)=\cup_{t=1}^{s}W^{s}(p_{t})$. Since the supports of physical
measures are ${\mathcal{W}}^{u}$-saturated and invariant, it follows that for
every $1\leq j\leq J$ there exists some $1\leq t\leq s$ such that
$\operatorname{supp}m_{j}$ intersects $W^{s}(p_{t})$. So, by Corollary 5.5,
$J\leq s$.

Let $\\{p_{s_{i}}\\}_{i=1}^{s/2}$ be periodic points in $\ell$ with stable
index $d_{s}$ (i.e. repellers for $f\mid\ell$) and let
$\\{p_{s_{i}}\\}_{i=s/2+1}^{s}$ be periodic points in $\ell$ with stable index
$d_{cs}$ (i.e. attractors for $f\mid\ell$). We claim that if
${\mathcal{W}}^{u}(p_{i})$ intersects
${\mathcal{W}}^{s}(\ell)\setminus\cup_{j=1}^{s}{\mathcal{W}}^{s}(p_{j})$ for
every $i$, then the support of every physical measure contains some $p_{i}$,
$s/2+1\leq i\leq s$. Indeed, by the previous observations the support must
intersect $W^{s}(p_{i})$ for some $i$, corresponding to either an attractor or
a repeller of $f\mid\ell$. In the former case, the claim is proved; in the
latter case, our assumption on $\ell$ implies that the support intersects the
stable set of some other periodic point $p_{j}$ which is an attractor, and so
the claim follows in just the same way. So, by the previous argument, the
number of physical measures can not exceed $s/2$ in this case. The proof of
Theorem 5.3 is complete. ∎

###  5.5. Statistical stability

We also want to analyze the dependence of the physical measures on the
dynamics. For this, we assume $N=M\times S^{1}$ and restrict ourselves to the
set ${\mathcal{S}}^{k}(N)\subset{\mathcal{P}}_{1}^{k}(N)$ of skew-product
maps. Notice that every $f\in{\mathcal{S}}^{k}(N)$ is dynamically coherent,
has compact one-dimensional center leaves, and absolutely continuous center
stable foliation. As pointed out before, partially hyperbolicity is an open
property and accessibility holds on an open and dense subset of
${\mathcal{S}}^{k}(N)$.

######  Theorem 5.6.

For any $k>1$ there exists a $C^{1}$ open and $C^{k}$ dense subset
${\mathcal{B}}^{k}(N)$ of ${\mathcal{S}}^{k}(N)$ such that every
$f\in{\mathcal{B}}^{k}(N)$ has mostly contracting center direction. Moreover,
on a $C^{k}$ open and dense subset of ${\mathcal{B}}^{k}(N)$ the number of
physical measures is locally constant and the physical measures depend
continuously on the diffeomorphisms.

###### Proof.

Notice that every $f\in{\mathcal{S}}^{k}(N)$ is dynamically coherent, has
compact one-dimensional center leaves, and absolutely continuous center stable
foliation. By a variation of an argument of Niţică, Török [32], one gets that
the set of diffeomorphisms in ${\mathcal{S}}^{k}(N)$ which are accessible on
all attractors are $C^{1}$ open and $C^{r}$ dense. Let us comment a bit on
this, since our setting is not exactly the same. The heart of the proof is to
show that the accessibility class of any point contains the corresponding
center leaf. This is done by considering 4-leg $su$-paths linking the point to
every nearby point in the center leaf; in this way one gets that every
accessibility class is open in the center leaf; then, connectivity gives the
conclusion. The only difference in our case is that we deal with accessibility
over each of the attractors, not the whole ambient manifold. However, the
arguments remains unchanged, just with the additional caution to choose the
corners of the $4$-leg $su$-path to be points over the attractor. It is easy
to see that the set of diffeomorphisms in ${\mathcal{S}}^{k}(N)$ which have a
center leaf containing some hyperbolic periodic point is $C^{1}$ open and
$C^{r}$ dense. Take ${\mathcal{B}}^{k}$ be the intersection of above two sets.
Then by Theorem D, any $f\in{\mathcal{B}}^{k}$ has mostly contracting center
bundle. By Andersson [9], for any partially hyperbolic diffeomorphism $f$ with
mostly contracting center direction there is a $C^{k}$, $k>1$ neighborhood
$\mathcal{U}$ of $f$ such that any $g\in\mathcal{U}$ has mostly contracting
center direction also, and on a $C^{k}$ open and dense subset of
$\mathcal{U}$, the number of physical measures is locally constant and the
physical measures depend continuously on the diffeomorphism. This ends the
proof of Theorem 5.6. ∎

##  6\. Absolute continuity for mostly contracting center

Throughout this section $f:N\to N$ is a partially hyperbolic, dynamically
coherent, $C^{k}$, $k>1$ diffeomorphism with mostly contracting center
direction. Recall the later is a robust (open) condition, by Andersson [9]. We
develop certain criteria for proving absolute continuity of the center stable,
center unstable, and center foliations and we apply these tools to exhibit
several robust examples of absolute continuity. In particular, this yields a
proof of Theorem E.

The starting point for our criteria is the observation that for maps with
mostly contracting center the Pesin stable manifolds are contained in, and
have the same dimension as the center stable leaves. Since the Pesin stable
lamination is absolutely continuous ([35, 37]), in this way one can get a
local property of absolute continuity for the center stable foliation. This
initial step of the construction is carried out in Section 6.2. Then one would
like to propagate this behavior to the whole ambient manifold, in order to
obtain actual absolute continuity. It is important to point out that this can
not possibly work without additional conditions. Example 6.1 below illustrates
some issues one encounters. A more detailed analysis, including explicit
robust counter-examples will appear in [49]. Suitable assumptions are
introduced in Section 6.1, where we also give the precise statements of our
criteria. In Section 6.3 we present the main tool for propagating local to
global behavior. The criteria are proved in Sections 6.4 through 6.6.

Before proceeding, let us give a simple example of a map whose center
foliation is leafwise absolutely continuous and locally absolutely continuous,
but not globally absolutely continuous. This kind of construction explains why
Pesin theory alone can not give (global) absolute continuity of center
foliations, even when the center direction is mostly contracting.

######  Example 6.1.

Let us start with $f_{0}:S^{1}\times[0,1]\to S^{1}\times[0,1]$,
$f_{0}(x,t)=(2x,g(t))$ where $g:[0,1]\to[0,1]$ is a $C^{2}$ diffeomorphism
such that $g(0)=0$, $g(1)=1$, $g(t)<t$ for all $t\in(0,1)$, and
$0<g^{\prime}(t)<2$ for every $t\in[0,1]$. Then $f_{0}$ is a partially
hyperbolic endomorphism of the cylinder, with the vertical segments as center
leaves. Next, let $f:S^{1}\times[0,1]\to S^{1}\times[0,1]$ be a $C^{2}$-small
perturbation, preserving the two boundary circles
${\mathcal{C}}_{i}=S^{1}\times\\{i\\}$, $i=0,1$ and the vertical line
$\\{0\\}\times[0,1]$ through the fixed point $(0,0)$. Moreover, the horizontal
derivatives of $f$ at the endpoints of this vertical line should be different:

(20) $\frac{\partial f}{\partial x}(0,0)\neq\frac{\partial f}{\partial
x}(0,1).$

By the stability of center foliations ([28], the new map $f$ has a center
foliation whose leaves are curve segments with endpoints in the two boundary
circles. Thus, they induce a holonomy map
$h:{\mathcal{C}}_{0}\to{\mathcal{C}}_{1}$ that conjugates the two expanding
maps $f\mid{\mathcal{C}}_{0}$ and $f\mid{\mathcal{C}}_{1}$. Condition (20)
implies that the conjugacy can not be absolutely continuous (see [43]). This
shows that the center foliation is not absolutely continuous. Yet, it is
absolutely continuous restricted to $S^{1}\times[0,1)$, as we are going to
explain. Notice that our assumptions imply that
$g^{\prime}(0)<1<g^{\prime}(1)$ and so the lower boundary component
${\mathcal{C}}_{0}$ is an attractor for $f_{0}$, with $S^{1}\times[0,1)$ as
its basin of attraction. Then the same is true for the perturbation $f$.
Moreover, restricted to this basin, the center leaves coincide with the Pesin
stable manifolds of the points in the attractor, and so they do form an
absolutely continuous foliation. In particular, this also shows that the
center foliation is leafwise absolutely continuous.

###  6.1. Criteria for absolute continuity

We assume that some small cone field around the strong unstable bundle has
been fixed. We call _$u$ -disk_ any embedded disk of dimension $d_{u}$ whose
tangent space is contained in that unstable cone field at every point.
Previously, we introduced the special case of ${\mathcal{W}}^{u}$-disks, which
are contained in strong unstable leaves. To begin with, in Section 6.4 we
prove that upper leafwise absolute continuity always holds in the present
context:

######  Proposition 6.2.

The center stable foliation of $f$ is upper leafwise absolutely continuous, if
it exists.

For the next criterion we assume the diffeomorphism is non-expanding along the
center direction. This notion is defined as follows. Assume also $f$ is
dynamically coherent. Given $r>0$ and $*\in\\{s,cs,c,cu,u\\}$, we denote by
${\mathcal{W}}^{*}_{r}(x)\subset{\mathcal{W}}^{*}(x)$ the ball of radius $r$
around $x$, relative to the distance induced by the Riemannian metric of $N$
on the leaf ${\mathcal{W}}^{*}(x)$. In what follows we always suppose $r$ is
small enough so that ${\mathcal{W}}^{*}_{r}(x)$ is an embedded disk of
dimension $d_{*}$ for all $x\in M$ and every choice of $*$. We use
$\widehat{W}^{s}(p)$ and $\widehat{W}^{u}(p)$ to denote the stable and
unstable sets of a periodic point $p$. We say that $f$ is _non-expanding along
the center direction_ if there exist $\rho>0$ and $\varepsilon>0$ such that

  * •

$f^{n}({\mathcal{W}}^{cs}_{\varepsilon}(x))\subset{\mathcal{W}}^{cs}_{\rho}(f^{n}(x))$
for every $n\geq 0$ and almost any $x$ in any $u$-disk.

  * •

the support of every ergodic Gibbs $u$-state $m$ contains some periodic point
$p$ such that $\widehat{W}^{s}(p)\supset{\mathcal{W}}^{cs}_{2\rho}(p)$

######  Proposition 6.3.

If $f$ is non-expanding along the center direction then the center stable
foliation is absolutely continuous.

The proof of this proposition is given in Sections 6.2 through 6.5. We will
see that the hypothesis holds for a classical construction of partially
hyperbolic, robustly transitive diffeomorphisms due to Mañé [30] (Section
7.1). It also holds for a more recent class of examples introduced by Bonatti,
Viana [19], which are not even partially hyperbolic (though they do admit a
dominated invariant splitting of the tangent bundle), but this fact will not
be proved here.

Let $f\in{\mathcal{P}}_{1}^{k}(N)$. Let $\ell$ be a periodic center leaf
$\ell$, with period $\kappa\geq 1$. For $*\in\\{s,u\\}$, we denote
${\mathcal{W}}^{*}(\ell)=\cup_{\zeta\in\ell}{\mathcal{W}}^{*}(\zeta)$. We call
_homoclinic leaf_ associated to $\ell$ any center leaf $\ell^{\prime}$
contained in ${\mathcal{W}}^{s}(\ell)\cap{\mathcal{W}}^{u}(\ell)$. Then there
exist strong stable and strong unstable holonomy maps

(21) $h^{s}:\ell\to\ell^{\prime}\quad\text{and}\quad
h^{u}:\ell\to\ell^{\prime}$

We say that $\ell$ is _in general position_ if

  * (a)

$f^{\kappa}\mid\ell$ is Morse-Smale with a single periodic attractor $a$ and a
single periodic repeller $r$;

  * (b)

$h^{s}(a\cup r)$ is disjoint from $h^{u}(a\cup r)$, for some homoclinic leaf
associated to the center leaf $\ell$.

Notice that
${\mathcal{W}}^{s}(\ell^{\prime})\setminus{\mathcal{W}}^{s}(h^{s}(r))$ is
contained in the stable manifold $\widehat{W}^{s}(a)$ of the attractor. Thus,
condition (b) implies that ${\mathcal{W}}^{u}(a)$ and ${\mathcal{W}}^{u}(r)$
intersect $\widehat{W}^{s}(a)$ transversely. Analogously,
${\mathcal{W}}^{s}(a)$ and ${\mathcal{W}}^{s}(r)$ intersect
$\widehat{W}^{u}(r)$ transversely.

######  Proposition 6.4.

Suppose $f\in{\mathcal{P}}^{k}_{1}(N)$ has some center leaf $\ell$ in general
position and such that every strong unstable leaf intersects
${\mathcal{W}}^{s}(\ell)$. Then the center stable foliation of $f$ is
absolutely continuous.

This proposition is proved in Section 6.6. In Section 7.2 we use it to prove
Theorems B and E, and in Section 7.3 we give an application to volume
preserving systems. Noticing that, apart from dynamical coherence, all the
hypotheses of Proposition 6.4 are robust, we get the following immediate
consequence:

######  Corollary 6.5.

Suppose $f\in{\mathcal{P}}_{1}^{k}(N)$ is robustly dynamically coherent and
has some periodic center leaf $\ell$ in general position and such that every
strong unstable leaf intersects ${\mathcal{W}}^{s}(\ell)$. Then the center
stable foliation is robustly absolutely continuous.

###  6.2. Local absolute continuity

The following lemma will allow us to obtain some property of local absolute
continuity:

######  Lemma 6.6.

For any ergodic $u$-state $m$ of $f$ and any disk $D$ contained in an unstable
leaf inside $\operatorname{supp}m$, there is a positive measure subset
$\Gamma$ such that the points in $\Gamma$ have (Pesin) stable manifolds with
uniform size. Moreover, these stable manifolds form an absolutely continuous
lamination, in the following sense: there is $K>0$ such that for any two
$u$-disks $D_{1}$, $D_{2}$ sufficiently close to $D$, the stable manifolds of
points in $\Gamma$ define a holonomy map between subsets of $D_{1}$ and
$D_{2}$, and this is absolutely continuous, with Jacobian between $1/K$ and
$K$.

###### Proof.

Because $f$ has mostly contracting center direction, $m$ is a hyperbolic
ergodic measure of $f$, by Pesin theory, there is a Pesin block $\Lambda$ with
positive $m$ measure such that every point $x\in\Lambda$ has uniform size of
stable manifold, and these stable manifolds on $\Lambda$ is uniformly
absolutely continuous. Notice that the stable manifolds are contained in the
center stable leaves. Since $m$ is a $u$-state, there is a disk $D_{0}$
contained in an unstable leaf inside the support and intersecting $\Lambda$ on
a $m^{u}$-positive measure subset $D_{0}^{*}$. Then the points in $D_{0}^{*}$
have stable manifolds of size bounded below by some $\delta_{0}>0$. Denote
$B_{0}=\cup_{x\in D_{0}^{*}}W^{s}_{\delta_{0}}(x)$. Since $m$ is a $u$-state,
$m(B_{0})=a_{0}>0$. We claim that there is $n_{0}>0$ such that
$(f^{n_{0}})_{*}\operatorname{vol}_{D}(B_{0})\neq 0$.

Let us prove this claim. Let $D_{\varepsilon}^{*}$ be the
$\varepsilon$-neighborhood of $D_{0}^{*}$ inside the corresponding unstable
leaf. Denote by $B_{\varepsilon}=\cup_{x\in
D_{\varepsilon}^{*}}W^{cs}_{\delta_{0}}(x)$, it is an open set, and
$m(B_{\varepsilon})\geq a_{0}>0$. Because every Cesaro accumulation point of
the iterates of Lebesgue measure on $D$ is a Gibbs $u$-state with support
contained in $\operatorname{supp}m$, and there is a unique ergodic $u$-state
with support contained in $\operatorname{supp}m$, then $m$ is the unique
Cesaro accumulation of the iterates of Lebesgue measure on $D$. Since
$B_{\varepsilon}$ is open, one has
$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^{n-1}(f^{i})_{*}\operatorname{vol}_{D}(B_{\varepsilon})\geq
m(B_{\varepsilon})=a_{0}$, so there is arbitrarily big $n$ such that
$f_{*}^{n}(\operatorname{vol}_{D})(B_{\varepsilon})>a_{0}/2$. For $\delta>0$
sufficiently small, denote by $D_{\delta}=\\{x\in
D,d^{u}(x,\partial(D))\geq\delta\\}$, one has $m^{u}(D\setminus
D_{\delta})<a_{0}/4$. Then there is $y\in D_{\delta}$ such that $f^{n}(y)\in
B_{\varepsilon}$ and $f^{n}(D)$ contains a disk $D_{y}$ around $y$ and for any
$x\in D_{\varepsilon}^{*}$ one has $W^{cs}_{\delta_{0}}(x)\cap
D_{y}\neq\emptyset$. Then the stable manifolds of $D_{0}^{*}$ define a
holonomy map between $D_{0}^{*}$ and $B_{0}\cap D_{y}$, by the uniform
absolute continuity of these stable manifolds,
$\operatorname{vol}_{D_{y}}(D_{y}\cap B_{0})>0$, then
$f_{*}^{n_{0}}(\operatorname{vol}_{D})(B_{0})>0$. This proves the claim.

This claim implies $\operatorname{vol}_{f^{n}(D)}(f^{n}(D)\cap B^{*}_{0})>0$,
let $\Gamma=D\cap f^{-n_{0}}(B^{*}_{0})$, then
$\operatorname{vol}_{D}(\Gamma)>0$, every point in $\Gamma$ has uniform size
of stable manifold, and these stable manifolds are uniformly absolutely
continuous. ∎

Suppose $f\in{\rm Diff}^{k}(N)$, $k>1$ admits dominated splitting
$E^{cu}\oplus E^{cs}$, and it is dynamically coherent, that is, it has center
stable and center unstable foliation. We call _$cs$ -block_ for $f$ the image
${\mathcal{B}}=h(\Sigma\times I^{d_{cs}})$ of any embedding $h:\Sigma\times
I^{d_{cs}}\to N$, with $\Sigma\subset I^{d_{cu}}$, satisfying the following
properties:

  1. (1)

$h(\\{a\\}\times I^{d_{cs}})$ is contained in ${\mathcal{W}}^{cs}(h(a,0))$,
for every $a\in\Sigma$

  2. (2)

$h(\\{a\\}\times I^{d_{cs}})$ is contained in the stable set of $h(a,0)$, for
every $a\in\Sigma$

  3. (3)

$h(\Sigma\times\\{0\\})$ is a positive measure subset of some disk $D$
transverse to ${\mathcal{W}}^{cs}$;

  4. (4)

there is $K>0$ such that for any $u$-disks $D_{1},D_{2}\subset N$ which cross
$h(\Sigma\times I^{u})$, that is, $D_{i}$ intersects $h(a\times I^{cs})$ for
every $a\in\Sigma$, there is a holonomy map $h^{cs}$ induced by
${\mathcal{W}}^{cs}$ from $D_{1}\cap h(a\times I^{cs})$ to $D_{2}\cap
h(a\times I^{cs})$, the Jacobian of the holonomy map between
$\operatorname{vol}_{D_{1}}$ and $\operatorname{vol}_{D_{2}}$ is bounded by
$K$ from above and $1/K$ from below.

We also say that ${\mathcal{B}}$ is a $cs$-block over the disk $D$ in (3). If
$D$ is contained in the unstable manifold of an index $d_{cs}$ periodic point
$p$, then we say the $cs$-block is associated with p.

######  Remark 6.7.

If $D$ is in the support of some ergodic Gibbs $u$-state $m$ then
$m({\mathcal{B}})>0$: this is a consequence of the absolute continuity
property (4) and the fact that Gibbs $u$-states have positive densities along
strong unstable leaves (Proposition 3.1).

We say that the $cs$-block has size $r>0$ if the _plaque_ $h(\\{a\\}\times
I^{d_{cs}})$ contains ${\mathcal{W}}^{cs}_{r}(h(a,0))$ for every $a\in\Sigma$.
If a map $\tilde{h}:\Sigma\times I^{d_{cs}}\to N$ satisfies

$\tilde{h}\mid{\Sigma\times\\{0\\}}\equiv
h\mid{\Sigma\times\\{0\\}}\quad\text{and}\quad\tilde{h}(a\times
I^{d_{cs}})\subset h(a\times I^{d_{cs}})$

for every $a\in\Sigma$ then $\tilde{\mathcal{B}}=\tilde{h}(\Sigma\times
I^{d_{cs}})$ is called a sub-block of ${\mathcal{B}}$.

######  Lemma 6.8.

Let $m$ be an ergodic $u$-state of $f$ and $p\in\operatorname{supp}m$ be a
periodic point of stable index $d_{cs}$ whose stable manifold
$\widehat{W}^{s}(p)$ has size $r$. Then there is a $cs$-block associated with
$p$ with size $r$.

###### Proof.

By Lemma 6.6, there is a $cs$-block over any $u$-disk
$D\subset{\mathcal{W}}^{u}(p)$. Let $\kappa$ be the period of $p$. For every
large $n$, the backward image $f^{-n\kappa}({\mathcal{B}})$ is a $cs$-block of
size $r$ over the $u$-disk $f^{-n}(D)$. ∎

###  6.3. Recurrence to $cs$-blocks

The next proposition is a key ingredient in the proof of our criteria for
absolute continuity.

######  Proposition 6.9.

Let $m_{i}$, $i=1,\dots,s$ be the ergodic Gibbs $u$-states of $f$ and $B_{i}$,
$i=1,\dots,n$ be $cs$-blocks over ${\mathcal{W}}^{u}$-disks
$D_{i}\subset\operatorname{supp}m_{i}$. Then for any positive Lebesgue measure
subset $D^{*}$ of any ${\mathcal{W}}^{u}$-disk $D$, there exists $n>0$
arbitrarily large and there exists $1\leq i\leq s$ such that
$\operatorname{vol}_{D}(D^{*}\cap f^{-n}(B_{i}))>0$.

###### Proof.

(For notational simplicity, we use $m^{u}$ to denote
$\operatorname{vol}_{f^{n}(\Gamma)}$ for any $u$-disk $\Gamma$ and any $n>0$.)
Let $D^{*}_{\varepsilon}=\cup_{x\in D^{*}}B_{\varepsilon}(x,D)$ where
$B_{\varepsilon}(x,D)$ is the ball in $D$ with radius $\varepsilon$ and center
in $x$, and $m$, $m_{\varepsilon}$ be Cesaro accumulation points of the
iterates of Lebesgue measure on $D^{*}$ and $D^{*}_{\varepsilon}$
respectively, such that there is $\\{n_{j}\\}_{j=1}^{\infty}$ satisfying

$\lim_{j\to\infty}\frac{1}{n_{j}m^{u}(D^{*})}\sum_{i=0}^{n_{j}-1}(f^{i})_{*}m^{u}\mid{D^{*}}=m;$
$\lim_{j\to\infty}\frac{1}{n_{j}m^{u}(D^{*}_{\varepsilon})}\sum_{i=0}^{n_{j}-1}(f^{i})_{*}m^{u}\mid{D^{*}_{\varepsilon}}=m_{\varepsilon}.$

Then they are $u$-states, denote by $m=a_{1}m_{1}+\cdots+a_{m}m_{s}$ the
ergodic decomposition of $m$, suppose $a_{1}\neq 0$. For $\varepsilon$
sufficiently small, $m^{u}(D^{*}_{\varepsilon})\approx m^{u}(D^{*})$, and then
$m_{\varepsilon}\approx m$, denote by
$m_{\varepsilon}=a_{1,\varepsilon}m_{1}+\cdots+a_{s,\varepsilon}m_{m}$ the
ergodic decomposition of $m_{\varepsilon}$, one has $a_{1,\varepsilon}\approx
a_{1}$. Denote $D_{1}^{*}=D_{1}\cap B_{1}$ and $D_{1,\delta}^{*}=\cup_{x\in
D_{1}^{*}}B^{u}_{\delta}(x)$ and

$B_{1,\delta}^{*}=\\{z:z\in{\mathcal{W}}^{cs}_{loc}(x)\cap{\mathcal{W}}^{u}_{loc}(y)\text{
for }x\in D_{1,\delta}^{*}\text{ and }y\in B_{1}\\}.$

Then there is $K_{1}>0$ such that for any $u$-disk $\Gamma$ which crosses
$B_{1,\delta}^{*}$, one has

$\frac{m^{u}(\Gamma\cap B_{1})}{m^{u}(\Gamma\cap B_{1,\delta}^{*})}>K_{1},$

that is because $m^{u}(\Gamma\cap B_{1})>\frac{1}{K}m^{u}(D_{1}\cap B_{1})>0$
and $m^{u}(\Gamma\cap B_{1,\delta}^{*})$ is bounded above, where $K$ is the
bound for the Jacobian of the center stable foliation in $B_{1}$. We can
choose $\varepsilon$ properly such that
$m^{u}(\partial(D^{*}_{\varepsilon}))=0$, by Remark 6.7, suppose
$m_{1}(B_{1})=b_{0}>0$. Because $B_{1,\delta}^{*}$ is open,

$\lim_{j\to\infty}\frac{1}{n_{j}m^{u}(D^{*}_{\varepsilon})}\sum_{i=0}^{n_{j}-1}\big{(}f^{i}_{*}m^{u}\mid
D\big{)}\big{(}f^{i}(D^{*}_{\varepsilon})\cap B_{1,\delta}^{*}\big{)}\geq
m_{\varepsilon}(B_{1,\delta})\gtrsim
b_{0}a_{\varepsilon}>\frac{a_{1}b_{0}}{2}.$

So there is $n_{j}$ arbitrarily big such that

$\big{(}f^{n_{j}}_{*}m^{u}\mid
D\big{)}\big{(}f^{n_{j}}(D^{*}_{\varepsilon})\cap
B_{1,\delta}^{*}\big{)}\geq\frac{a_{1}b_{0}}{4}m^{u}(D_{\varepsilon}^{*}).$

We claim that there is $b_{1}>0$ such that, for every $\varepsilon>0$
sufficiently small,

$\big{(}f^{n_{j}}_{*}m^{u}\mid
D\big{)}\big{(}f^{n_{j}}({D^{*}_{\varepsilon}})\cap B_{1}\big{)}\geq
2b_{1}m^{u}(D_{\varepsilon}^{*}).$

Let us prove the claim. For $\varepsilon_{1}<\varepsilon$, denote
$D^{*}_{\varepsilon,\varepsilon_{1}}=\\{x\in
D^{*}_{\varepsilon};d_{D}(x,\partial(D^{*}_{\varepsilon}))>\varepsilon_{1}\\}$.
Then, for $\varepsilon_{1}$ sufficiently small, one has
$m^{u}({D^{*}_{\varepsilon,\varepsilon_{1}}})>m^{u}(D^{*}_{\varepsilon})-a_{1}b_{0}/8$.
It follows that

$\big{(}f^{n_{j}}_{*}m^{u}\mid
D\big{)}\big{(}B_{1,\delta}^{*}\big{)}\geq\frac{a_{1}b_{0}}{8}m^{u}(D^{*}_{\varepsilon})$

and for any $x\in f^{n_{j}}(D^{*}_{\varepsilon,\varepsilon_{1}})\cap
B_{1,\delta}^{*}$, there is a $u$-disk $D_{x}\subset
f^{n_{j}}(D^{*}_{\varepsilon})$ containing $x$ such that
$D_{x}\cap{\mathcal{W}}^{cs}_{loc}(y)\neq\emptyset$ for any $y\in
D_{1,\delta}^{*}$. Since

$\frac{m^{u}(D_{x}\cap B_{1})}{m^{u}(D_{x}\cap B_{1,\delta}^{*})}>K_{1},$

and the distortion property on ${\mathcal{W}}^{u}$, there is $K_{2}>0$ such
that

$\frac{m^{u}(f^{-n_{j}}(D_{x}\cap B_{1}))}{m^{u}(f^{-n_{j}}(D_{x}\cap
B_{1,\delta}^{*}))}>K_{2}.$

Then, taking $b_{1}={K_{2}a_{1}b_{0}}/{16}$, we get our claim:

$m^{u}(D^{*}_{\varepsilon}\cap
f^{-n_{j}}(B_{1}))>K_{2}m^{u}(D^{*}_{\varepsilon,\varepsilon_{1}}\cap
f^{-n_{j}}(B_{1,\delta}^{*}))>2b_{1}m^{u}(D^{*}_{\varepsilon}),$

Since $\lim_{\varepsilon\to 0}m^{u}(D^{*}_{\varepsilon}\setminus D^{*})=0$,
this proves that $m^{u}(D^{*}\cap f^{-n_{j}}(B_{1}))>b_{1}m^{u}(D^{*})$. This
completes the proof of the proposition. ∎

######  Remark 6.10.

Assuming there exists a unique Gibbs $u$-state, the arguments in the proof of
Proposition 6.9 yield a slightly stronger conclusion that will be useful in
the sequel: there exists $b_{1}>0$ such that for any positive Lebesgue measure
subset $D^{*}$ of any ${\mathcal{W}}^{u}$-disk $D$ and for any $1\leq i\leq s$
there exist arbitrarily large values of $n>0$ such that
$\operatorname{vol}_{D}(D^{*}\cap f^{-n}(B_{i}))\geq
b_{1}\operatorname{vol}_{D}(D^{*})$.

###  6.4. Upper leafwise absolute continuity

Here we prove Proposition 6.2. Suppose there exists some measurable set $Y$
with $\operatorname{vol}(Y)>0$ that meets almost every center stable leaf
${\mathcal{W}}^{cs}(z)$ on a zero $\operatorname{vol}^{cs}$-measure subset. Up
to replacing $Y$ by some full measure subset, we may suppose that every $x\in
f^{n}(Y)$ is a Lebesgue density point of $f^{n}(Y)$ for every $n\geq 0$:

(22) $\lim_{\rho\to 0}\frac{\operatorname{vol}(B_{\rho}(x)\cap
f^{n}(Y))}{\operatorname{vol}(B_{\rho}(x))}=1.$

Since $f$ has finitely many ergodic $u$-states and their basins cover a full
measure subset of $N$ (see [19]), it is no restriction to suppose that $Y$ is
contained in the basin of some ergodic Gibbs $u$-state $m$. Let
${\mathcal{B}}$ be a $cs$-block over some $u$-disk contained in the support of
$m$ (recall Proposition 3.1 and Section 6.2). Since the strong unstable
foliation is absolutely continuous (see [20]), we can find a $u$-disk $D$ such
that $D^{*}=D\cap Y$ has positive $\operatorname{vol}_{D}$-measure. By
Proposition 6.9, there exists $n>0$ such that
$\operatorname{vol}_{f^{n}(D)}(f^{n}(D^{*})\cap{\mathcal{B}})>0$. Take $y\in
D^{*}$ such that $f^{n}(y)\in{\mathcal{B}}$ and $f^{n}(y)$ is a Lebesgue
density point for $f^{n}(D^{*})\cap{\mathcal{B}}$ inside $f^{n}(D)$. Then, for
every small $\rho>0$,

$\frac{\operatorname{vol}_{f^{n}(D)}\big{(}B^{u}_{\rho}(f^{n}(y))\cap{\mathcal{B}}\big{)}}{\operatorname{vol}_{f^{n}(D)}\big{(}B^{u}_{\rho}(f^{n}(y)\big{)}}\geq\frac{\operatorname{vol}_{f^{n}(D)}\big{(}B^{u}_{\rho}(f^{n}(y))\cap
f^{n}(D^{*})\cap{\mathcal{B}}\big{)}}{\operatorname{vol}_{f^{n}(D)}\big{(}B^{u}_{\rho}(f^{n}(y)\big{)}}\approx
1,$

where $B^{u}_{\rho}(x)$ denotes the connected component of
$B_{\rho}(x)\cap{\mathcal{W}}^{u}(x)$ that contains $x$. Then, since the
center stable foliation is uniformly absolutely continuous on the $cs$-block,
there exists $c>0$ such that

$\frac{\operatorname{vol}\big{(}B_{\rho}(f^{n}(y))\cap{\mathcal{B}}\big{)}}{\operatorname{vol}\big{(}B_{\rho}(f^{n}(y)\big{)}}\geq
c\quad\text{for all small }\rho>0.$

Together with (22), this implies that
$\operatorname{vol}(f^{n}(Y)\cap{\mathcal{B}})>0$. On the other hand, the
hypothesis implies that $f^{n}(Y)$ intersects almost every center stable leaf
on a zero Lebesgue measure subset. Using, once more, that the center stable
leaf is absolutely continuous on the $cs$-block, we get that
$\operatorname{vol}(f^{n}(Y)\cap{\mathcal{B}})=0$. This contradicts the
previous conclusion, and that contradiction completes the proof of Proposition
6.2.

###  6.5. Non-expansion along the center

Now we prove Proposition 6.3. Let the ergodic $u$-states
$\\{m_{i}\\}_{i=1}^{m}$, periodic points $\\{p_{i}\\}_{i=1}^{m},$ and
$\rho,\varepsilon$ given in the definition of non-expansion along the center.
By Lemma 6.8, we can choose $cs$-blocks $\\{{\mathcal{B}}_{i}\\}_{i=1}^{m}$
associated with $m_{i}$ with size $\rho$.

In order to prove the center stable foliation is absolutely continuous, we
just need show that for any two $u$-disks $D_{1},D_{2}$ which are
$\varepsilon$ near, the holonomy map induced by ${\mathcal{W}}^{cs}$ between
$D_{1}$ and $D_{2}$ maps Lebesgue positive measure subset to a Lebesgue
positive measure subset, where two $u$-disks $D_{1},D_{2}$ are $\varepsilon$
near if for any $x\in D_{1}$, there is $y\in D_{2}$ belonging to
${\mathcal{W}}^{cs}_{\varepsilon}(x)$.

Suppose $D_{1}^{*}\subset D_{1}$ is a positive measure subset, denote by
$D_{2}^{*}\subset D_{2}$ the image of $D_{1}^{*}$ under cs-holonomy map. Since
$f$ is non-expanding along the center, we can assume that for any $x\in
D_{1}^{*}$, one has
$f^{n}({\mathcal{W}}^{cs}_{\varepsilon}(x))\subset{\mathcal{W}}^{cs}_{\rho}(f^{n}(x))$
for $n>0$. Choose $\tilde{B}_{i}$ a sub-block of $B_{i}$ with arbitrarily
small size, then by Proposition 6.9, there is $n$ and $j$ such that
$m^{u}(f^{n}(D_{1}^{*})\cap\tilde{B}_{j})>0$. Because $f^{n}(D_{1}^{*})$ and
$f^{n}(D_{2}^{*})$ are $\rho$ near, and the cs-holonomy map in $B_{i}$ is
absolutely continuous, one has that $m^{u}(f^{n}(D_{2}^{*})\cap B_{j})>0$,
this implies $m^{u}(D_{2}^{*})>0$, so ${\mathcal{W}}^{cs}$ is absolutely
continuous.

###  6.6. Center leaves in general position

We are going to prove Proposition 6.4. Let us start by giving an overview of
the argument. We need to compare a set on any $u$-disk with its projection to
another $u$-disk under $cs$-holonomy. The idea is to consider appropriate
iterates of both $u$-disks intersecting a given $cs$-block, and then take
advantage of the uniform structure on the $cs$-block. The problem is that,
because $cs$-blocks have gaps along the center direction, one can not
immediately ensure that iterates of both disks intersect the same $cs$-block.
To this end, we use the twisting property in the assumption of general
position to find a pair of $cs$-blocks whose union covers the whole center
direction, in the sense that it intersects any large iterate of any $u$-disk.
Then, we show that some iterate of any of the disks intersects both
$cs$-blocks, which gives the required property.

Now we fill in the details in the proof. Let $f$ and $\ell$ be as in the
statement of the proposition. For simplicity, consider the center leaf $\ell$
to be fixed (in other words, $\kappa=1$) and we also take the attractor $a$
and repeller $r$ of $f\mid\ell$ to be fixed. Extension to the general case is
straightforward.

######  Lemma 6.11.

The diffeomorphism $f$ has a unique ergodic $u$-state and its support contains
the attractor $a$.

###### Proof.

By Lemma 4.2, the supports of all ergodic Gibbs $u$-states are pairwise
disjoint. Thus, it suffices to show that the support of any ergodic $u$-state
contains $a$. By Proposition 3.1, the support of $m$ consists of entire
unstable leaves. So, it suffices to prove that every strong unstable leaf
intersects the stable manifold $\widehat{W}^{s}(a)$ of the attractor. By
hypothesis, every strong unstable leaf intersects ${\mathcal{W}}^{s}(\ell)$.
Moreover, ${\mathcal{W}}^{s}(\ell)$ is the union of $\widehat{W}^{s}(a)$ with
the strong stable leaf through the repeller $r$. If a strong unstable leaf $L$
intersects ${\mathcal{W}}^{s}(r)$ then its forward orbit accumulates on
${\mathcal{W}}^{u}(r)$ and, in particular, on $h^{u}(r)$. Since $\ell$ is in
general position, $h^{u}(r)\neq h^{s}(r)$ and so $h^{u}(r)$ belongs to the
stable manifold of $a$. Hence, in any case, $L$ does intersect
$\widehat{W}^{s}(a)$. This completes the argument. ∎

Consider the four points $a_{s}=h^{s}(a)$, $a_{u}=h^{u}(a)$, $r_{s}=h^{s}(r)$,
$r_{u}=h^{u}(r)$ in $\ell^{\prime}$. For $\rho$, $\varepsilon>0$ small, and
$\zeta\in\ell^{\prime}$, denote

${\mathcal{W}}_{\rho}^{s}(\ell^{\prime})=\cup_{\xi\in\ell^{\prime}}{\mathcal{W}}^{s}_{\rho}(\xi)\quad\text{and}\quad
V^{cs}_{\varepsilon}(\zeta)=\cup_{\xi\in
B^{c}_{\varepsilon}(\zeta)}{\mathcal{W}}^{s}_{\rho}(\xi).$

Let $\tilde{\mathcal{B}}$ be a $cs$-block over ${\mathcal{W}}^{u}_{loc}(a)$
(Lemma 6.8). Then for $n$ large, $f^{-n}(\tilde{\mathcal{B}})$ intersects
${\mathcal{W}}^{u}_{loc}(a_{s})$ in a set $\tilde{D}_{1}^{*}$ with positive
Lebesgue measure, and we may choose a $cs$-block ${\mathcal{B}}_{1}\subset
f^{-n}(\tilde{\mathcal{B}})$ over a $u$-disk
$\tilde{D}_{1}\supset\tilde{D}_{1}^{*}$ such that

${\mathcal{W}}^{u}_{2\tau}(\zeta)\cap{\mathcal{B}}_{1}\neq\emptyset\quad\text{for
all }\zeta\in{\mathcal{W}}^{s}_{\rho}(\ell^{\prime})\setminus
V^{cs}_{\varepsilon}(r_{s}).$

We think of the union $W^{u}_{2\tau}(V^{cs}_{\varepsilon}(r_{s}))$ of the
local unstable manifolds through the local center stable manifold of $r_{s}$
as the gap of ${\mathcal{B}}_{1}$ along the center direction. See Figure 3.

Figure 3.

Dually, consider a $cs$-block ${\mathcal{B}}_{2}\subset
f^{-n}(\tilde{\mathcal{B}})$ over a $u$-disk
$\tilde{D}_{2}\subset{\mathcal{W}}^{u}(a)$ such that

${\mathcal{W}}^{u}_{2\tau}(\zeta)\cap{\mathcal{B}}_{2}\neq\emptyset\quad\text{for
all }\zeta\in{\mathcal{W}}^{s}_{\rho}(\ell^{\prime})\setminus
V^{cs}_{\varepsilon}(r_{u}).$

Again, the union $W^{u}_{2\tau}(V^{cs}_{\varepsilon}(r_{u}))$ of the local
unstable manifolds through the local center stable manifold of $r_{u}$ is the
gap of ${\mathcal{B}}_{2}$ along the center direction. Moreover, we may fix
$\delta_{0}>0$ such that, for any
$\zeta\in{\mathcal{W}}^{s}_{\rho}(\ell^{\prime})$, either

$m^{u}({\mathcal{W}}^{u}_{2\tau}(\zeta)\cap{\mathcal{B}}_{1})>\delta_{0}\quad\text{or}\quad
m^{u}({\mathcal{W}}^{u}_{2\tau}(\zeta)\cap{\mathcal{B}}_{2})>\delta_{0}.$

This is, in precise terms, what we meant when we announced that the union
${\mathcal{B}}_{1}\cup{\mathcal{B}}_{2}$ of the two $cs$-blocks would cover
the whole center direction.

Now consider a new $cs$-block ${\mathcal{B}}$ defined as the union of

$({\mathcal{W}}^{u}_{2\tau}(\xi)\cap{\mathcal{B}}_{2})\cup({\mathcal{W}}^{u}_{2\tau}(\xi)\cap{\mathcal{B}}_{1})$

over all $\xi\in
W_{\rho}^{s}(\ell^{\prime})\setminus\big{(}V^{cs}_{\varepsilon}(r_{s})\cup
V^{cs}_{\varepsilon}(r_{u})\big{)}$. In other words, ${\mathcal{B}}$ is
obtained from ${\mathcal{B}}_{1}\cup{\mathcal{B}}_{2}$ by removing the two
gaps. Thus, ${\mathcal{B}}={\mathcal{B}}^{1}\cup{\mathcal{B}}^{2}$ with
${\mathcal{B}}^{1}\subset{\mathcal{B}}_{1}$ and
${\mathcal{B}}^{2}\subset{\mathcal{B}}_{2}$. We are going to show that
arbitrarily large iterates of any $u$-disk intersect both connected components
of ${\mathcal{B}}$ on positive measure subsets.

######  Lemma 6.12.

Given any $u$-disk $D$ and any positive $\operatorname{vol}_{D}$-measure
subset $D^{*}$ there exists $\zeta\in D^{*}$ and $k$ arbitrarily large such
that

$\operatorname{vol}_{f^{k}(D)}\big{(}{\mathcal{W}}^{u}_{2\tau}(f^{n}(\zeta))\cap
f^{k}(D^{*})\cap{\mathcal{B}}^{i}\big{)}>0\quad\text{for both }i=1,2.$

###### Proof.

It is no restriction to suppose every point of $D^{*}$ is a Lebesgue density
point. Fix $\varepsilon>0$ small (the precise choice will be given later).
Take any point $x\in D^{*}$ and let $r>0$ small enough so that
$\operatorname{vol}_{D}(D_{r}^{*})>(1-\varepsilon)\operatorname{vol}_{D}(D_{r})$,
where $D_{r}$ is the disk of radius $r$ around $x$ and $D_{r}^{*}=D_{r}\cap
D^{*}$. By Proposition 6.9 and Remark 4.3 there exists $b_{1}>0$, independent
of $x$ and $r$, such that

$\operatorname{vol}_{D}(D_{r}^{*}\cap f^{-n_{i}}({\mathcal{B}}^{1}))\geq
b_{1}\operatorname{vol}_{D}(D_{r}^{*})\geq
b_{1}(1-\varepsilon)\operatorname{vol}_{D}(D_{r})$

for a sequence $n_{i}\to\infty$. Let $\rho>0$ be slightly smaller than $r$, so
that

$\operatorname{vol}_{D}(D_{\rho})>(1-\varepsilon)\operatorname{vol}_{D}(D_{r}).$

Then, for any $n_{i}$ sufficiently large and any $y\in D_{\rho}$, we have
$f^{-n_{i}}(W^{u}_{loc}(f^{n_{i}}(y)))\subset D_{r}$. Since the local unstable
manifold of $f^{n_{i}}(y)$ cuts across both ${\mathcal{B}}^{1}$ and
${\mathcal{B}}^{2}$, this means that we can associate to $y\in
D^{*}_{\rho}\cap f^{-n_{i}}({\mathcal{B}}_{1})$ the following subsets of
$D_{r}$:

$D^{1}_{i}(y)=f^{-n_{i}}(W^{u}_{loc}(f^{n_{i}}(y))\cap{\mathcal{B}}^{1})\quad\text{and}\quad
D^{2}_{i}(y)=f^{-n_{i}}(W^{u}_{loc}(f^{n_{i}}(y))\cap{\mathcal{B}}^{2}).$

By bounded distortion, there exists $\kappa=\kappa(f)>0$ such that

$\operatorname{vol}_{D}(D^{2}_{i}(y))\geq\kappa\operatorname{vol}_{D}(D^{1}_{i}(y))\quad\text{for
every $y$ and every $i$.}$

We also denote by $D^{1}_{i}$ and $D^{2}_{i}$ the (disjoint) unions of
$D^{1}_{i}(y)$ and $D^{2}_{i}(y)$, respectively, over all $y\in
D^{*}_{\rho}\cap f^{-n_{i}}({\mathcal{B}}_{1})$. Then, the previous inequality
gives

$\operatorname{vol}_{D}(D^{2}_{i})\geq\kappa\operatorname{vol}_{D}(D^{1}_{i})\quad\text{for
every $i$.}$

By Proposition 6.9 and Remark 6.10, there exists a sequence $(n_{i})_{i}$ of
positive integers and there exists $b_{1}>0$ such that

$\operatorname{vol}_{D}(D_{\rho}^{*}\cap f^{-n_{i}}({\mathcal{B}}^{1}))\geq
b_{1}\operatorname{vol}_{D_{\rho}}(D_{\rho}^{*})\geq
b_{1}(1-\varepsilon)^{2}\operatorname{vol}_{D}(D_{\rho}).$

Consequently,

$\operatorname{vol}_{D}(D^{1}_{i})\geq
b_{1}(1-\varepsilon)^{2}\operatorname{vol}_{D}(D_{\rho})\geq
b_{1}(1-\varepsilon)^{3}\operatorname{vol}_{D}(D_{r}).$

This implies that $\operatorname{vol}_{D}(D^{2}_{i})\geq
b_{2}\operatorname{vol}_{D}(D_{r})$, where the constant $b_{2}>0$ is
independent of $i$ and the choice of $r$. Now, suppose the lemma is false.
Then $D^{2}_{i}(y)\cap D^{*}$ is empty, for every $y\in D_{\rho}^{*}\cap
f^{-n_{i}}({\mathcal{B}}^{1})$, that is, $D^{2}_{i}\cap D^{*}=\emptyset$. It
follows that
$\operatorname{vol}_{D}(D_{r}^{*})\leq(1-b_{2})\operatorname{vol}_{D}(D_{r})$.
This contradicts the choice of $D_{r}^{*}$ at the beginning of the proof, as
long as we fix $\varepsilon<b_{2}$. The proof of the lemma is complete. ∎

###### Proof of Proposition 6.4.

Let $h^{cs}:D^{1}\to D^{2}$ be a $cs$-holonomy between $u$-disks $D_{1}$ and
$D_{2}$. Let $D_{1}^{*}\subset D_{1}$ be a positive
$\operatorname{vol}_{D_{1}}$-measure subset and $D_{2}^{*}=h^{cs}(D_{1}^{*})$.
We want to prove that $\operatorname{vol}_{D_{2}}(D_{2}^{*})$ is also
positive. By Lemma 6.12, there exists $\zeta\in D^{*}_{1}$ and $k\geq 1$ such
that

(23)
$\operatorname{vol}_{f^{k}(D)}\big{(}{\mathcal{W}}^{u}_{2\tau}(f^{k}(\zeta))\cap
f^{k}(D^{*}_{1})\cap{\mathcal{B}}^{i}\big{)}>0\quad\text{for both }i=1,2.$

Notice that for $k$ big enough, ${\mathcal{W}}^{u}_{2\tau}(f^{k}(\zeta))$ and
${\mathcal{W}}^{u}_{2\tau}(f^{k}(h^{cs}(\zeta)))$ are contained in nearby
$cu$-disks. That is because the stable foliation is uniformly contracting.
Then
${\mathcal{W}}^{u}_{2\tau}(h^{cs}(\zeta))\cap{\mathcal{W}}^{s}_{\rho}(\ell^{\prime})\neq\emptyset$.
This implies
${\mathcal{W}}^{u}_{2\tau}(h^{cs}(\zeta))\cap\tilde{{\mathcal{B}}}_{1}\neq\emptyset$
or
${\mathcal{W}}^{u}_{2\tau}(h^{cs}(\zeta))\cap{\mathcal{B}}_{2}\neq\emptyset$.
Since $\tilde{{\mathcal{B}}}_{1},{{\mathcal{B}}}_{2}$ are $cs$-blocks, whose
$cs$-foliations are uniformly absolutely continuous, from (23) one gets that

$\operatorname{vol}_{f^{k}(D_{2})}\big{(}{\mathcal{W}}^{u}_{2\tau}(f^{k}(h^{cs}(\zeta)))\cap
f^{k}(D_{2}^{*})\cap{{\mathcal{B}}^{i}}\big{)}>0$

for either $i=1$ or $i=2$. This implies that
$\operatorname{vol}_{D_{2}}(D_{2}^{*})>0$. Thus, the center stable foliation
is absolutely continuous, as claimed. ∎

##  7\. Robust absolute continuity

Here we use the results in the previous section to give examples of open sets
of diffeomorphisms with absolutely continuous center stable/unstable
foliations.

###  7.1. Mañé’s example

Mañé [30] constructed a $C^{1}$ open set of diffeomorphisms ${\mathcal{U}}$
such that every $f\in{\mathcal{U}}$ is partially hyperbolic (but not
hyperbolic), dynamically coherent, and transitive. From Proposition 6.3 one
gets that every $C^{k}$, $k>1$ diffeomorphism $f$ in some non-empty $C^{1}$
open subset ${\mathcal{U}}^{\prime}$ has absolutely continuous center stable
foliation. To explain this, let us recall some main features in Mañé’s
construction.

One starts from a convenient linear Anosov map
$A:{\mathbb{T}}^{3}\to{\mathbb{T}}^{3}$ with eigenvalues
$0<\lambda_{1}<\lambda_{2}<1<\lambda_{3}$. Let $p$ be a fixed point of $A$ and
$\rho>0$ be small. One deforms $A$ inside the $\rho$-neighborhood of $p$, so
as to create some fixed point with stable index $1$, while keeping the
diffeomorphism unchanged outside $B_{\rho}(p)$. Mañé [30] shows that this can
be done in such a way that the diffeomorphism
$f_{0}:{\mathbb{T}}^{3}\to{\mathbb{T}}^{3}$ thus obtained is partially
hyperbolic, with splitting $E^{s}\oplus E^{c}\oplus E^{s}$ where all factors
have dimension $1$, and every diffeomorphism in some $C^{1}$ neighborhood
${\mathcal{U}}$ is dynamically coherent and transitive. The presence of
periodic points with both stable indices $1$ and $2$ ensures that $f_{0}$ is
not Anosov. Bonatti, Viana [19] observed that every $C^{k}$, $k>1$
diffeomorphism $f\in{\mathcal{U}}$ has mostly contracting center direction.
Here, as well as in the steps that follow, one may have to reduce the
neighborhood ${\mathcal{U}}$. Then Bonatti, Díaz, Ures [16] showed that the
unstable foliation of every $f\in{\mathcal{U}}$ is minimal. According to [19],
this implies that every $C^{k}$, $k>1$ diffeomorphism $f\in{\mathcal{U}}$
admits a unique physical measure, whose basin contains Lebesgue almost every
point. The non-expansion condition in Proposition 6.3 can be checked as
follows.

A crucial observation is that the center stable bundle $E^{c}\oplus E^{s}$ is
uniformly contracting outside $B_{\rho}(p)$, for all diffeomorphisms in a
neighborhood, because $f_{0}=A$ outside $B_{\rho}(p)$. Let $q$ be another
fixed or periodic point of $A$ and assume $\rho$ was chosen much smaller than
the distance from $p$ to the orbit of $q$. Then $q$ remains a periodic point
for $f_{0}$, with stable index $2$ and stable manifold of size $\geq 5\rho$.
Let $q_{f}$ denote the hyperbolic continuation of $q$ for every $f$ in a
neighborhood of $f_{0}$: $q_{f}$ is a periodic point with stable index $2$ and
stable manifold of size $\geq 4\rho$. The fact that $E^{c}\oplus E^{s}$ is
uniformly contracting outside $B_{\rho}(p)$ also implies that
$f^{n}({\mathcal{W}}^{cs}_{\rho}(x))\subset{\mathcal{W}}^{cs}_{2\rho}(f^{n}(x))$
for all $x\in{\mathbb{T}}^{3}$ and $n\geq 0$. This proves that $f$ is non-
expanding along the center direction, and so we may apply Proposition 6.3 to
conclude that the center stable foliation of every $f$ near $f_{0}$ is
absolutely continuous.

We ignore whether the center unstable foliation and the center foliation are
absolutely continuous or not in this case. However, in the next section, a
different construction allows us to give examples where all three invariant
foliations are robustly absolutely continuous.

###  7.2. Robust absolute continuity for all invariant foliations

Here we prove Theorem E and use it to deduce Theorem B. We begin with an
intermediate result:

######  Proposition 7.1.

Let $f_{0}:N\to N$ be a $C^{k}$, $k>1$ skew-product
$f_{0}(x,\theta)=(g_{0}(x),h_{0}(x,\theta))$, where $g_{0}$ is a transitive
Anosov diffeomorphism. Assume that $f_{0}$ is accessible and has some periodic
center leaf in general position. Then there exists a $C^{k}$ neighborhood
${\mathcal{V}}$ of $f_{0}$ such that for every $f\in{\mathcal{V}}$, the center
stable, center unstable, and center foliation are absolutely continuous.

###### Proof.

Every skew-product has absolutely continuous center stable and center unstable
foliation and is robustly dynamically coherent (by [28]; the center foliation
of a partially hyperbolic skew-product is always plaque expansive). In
particular, $f_{0}$ satisfies all the hypotheses of Theorem D. The presence of
a Morse-Smale center leaf prevents $f_{0}$ from being conjugate to a rotation
extension. Thus, the center direction is mostly contracting in a whole
neighborhood of $f_{0}$. The assumption that $g_{0}$ is transitive also
ensures that every strong unstable leaf intersects ${\mathcal{W}}^{s}(\ell)$.
So, we are in a position to apply Corollary 6.5 to conclude that the center
stable foliation is robustly absolutely continuous. The same reasoning applied
to the inverse of $f_{0}$ gives that the center unstable foliation is also
robustly absolutely continuous. From the following general fact we get that
the center foliation is also robustly absolutely continuous:

######  Lemma 7.2 (Pugh, Viana, Wilkinson [39]).

Let ${\mathcal{F}}^{1}$, ${\mathcal{F}}^{2}$, ${\mathcal{F}}^{3}$ be foliation
in some smooth manifold $N$ such that ${\mathcal{F}}^{1}$ and
${\mathcal{F}}^{2}$ are transverse at every point and the leaves of
${\mathcal{F}}^{3}$ are coincide with the intersections of leaves of
${\mathcal{F}}^{1}$ and ${\mathcal{F}}^{2}$: for every point $x\in N$,
${\mathcal{F}}^{3}(x)={\mathcal{F}}^{1}(x)\cap{\mathcal{F}}^{2}(x)$. If
${\mathcal{F}}_{1}$ and ${\mathcal{F}}_{2}$ are absolutely continuous then so
is ${\mathcal{F}}_{3}$.

###### Proof.

Suppose $D_{1},D_{2}$ are two disks transverse with ${\mathcal{F}}^{3}$, and
$h^{3}:D_{1}\rightarrow D_{2}$ is the holonomy map induced by
${\mathcal{F}}^{3}$. Then ${\mathcal{F}}^{1}$ and ${\mathcal{F}}^{2}$ induce
two foliations $\hat{{\mathcal{F}}}^{1}_{i}$ and $\hat{{\mathcal{F}}}^{2}_{i}$
on $D_{i}$, $i=1,2$, and these two foliations absolutely continuous in
$D_{i}$. Fix $l_{1}\subset D_{1}$ a leaf of $\hat{{\mathcal{F}}}^{2}_{1}$, and
denote by $l_{2}=h^{3}(l_{1})$, then $l_{2}$ is a leaf of
$\hat{{\mathcal{F}}}^{2}_{2}$. Since the foliations
$\hat{{\mathcal{F}}}^{1}_{i}$, $i=1,2$ are absolutely continuous, one has that
the disintegration of the Lebesgue measure $\operatorname{vol}_{D_{1}}$ along
the foliation $\hat{{\mathcal{F}}}^{1}_{1}$ is

$\operatorname{vol}_{D_{1}}=\varphi_{x}(y)d\operatorname{vol}_{\hat{{\mathcal{F}}}^{1}_{1}(x)}(y)d\operatorname{vol}_{l_{1}}(x),\quad\text{where
}\varphi_{x}(y)>0$

and the disintegration of the Lebesgue measure of $D_{2}$ along the foliation
$\hat{{\mathcal{F}}}^{1}_{2}$ is

$\operatorname{vol}_{D_{2}}=\phi_{x}(y)d\operatorname{vol}_{\hat{{\mathcal{F}}}^{1}_{2}(x)}(y)d\operatorname{vol}_{l_{2}}(x),\quad\text{where
}\phi_{x}(y)>0.$

Now for any set $\Delta_{1}\subset D_{1}$ with
$\operatorname{vol}_{D_{1}}(\Delta_{1})>0$, denote its image for $h^{3}$ by
$\Delta_{2}$. By the above formulas for the disintegration, there is a
positive $\operatorname{vol}_{l_{1}}$ measure subset $\Gamma_{1}\subset l_{1}$
such that for any $x\in\Gamma_{1}$, one has

$\operatorname{vol}_{\hat{F}^{1}_{1}(x)}(\Delta_{1}\cap\hat{F}^{1}_{1}(x))>0.$

Denote $\Gamma_{2}=h^{3}(\Gamma_{1})\subset l_{2}$. By the absolute continuity
of ${\mathcal{F}}^{1}$ and ${\mathcal{F}}^{2}$,
$\operatorname{vol}_{l_{2}}(\Delta_{2})>0$ and
$\operatorname{vol}_{\hat{F}^{1}_{2}(x)}(\Delta_{2}\cap\hat{F}^{1}_{2}(x))>0$
for any $x\in\Gamma_{2}$. This implies
$\operatorname{vol}_{D_{2}}(\Delta_{2})>0$, and so the proof is complete. ∎

This completes the proof of Proposition 7.1. ∎

To complete the proof of Theorem E it suffices to note that any skew-product
$f_{0}$ with a Morse-Smale center leaf, as in the statement of the theorem, is
approximated by skew-products with center leaves in general position: all that
is missing is property (b) in the definition of general position, and this can
be achieved by a $C^{k}$ small perturbation inside the space of skew-products.
Then Theorem E follows from Proposition 7.1.

Now Theorem B is deduced as follows. For any skew-product $f_{0}$ as in the
statement, Let ${\mathcal{U}}$ be an open set that accumulates on $f_{0}$ as
given by Theorem E: for any $f\in{\mathcal{U}}$ the center stable, center
unstable, and center foliations are absolutely continuous. Then let
${\mathcal{V}}\subset{\mathcal{U}}$ be an open subset such that every
$f\in{\mathcal{V}}$ is accessible ([32]). Then every $f\in{\mathcal{V}}$ has
finitely many physical measures, with basins containing almost every point.
The Morse-Smale behavior on the center leaf $\ell$ prevents $f$ from being
conjugate to a rotation extension. Thus, we are in case (b) of Theorem D. From
the fact that $\ell$ contains a unique periodic attractor we also get that the
physical measure is unique (see Theorem 5.6). The same argument applies for
$f^{-1}$. This finishes the proof of Theorem B.

###  7.3. Volume preserving systems

Here we prove Theorem C and a pair of related results. Based on these, we also
describe a, partially conjectural, scenario for absolute continuity of
foliations of conservative and dissipative systems.

Part (1) of Theorem C is a direct consequence of the main result of Baraviera,
Bonatti [15]. Part (2) is given by the following result:

######  Lemma 7.3.

For any $f\in{\mathcal{W}}_{0}$ with $\lambda^{c}(f)>0$, the center foliation
and the center stable foliation are not upper leafwise absolutely continuous.

###### Proof.

Fix $c\in(0,\lambda^{c}(f))$. Then, by the Birkhoff ergodic theorem, the set

$\Gamma_{c,1}=\\{x\in
N:\lim\frac{1}{n}\sum_{i=1}^{n}\log\|Df^{-1}\mid{E^{c}(f^{i}(x))}\|^{-1}\geq
c\\}$

has positive volume. Then, by Proposition 3.6, there is $n_{0}\geq 1$ such
that the intersection of any center leaf with $\Gamma_{c,1}$ has at most
$n_{0}$ points. In particular, the intersection has zero volume inside the
center leaf. So, the center foliation of $f$ is not upper leafwise absolutely
continuous. Next, observe that the set $\Gamma_{c,1}$ consists of entire
strong stable leaves. So, the intersection of $\Gamma_{c,1}$ with any center
stable leaf consists of no more than $n_{0}$ strong stable leaves. This
implies that the intersection has zero volume inside the center stable leaf.
Consequently, the center stable foliation is not upper leafwise absolutely
continuous. In particular, we get that the center foliation and the center
stable foliation are not absolutely continuous, as claimed. ∎

Now we prove part (3) of the theorem. Let $p\in M$ be a periodic point of
$g_{0}$ and $a\in M$ be a homoclinic point associated to $p$. For simplicity,
we take the periodic point to be fixed. Let us begin by constructing
${\mathcal{W}}_{1}$. The first step is to approximate $f_{0}$ by some
diffeomorphism $f_{1}$ such that $\lambda^{c}(g)>0$ for any $g$ in a $C^{1}$
neighborhood. This can be done by the perturbation method in [15]; the
perturbation may be chosen such that $f_{1}=f_{0}$ on a neighborhood of
$\\{p\\}\times S^{1}$, and we assume that this is the case in what follows.
The second step is to find $f_{2}$ arbitrarily close to $f_{1}$ such that,
denoting by $\ell_{p}$ and $\ell_{a}$ the center leaves associated to the
continuation of $p$ and $a$,

  * •

every strong unstable leaf of $f_{2}$ intersects $W^{s}(\ell_{p})$;

  * •

the restriction of $f_{2}$ to $\ell_{p}$ is a Morse-Smale diffeomorphism, with
a single attractor $\xi$ and a single repeller $\eta$

  * •

and ${\mathcal{W}}^{u}(\eta)$ and $W^{s}(\xi)$ are in general position (we
call this non-strong connection).

These properties remain valid in a small neighborhood of $f_{2}$. As a final
step, we use [16, 27, 26] to find a diffeomorphism $f_{3}$ arbitrarily close
to $f_{2}$ and such that the strong stable and the strong unstable foliations
are minimal in a whole $C^{1}$ neighborhood of $f_{3}$. We take
${\mathcal{W}}_{1}$ to be such a neighborhood. By [19], for every
diffeomorphism $f\in{\mathcal{W}}$ the inverse $f^{-1}$ has mostly center
direction. Then, by [9], the same is true in a whole $C^{k}$ neighborhood
${\mathcal{W}}_{f}$ in the space of all (possibly dissipative).
diffeomorphisms. Hence, we are in a position to apply Corollary 6.5 to
conclude that the center unstable foliation is absolutely continuous for every
diffeomorphism in ${\mathcal{W}}_{f}$.

This finishes the proof of Theorem C. The next proposition is a variation of
results in [14] where center foliations are replaced by center stable or
center unstable foliations.

######  Proposition 7.4.

Let $f_{0}$ be as in Theorem A, where $M$ is a surface, and let $f$ be any
$C^{1}$ nearby accessible, volume preserving diffeomorphism with
$\lambda^{c}(f)=0$. If either the center stable foliation or the center
unstable foliation is absolutely then $f$ is smoothly conjugate to a rotation
extension and the center foliation is a smooth foliation.

###### Proof.

Suppose ${\mathcal{W}}^{cs}$ is absolutely continuous. Then we may apply
Theorem D. In this case Lebesgue measure is a Gibbs $u$-state with zero center
exponent, and so we are in the elliptic case (a) of the theorem. In
particular, the center foliation is leafwise absolutely continuous. Then we
can apply [14] to conclude that the center foliation is smooth and $f$ is
smoothly conjugate to a rigid model. In present case, where the center fiber
bundle is trivial, we get that $f$ is topologically conjugate to a rotation
extension (cf. Remark 4.3). ∎

######  Remark 7.5.