# On invariant holonomies between centers

Radu Saghin  Instituto de Matemática, Pontificia Universidad Católica de
Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso-Chile.
<EMAIL_ADDRESS>
###### Abstract.

We prove that for $C^{1+\theta}$, $\theta$-bunched, dynamically coherent
partially hyperbolic diffeomorphisms, the stable and unstable holonomies
between center leaves are $C^{1}$ and the derivative depends continuously on
the points and on the map. Also for $C^{1+\theta}$, $\theta$-bunched partially
hyperbolic diffeomorphism, the derivative cocycle restricted to the center
bundle has invariant continuous holonomies which depend continuously on the
map. This generalizes previous results by Pugh-Shub-Wilkinson, Burns-
Wilkinson, Brown, Obata, Avila-Santamaria-Viana, Marin.

partially supported by ANID

###### Contents

  1. 1 Introduccion
    1. 1.1 Ideas of the proofs
    2. 1.2 Several applications
    3. 1.3 Organization of the paper
  2. 2 Preparations
    1. 2.1 Regularity of holonomy along a subfoliation: some general comments.
    2. 2.2 Smooth approximations of invariant submanifolds.
      1. 2.2.1 Smooth uniform approximation of center manifolds
      2. 2.2.2 Smooth uniform approximation of center-unstable manifolds and of unstable foliation
  3. 3 Proofs
    1. 3.1 Approximation of the unstable holonomies: construction of $h^{n}_{p,q}$
    2. 3.2 $h^{n}_{p,q}$ converges uniformly to $h^{u}_{p,q}$.
    3. 3.3 $h^{u}_{p,q}$ is Lipschitz.
    4. 3.4 Estimate on the Lipschitz jet of $h^{u}_{p,q}$.
    5. 3.5 $h^{u}_{p,q}$ is differentiable.
    6. 3.6 $Dh^{u}_{p,q}$ is continuous in $p,q,f$.
    7. 3.7 Proof of Corollary 1.1
    8. 3.8 Proof of Theorem 1.2

###### Contents

  1. 1 Introduccion
  2. 2 Preparations
  3. 3 Proofs

##  1\. Introduccion

Let $M$ be a compact smooth Riemannian manifold.

######  Definition 1.1.

A diffeomorphism $f:M\to M$ of the compact Riemannian manifold $M$ is called
_partially hyperbolic_ if the tangent bundle admits a continuous
$Df$-invariant splitting $TM=E^{s}\oplus E^{c}\oplus E^{u}$ such that there
exist continuous functions
$0<\lambda_{s}(x)<\lambda_{c}^{-}(x)\leq\lambda_{c}^{+}(x)<\lambda_{u}(x)$,
with $\lambda_{s}(x)<1<\lambda_{u}(x)$, satisfying the following conditions:

  * (1)

$\|Df(x)v^{s}\|\leq\lambda_{s}(x)$,

  * (2)

$\lambda_{c}^{-}(x)\leq\|Df(x)v^{c}\|\leq\lambda_{c}^{+}(x)$,

  * (3)

$\|Df(x)v^{u}\|\geq\lambda_{u}(x)$,

for every $x\in M$ and unit vectors $v^{*}\in E^{*}(x)$($*=s,c,u$).

$E^{s}$ and $E^{u}$ are uniquely integrable, generating the stable and
unstable foliations ${\mathcal{W}}^{s}$ and ${\mathcal{W}}^{u}$. A partially
hyperbolic diffeomorphism is called _dynamically coherent_ if there exist
invariant foliations ${\mathcal{W}}^{cs}$ and ${\mathcal{W}}^{cu}$ tangent to
$E^{cs}=E^{c}\oplus E^{s}$ and $E^{cu}=E^{c}\oplus E^{u}$. In this case
${\mathcal{W}}^{cs}$ is subfoliated by the stable and central foliations
${\mathcal{W}}^{s}$ and ${\mathcal{W}}^{c}$, while ${\mathcal{W}}^{cu}$ is
subfoliated by the unstable and center foliations ${\mathcal{W}}^{u}$ and
${\mathcal{W}}^{c}$.

######  Definition 1.2.

A partially hyperbolic diffeomorphism is $\theta$-unstable bunched, $\theta>0$
if

(1) $\lambda_{u}^{\theta}>\frac{\lambda_{c}^{+}}{\lambda_{c}^{-}}.$

Similarly we define $\theta$-stable bunching if
$\lambda_{s}^{\theta}<\frac{\lambda_{c}^{-}}{\lambda_{c}^{+}}$, and
$\theta$-bunched means both stable and unstable bunched.

Given $f:M\rightarrow M$ partially hyperbolic and dynamically coherent, $p\in
M$, $q\in{\mathcal{W}}^{u}(x,f)$, we can define the unstable holonomy
$h_{p,q,f}^{u}:{\mathcal{W}}^{c}_{loc}(p)\rightarrow{\mathcal{W}}^{c}(q)$
between the center leaves. We are addressing the question of differentiability
of the holonomy along the center leaves, and the continuity of the derivative
with respect to the points and the map.

######  Theorem 1.1.

Suppose that $f$ is a $C^{1+\theta}$ partially hyperbolic diffeomorphism which
is dynamically coherent and $\theta$-unstable bunched, $\theta\in(0,1]$. Then
$h^{u}_{p,q,f}$ is $C^{1}$ and its derivative depends continuously on $f,p,q$
with $q\in{\mathcal{W}}^{u}(p)$. A similar statement holds for the stable
holonomy under the $\theta$-stable bunching condition.

######  Remark 1.1.

The continuity means that if $f_{n}$ is inside a $C^{1+\theta}$ neighborhood
of $f$ and converges to $f$ in the $C^{1}$ topology, $x_{n}$ converges to $x$,
$y_{n}\in{\mathcal{W}}^{u}_{loc}(x_{n})$ and $y_{n}$ converges to $y$, then
$Dh^{u}_{x_{n},y_{n},f_{n}}$ converges to $Dh^{u}_{x,y,f}$.

Even if $f$ is not dynamical coherent, one can always construct fake
foliations which are locally invariant under $f$ and are almost tangent to the
invariant bundles (see [4] for example). The fake foliations are a fundamental
tool for the study of ergodic properties of partially hyperbolic
diffeomorphisms.

######  Corollary 1.1.

Suppose that $f$ is a $C^{1+\theta}$ partially hyperbolic diffeomorphism which
is $\theta$-unstable bunched, $\theta\in(0,1]$. Then the fake unstable
holonomy between fake center leaves is uniformly $C^{1}$ (Lipschitz). A
similar statement holds for the stable holonomy under the $\theta$-stable
bunching condition.

Independently if $f$ is dynamically coherent or not, one can have invariant
holonomies of the continuous cocycle defined by $Df|_{E^{c}}$.

######  Definition 1.3.

Let $\mathcal{E}$ be a continuous vector bundle over $M$ and
$F:\mathcal{E}\rightarrow\mathcal{E}$ a continuous linear cocycle over the
partially hyperbolic diffeomorphism $f:M\rightarrow M$. An invariant unstable
holonomy for $F$ is a family of linear maps
$\\{H^{u}_{x,y}:\mathcal{E}(x)\rightarrow\mathcal{E}(y):\ \ x\in
M,y\in{\mathcal{W}}^{u}(x)\\}$ satisfying the following conditions:

  1. (1)

$H^{u}_{x,x}=Id$, $H^{u}_{y,z}\circ H^{u}_{x,y}=H^{u}_{x,z}$;

  2. (2)

$F\circ H^{u}_{x,y}=H^{u}_{f(x),f(y)}\circ F$;

  3. (3)

$H^{u}_{x,y}$ is continuous in $x,y$ under the condition
$y\in{\mathcal{W}}^{u}_{loc}(x)$;

The invariant stable holonomy is defined in a similar manner.

One can also consider the projectivized bundle $\mathbb{P}\mathcal{E}$ over
$M$, with fibers $\mathbb{P}\mathcal{E}(x)$ (the projective space of
$\mathcal{E}(x)$), which is also a continuous bundle (with smooth fibers) over
$M$. The projectivization of the cocycle $F$, $\mathbb{P}F$, is a continuous
cocycle in $\mathbb{P}\mathcal{E}$. If $H$ is an invariant unstable holonomy
for the cocycle $F$, then its projectivization $\mathbb{P}H$ is an invariant
unstable holonomy for the cocycle $\mathbb{P}F$ (see for example [1] for more
details on cocycles with holonomy and applications to the study of central
Lyapunov exponents).

If $f$ is partially hyperbolic, then the center bundle forms a continuous (in
fact Hölder if $f$ is $C^{1+\theta}$) vector bundle $\mathcal{E}^{c}(f)$ over
$M$ and $Df|_{E^{c}(f)}$ is a continuous (Hölder) linear cocycle over $f$. A
by-product of the proof of Theorem 1.1 is the following result.

######  Theorem 1.2.

Suppose that $f$ is a $C^{1+\theta}$ partially hyperbolic diffeomorphism which
is $\theta$-unstable bunched, $\theta\in(0,1]$. Then $Df|_{E^{c}}$ and
$\mathbb{P}Df|_{\mathbb{P}E^{c}}$ have invariant unstable holonomies. The
holonomies are also continuous with respect to the map in the $C^{1}$ topology
restricted to a $C^{1+\theta}$ neighborhood of $f$. A similar statement holds
for the stable holonomy under the $\theta$-stable bunching condition. If $f$
is dynamically coherent then the invariant holonomy coincides with the
derivative of the holonomy between center leaves.

######  Remark 1.2.

Theorem 1.1 and Theorem 1.2 work in particular for $C^{2}$ maps and the
regular (1-) bunching condition.

Let us make some historical remarks about these results. The differentiability
of the holonomies along center leaves was established in [15] for $C^{2}$
partially hyperbolic diffeomorphisms which are 1-bunched, however the
continuity of the derivative with respect to the points or the maps was not
considered. The continuity of the derivative with respect to the points was
proven in [13] under the additional assumptions of $\alpha$-bunching and
$\alpha$-pinching for some $\alpha>0$. The case of $C^{1+\theta}$ partially
hyperbolic diffeomorphisms was addressed in several papers like [3], [2]. The
differentiability of the holonomy and the continuity of the derivative with
respect to the point was obtained under the assumption of $\theta$-bunching
together with more restrictive assumptions of pinching. The continuity of the
derivative of the holonomy with respect to the map was not addressed to our
knowledge.

Regarding the invariant holonomies, there are also various works establishing
the existence and the continuity with respect to the map (the continuity with
respect to the points is included in the definition) under the assumptions of
$C^{2}$ smoothness, $\theta$-bunching and $\theta$-pinching (see for example
[1],[12],[9],[10]). It seems to follow from the construction that in the
dynamical coherent case the invariant holonomy of the center bundle cocycle
coincides with the derivative of the regular holonomy between the centers of
the original partially hyperbolic diffeomorphism.

Our contribution is to get rid of the unnecessary and restrictive pinching
conditions, and to establish the full continuity (including with respect to
the map) of the derivative of the holonomy and of the invariant holonomy,
assuming only $\theta$-bunching and $C^{1+\theta}$ regularity of the map. We
also give a unifyed presentation of both the differentiability of the holonomy
between centers and the existence of invariant holonomies for the center
derivative cocycle.

###  1.1. Ideas of the proofs

The main difficulty in the proof is the lack of sufficient regularity of the
invariant bundles. The center bundle is Hölder continuous, but the Hölder
exponent is smaller than $\theta$ in general, and this makes difficult to use
the control which comes from the $\theta$-bunching and the $C^{\theta}$
regularity of the derivative. A first idea which we use is to consider the
invariant holonomy together with a correction of the potential error coming
from the variation of the center bundle with respect to the points (the
projection from one bundle to the other, roughly along the unstable leaf is
good enough). We can expect that the difference has better regularity along
the unstable leaves. This observation together with a (more or less) standard
application of the invariant section theorem [6] gives us the existence and
continuity of the invariant holonomies (Theorem 1.2 without the identification
with the derivative of the regular holonomy in the dynamically coherent case).

The differentiability of the regular holonomy requires more work. Previous
works usually start with a good approximations of ${\mathcal{W}}^{u}$ inside
${\mathcal{W}}^{cu}$-leaves, and iterate it forward. Unfortunately again the
leaves of ${\mathcal{W}}^{cu}$ and ${\mathcal{W}}^{c}$ are only $C^{1+\alpha}$
for some $\alpha<\theta$, and this fact limits the regularity of the
approximation to $C^{1+\alpha}$, and consequently we loose the control when we
iterate forward. The second idea of this paper is to start with a smooth
approximation of both ${\mathcal{W}}^{u}$ and ${\mathcal{W}}^{cu}$-leaves and
iterate it forward. It is important that these approximations are uniformly
smooth, which makes the construction a bit more technical. When we iterate
forward the approximation of ${\mathcal{W}}^{cu}$-leaves and its subfoliation,
the bunching condition helps us keep uniform $C^{1+\theta}$ control of the
holonomy along the subfoliation. This argument will give us that the holonomy
is Lipschitz, with uniform bounds on the Lipschitz constants.

In order to upgrade to differentiability we use the ideas from [6] on
Lipschitz jets. The continuity of the derivative and the identification with
the invariant holonomy is obtained again using the invariant section theorem.

###  1.2. Several applications

We list a couple of applications of the above results.

  1. (1)

The ergodicity of $C^{1+\theta}$ accessible $\theta$-center bunched partially
hyperbolic diffeomorphisms can be obtained under weaker assumptions, without
the condition that the invariant bundles are $C^{\theta}$ ([4],[16]).

  2. (2)

The existence of invariant holonomies for the derivative cocycle on the center
bundle for partially hyperbolic diffeomorphisms can be also obtained with
weaker assumptions, without the $\theta$-pinching condition (and in
$C^{1+\theta}$ regularity). This applies for example to various results
concerning the continuity and the non-vanishing of central exponents of
partially hyperbolic diffeomorphisms with two dimensional center ([1], [12],
[9], [10], [7])

  3. (3)

We establish the continuity of the derivative of the holonomies with respect
to the points and the map, under more general conditions. This is a useful
tool which can be applied in order to obtain perturbation results related to
the uniqueness of u-Gibbs or MMEs for some classes of partially hyperbolic
diffeomorphisms (for example along the lines of [14], [5], [11]) or related to
the accessibility of partially hyperbolic diffeomorphisms ([8]).

###  1.3. Organization of the paper

In Section 2 we present some tools which we will use in the proof. In
particular we discuss the regularity of the holonomy along a subfoliation of a
submanifold, and how to approximate immersed submanifolds with smooth ones. In
Section 3 we present the proofs.

##  2\. Preparations

###  2.1. Regularity of holonomy along a subfoliation: some general comments.

We will start with a discussion about the regularity of the (derivative of)
holonomy along a subfoliation of a submanifold in $\mathbb{R}^{d}$.

Assume that we have a $C^{1}$ embedded submanifold ${\mathcal{W}}$ inside
$\mathbb{R}^{d}$. Assume that $\mathcal{F}$ is a $C^{1}$ subfoliation of
${\mathcal{W}}$. Given two points $x,y$ on the same leaf of $\mathcal{F}$, and
two transversals $T_{x}$, $T_{y}$ to ${\mathcal{F}}$ inside ${\mathcal{W}}$
passing through $x$ and $y$, let
$h^{\mathcal{F}}_{T_{x},T_{y}}:T_{x}\rightarrow T_{y}$ be the holonomy given
by $\mathcal{F}$.

Let $D_{x}=T_{x}T_{x}$ and $D_{y}=T_{y}T_{y}$ the tangent planes to $T_{x}$,
$T_{y}$ in $x$ and $y$. Let $Dh^{{\mathcal{F}}}_{T_{x},T_{y}}:D_{x}\rightarrow
D_{y}$ be the derivative of the holonomy $h^{\mathcal{F}}_{T_{x},T_{y}}$.
Clearly it depends only on $D_{x}$ and $D_{y}$ and not on the transversals
$T_{x}$ and $T_{y}$, this is why we will also use the notation
$Dh^{{\mathcal{F}}}_{D_{x},D_{y}}$. Given a decomposition $A\oplus
B=\mathbb{R}^{d}$, we denote by $p_{A}^{B}:\mathbb{R}^{d}\rightarrow A$ the
projection to $A$ parallel to $B$. If we want to specify that we consider the
restriction of $p_{A}^{B}$ to a subspace $A^{\prime}$ we will denote it
$p_{A^{\prime},A}^{B}$.

Let $d_{\mathcal{F}}$ be the distance induced on the leaves of $\mathcal{F}$.

######  Definition 2.1.

Let $x\in{\mathcal{W}}$, $\Delta$ be a continuous cone field inside
$T{\mathcal{W}}$ uniformly transverse to ${\mathcal{F}}$, $E_{x}$ transverse
to $\Delta_{x}$ and $\delta>0$. We say that $Dh^{\mathcal{F}}$ is
$(C_{\mathcal{F}},\theta)$-Hölder along $\mathcal{F}$ at $x$ with respect to
$\Delta,E_{x}$ and at scale $\delta$ if

(2)
$\left\|Dh^{\mathcal{F}}_{D_{x},D_{y}}(x)-p_{D_{x},D_{y}}^{E_{x}}\right\|\leq
C_{\mathcal{F}}d_{\mathcal{F}}(x,y)^{\theta},\ \ \forall
y\in\mathcal{F}_{\delta}(x),\ \forall D_{x}\in\Delta_{x},D_{y}\in\Delta_{y}.$

If instead of $\mathbb{R}^{d}$ we are in a smooth Riemannian manifold, the
definition is similar, with the requirement that the condition 2 holds in an
exponential chart at $x$ of size $\delta$.

Let us remark that given a $C^{2}$ submanifold ${\mathcal{W}}$ with a $C^{2}$
subfoliation ${\mathcal{F}}$, the continuous cone field $\Delta$, and a
subspace $E_{x}$ containing $T_{x}{\mathcal{F}}(x)$, there exist
$C_{{\mathcal{F}}},\delta>0$ such that $Dh^{\mathcal{F}}$ is
$(C_{\mathcal{F}},\theta)$-Hölder along $\mathcal{F}$ at $x$ with respect to
$\Delta,E_{x}$ and at scale $\delta$ (we can actually take $\theta=1$). The
following lemma explains this fact in more detail.

We need a bound on the transversality between $E_{x}$ and $\Delta$ at the
scale $\delta$:

$t(E_{x},\Delta,\delta)=\sup\left\\{\frac{1}{\sin(\angle(E_{x},D_{y}))}:\
y\in{\mathcal{W}}_{\delta}(x),D_{y}\in\Delta_{y}\right\\}.$

In particular we have

$\|p_{D_{y}}^{E_{x}}\|\leq t(E_{x},\Delta,\delta)\hbox{ for all $y$ such that
}d(x,y)<\delta.$

We aslo consider a bound on the transversality between $\Delta$ and
${\mathcal{F}}$:

$t({\mathcal{F}},\Delta)=\sup\left\\{\frac{1}{\sin(\angle(T_{y}{\mathcal{F}}(y),D_{y}))}:\
D_{y}\in\Delta_{y}\right\\}.$

We say that $\phi:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ is a linear
parametrization of (${\mathcal{W}},{\mathcal{F}}$) if
$\phi(\mathbb{R}^{\dim{\mathcal{W}}}\times\\{0\\}^{d-\dim{\mathcal{W}}})={\mathcal{W}}$
and
$\phi(\mathbb{R}^{\dim{\mathcal{F}}}\times\\{b\\}\times\\{0\\}^{d-\dim{\mathcal{W}}})={\mathcal{F}}(\phi(0,b,0)),\forall
b\in\mathbb{R}^{\dim{\mathcal{W}}-\dim{\mathcal{F}}}$ ($\phi$ basically
straightens both ${\mathcal{W}}$ and ${\mathcal{F}}$). If $\phi$ is defined
only between balls of radius $\delta$ at the origin and $x$ we say that it is
a $\delta$-linear parametrization of (${\mathcal{W}},{\mathcal{F}}$) at $x$.

######  Lemma 2.1.

Let ${\mathcal{W}}$ be a $C^{2}$ submanifold in $\mathbb{R}^{d}$ and
${\mathcal{F}}$ a $C^{2}$ subfoliation of ${\mathcal{W}}$. Let $\Delta$ be a
continuous cone field in $T{\mathcal{W}}$ transferse to ${\mathcal{F}}$,
$x\in{\mathcal{W}}$ and $E_{x}$ a subspace containing $T_{x}{\mathcal{F}}(x)$
and transverse to $\Delta_{y}$ for all $y\in{\mathcal{F}}_{\delta}(x)$ for
some $\delta>0$. Let $\phi$ be a $C^{2}$ $\delta$-linear parametrization of
(${\mathcal{W}},{\mathcal{F}}$) at $x$. Then $Dh^{\mathcal{F}}$ is
$(C_{\mathcal{F}},\theta)$-Hölder along $\mathcal{F}$ at $x$ with respect to
$\Delta,E_{x}$ and at scale $\delta$ for
$C_{{\mathcal{F}}}=\|\phi\|_{C^{1+\theta}}^{2}\cdot\|\phi^{-1}\|_{C^{1}}^{2+\theta}\cdot
t(E_{x},\Delta,\delta)\cdot t({\mathcal{F}},\Delta)\cdot\delta^{\theta}$.

###### Proof.

Denote $\tilde{*}$ the pushed forward under $\phi^{-1}$ of the objects $*$.
Observe that

$Dh^{\tilde{\mathcal{F}}}_{\tilde{D}_{x},\tilde{D}_{y}}=p^{\tilde{E}_{x}}_{\tilde{D}_{x},\tilde{D}_{y}}$

because $\tilde{E}_{x}$ contains the plane parallel to the linear foliation
$\tilde{\mathcal{F}}$. Denote $D^{\prime}=D\phi(\tilde{x})\tilde{D}_{y}$ and
$\tilde{D}^{\prime}=D\phi^{-1}(x)D_{y}$. Since $D\phi\tilde{E}_{x}=E_{x}$ we
have

$\displaystyle p^{E_{x}}_{D_{x},D_{y}}$ $\displaystyle=$ $\displaystyle
D\phi(\tilde{x})|_{\tilde{D}^{\prime}}\circ
p^{\tilde{E}_{x}}_{\tilde{D}_{x},\tilde{D}^{\prime}}\circ
D\phi^{-1}(x)|_{D_{x}}$ $\displaystyle=$ $\displaystyle
D\phi(\tilde{x})|_{\tilde{D}^{\prime}}\circ
p^{\tilde{E}_{x}}_{\tilde{D}_{y},\tilde{D}^{\prime}}\circ
p^{\tilde{E}_{x}}_{\tilde{D}_{x},\tilde{D}_{y}}\circ D\phi^{-1}(x)|_{D_{x}}$
$\displaystyle=$ $\displaystyle p^{E_{x}}_{D^{\prime},D_{y}}\circ
D\phi(\tilde{x})|_{\tilde{D}_{y}}\circ
p^{\tilde{E}_{x}}_{\tilde{D}_{x},\tilde{D}_{y}}\circ D\phi^{-1}(x)|_{D_{x}}.$

Then

$\displaystyle\|Dh^{{\mathcal{F}}}_{D_{x},D_{y}}-p^{E_{x}}_{D_{x},D_{y}}\|$
$\displaystyle=$ $\displaystyle\|D\phi(\tilde{y})|_{\tilde{D}_{y}}\circ
Dh^{\tilde{\mathcal{F}}}_{\tilde{D}_{x},\tilde{D}_{y}}\circ
D\phi^{-1}(x)|_{D_{x}}-p^{E_{x}}_{D_{x},D_{y}}\|$ $\displaystyle=$
$\displaystyle\|\left(D\phi(\tilde{y})|_{\tilde{D}_{y}}-p^{E_{x}}_{D^{\prime},D_{y}}\circ
D\phi(\tilde{x})|_{\tilde{D}_{y}}\right)\circ
p^{\tilde{E}_{x}}_{\tilde{D}_{x},\tilde{D}_{y}}\circ D\phi^{-1}(x)|_{D_{x}}\|$
$\displaystyle\leq$
$\displaystyle\|p^{E_{x}}_{D_{y}}\|\cdot\|D\phi(\tilde{y})-D\phi(\tilde{x})\|\cdot\|p^{\tilde{E}_{x}}_{\tilde{D}_{y}}\|\cdot\|D\phi^{-1}(x)\|$
$\displaystyle\leq$
$\displaystyle\frac{t(E_{x},\Delta,\delta)\cdot\|\phi\|_{C^{1+\theta}}\cdot\|\phi^{-1}\|_{C^{1}}^{1+\theta}\cdot\delta^{\theta}}{\sin(\angle(\tilde{D}_{y},T\tilde{F}))}$
$\displaystyle\leq$
$\displaystyle\|\phi\|_{C^{1+\theta}}^{2}\cdot\|\phi^{-1}\|_{C^{1}}^{2+\theta}\cdot
t(E_{x},\Delta,\delta)\cdot t({\mathcal{F}},\Delta)\cdot\delta^{\theta}.$

We used the fact that

$\sin(\angle(D_{y},T_{y}F))\leq\sin(\angle(\tilde{D}_{y},T\tilde{F}))\|D\phi\|\cdot\|D\phi^{-1}\|.$

∎

We want to study the behavior of the regularity of foliations under the push-
forward of a diffeomorphism. Assume that ${\mathcal{W}}$ is contained in the
open set $U$ and $f:U\rightarrow F(U)$ is a $C^{1+\theta}$ diffeomorphism. We
will use the following notations for the bounds of $Df$ along $\Delta$ and
$T\mathcal{F}$:

$\displaystyle\lambda_{\Delta}^{+}(f,x,\delta)$ $\displaystyle:=$
$\displaystyle\sup_{d(x,y)<\delta}\|Df(y)|_{\Delta_{y}}\|;$
$\displaystyle\lambda_{\Delta}^{-}(f,x,\delta)$ $\displaystyle:=$
$\displaystyle\left(\sup_{d(x,y)<\delta}\|(Df(y)|_{\Delta_{y}})^{-1}\|\right)^{-1};$
$\displaystyle\lambda_{\mathcal{F}}(f,x,\delta)$ $\displaystyle:=$
$\displaystyle\left(\sup_{d(x,y)<\delta}\|(Df(y)|_{T_{y}\mathcal{F}(x)})^{-1}\|\right)^{-1};$

The following lemma is one of the main tools behind our proof. It keeps track
on how the constant $C_{{\mathcal{F}}}$ changes under iterations.

######  Lemma 2.2.

Let $\mathcal{F}$ be a foliation as above such that $Dh^{\mathcal{F}}$ is
$(C_{\mathcal{F}},\theta)$-Hölder along $\mathcal{F}$ at $x\in\mathbb{R}^{n}$
with respect to $\Delta,E_{x}$ and at scale $\delta$. Let $f:U\rightarrow
f(U)\subset\mathbb{R}^{n}$ be a $C^{1+\theta}$ diffeomorphism. Then for any
$\Delta^{\prime}\subset f_{*}\Delta$ and
$\delta^{\prime}<\lambda_{{\mathcal{F}}}(f,x,\delta)\delta$,
$Dh^{f_{*}\mathcal{F}}$ is $(C_{f_{*}\mathcal{F}},\theta)$-Hölder along
$f_{*}\mathcal{F}$ at $f(x)$ with respect to $\Delta^{\prime},f_{*}E_{x}$ and
at scale $\delta^{\prime}$, where

(3)
$C_{f_{*}\mathcal{F}}=\frac{\lambda_{\Delta}^{+}(f,x,\delta)C_{\mathcal{F}}+t(E_{x},\Delta,\delta)t(f_{*}E_{x},\Delta^{\prime},\delta^{\prime})\|Df\|_{C^{\theta}}}{\lambda_{\Delta}^{-}(f,x,\delta)\lambda_{\mathcal{F}}(f,x,\delta)^{\theta}},$

###### Proof.

Denote
$E^{\prime}_{x}=Df(x)E_{x},D^{\prime}_{x}=Df(x)D_{x},D^{\prime}_{y}=Df(y)D_{y},\tilde{D}=Df(x)D_{y}$.

Since $Df(x)$ takes the decomposition $E_{x}\oplus D_{x}$ to
$E^{\prime}_{x}\oplus D^{\prime}_{x}$, we have that

$p_{D^{\prime}_{x},D^{\prime}_{y}}^{E^{\prime}_{x}}\circ
Df(x)|_{D_{x}}=p_{\tilde{D},D^{\prime}_{y}}^{E^{\prime}_{x}}\circ
Df(x)|_{D_{y}}\circ p_{D_{x},D_{y}}^{E_{x}}.$

We also have

$d_{f_{*}\mathcal{F}}(f(x),f(y))\geq\lambda_{\mathcal{F}}(f)d(x,y).$

For simplicity we will use the notation
$\lambda_{\Delta}^{\pm},\lambda_{{\mathcal{F}}}$. We have

$\displaystyle\left\|Dh^{f_{*}\mathcal{F}}_{D^{\prime}_{x},D^{\prime}_{y}}-p_{D^{\prime}_{x},D^{\prime}_{y}}^{E^{\prime}_{x}}\right\|=$
$\displaystyle\left\|Df(y)|_{D_{y}}\circ
Dh^{\mathcal{F}}_{D_{x},D_{y}}(x)\circ\left(Df(x)|_{D_{x}}\right)^{-1}-p_{D^{\prime}_{x},D^{\prime}_{y}}^{E^{\prime}_{x}}\right\|$
$\displaystyle\leq$
$\displaystyle\left\|Df(y)|_{D_{y}}\circ\left(Dh^{\mathcal{F}}_{D_{x},D_{y}}(x)-p_{D_{x},D_{y}}^{E_{x}}\right)\circ\left(Df(x)|_{D_{x}}\right)^{-1}\right\|+$
$\displaystyle+\left\|Df(y)|_{D_{y}}\circ
p_{D_{x},D_{y}}^{E_{x}}\circ\left(Df(x)|_{D_{x}}\right)^{-1}-p_{D^{\prime}_{x},D^{\prime}_{y}}^{E^{\prime}_{x}}\right\|$
$\displaystyle\leq$ $\displaystyle\left\|\left(Df(y)|_{D_{y}}\circ
p_{D_{x},D_{y}}^{E_{x}}-p_{D^{\prime}_{x},D^{\prime}_{y}}^{E^{\prime}_{x}}\circ
Df(x)|_{D_{x}}\right)\left(Df(x)|_{D_{x}}\right)^{-1}\right\|+$
$\displaystyle+\frac{\lambda_{\Delta}^{+}}{\lambda_{\Delta}^{-}}C_{\mathcal{F}}d_{\mathcal{F}}(x,y)^{\theta}$
$\displaystyle\leq$
$\displaystyle\frac{\lambda_{\Delta}^{+}}{\lambda_{\Delta}^{-}}C_{\mathcal{F}}d_{\mathcal{F}}(x,y)^{\theta}+\frac{1}{\lambda_{\Delta}^{-}}\left\|Df(y)|_{D_{y}}\circ
p_{D_{x},D_{y}}^{E_{x}}-p_{\tilde{D},D^{\prime}_{y}}^{E^{\prime}_{x}}\circ
Df(x)|_{D_{y}}\circ p_{D_{x},D_{y}}^{E_{x}}\right\|$ $\displaystyle\leq$
$\displaystyle\frac{\lambda_{\Delta}^{+}}{\lambda_{\Delta}^{-}}C_{\mathcal{F}}d_{\mathcal{F}}(x,y)^{\theta}+\frac{1}{\lambda_{\Delta}^{-}}\|p_{D^{\prime}_{y}}^{E^{\prime}_{x}}\|\cdot\|Df(y)|_{D_{y}}-Df(x)|_{D_{y}}\|\cdot\|p_{D_{x},D_{y}}^{E_{x}}\|$
$\displaystyle\leq$
$\displaystyle\left(\frac{\lambda_{\Delta}^{+}C_{\mathcal{F}}}{\lambda_{\Delta}^{-}\lambda_{\mathcal{F}}^{\theta}}+\frac{t(E_{x},\Delta,\delta)t(f_{*}E_{x},\Delta^{\prime},\delta^{\prime})\|Df\|_{C^{\theta}}}{\lambda_{\Delta}^{-}\lambda_{\mathcal{F}}^{\theta}}\right)d_{f_{*}\mathcal{F}}(f(x),f(y))^{\theta}.$

∎

###  2.2. Smooth approximations of invariant submanifolds.

The center-unstable leaves ${\mathcal{W}}^{cu}$ of the partially hyperbolic
diffeomorphism $f$ are subfoliated by the unstable leaves ${\mathcal{W}}^{u}$,
but unfortunately they are not smooth enough in order to carry out the ideas
from the previous sub-section. This is why we need to construct smooth
approximations of the center-unstable leaves, together with a smooth
approximation of the unstable subfoliation. We need to approximate pieces of
${\mathcal{W}}^{cu}$ which are arbitrarily large in the center direction,
while making sure that the $C^{2}$ bounds of the approximations are uniform.
The reader can keep in mind some specific examples where the smooth
approximations are more or less straightforward: fake foliations –
$cu$-subspace subfoliated by $u$-subspaces; perturbations of linear maps – the
linear foliations of the original linear map. The case of partially hyperbolic
diffeomorphisms which are fibered over hyperbolic homeomorphisms is also
easier, because the center leaves are uniform $C^{1+\alpha}$ embeddings of the
same compact fiber, and one can use standard smooth approximation. Our
construction is a bit more technical because we want to include possible large
pieces of the center manifolds with possible complicated topology.

Let us make some preparations.

######  Definition 2.2.

A $C^{r}$ submanifold ${\mathcal{W}}$ has size greater than $\delta$ at $x$ if
within the exponential chart at $x$, $W$ contains the graph of a $C^{r}$
function $g$ from the ball of radius $\delta$ in $T_{x}{\mathcal{W}}$ to the
orthogonal complement $T_{x}{\mathcal{W}}^{\perp}$.

If the ball of radius $\delta$ at $x$ in the $C^{r}$ submanifold
${\mathcal{W}}$ can be written, in an exponential chart at $x\in M$, as the
graph of a $C^{r}$ function $g$ from an open subset of $T_{x}{\mathcal{W}}$ to
the orthogonal complement $T_{x}{\mathcal{W}}^{\perp}$, then the
$(C^{r},x,\delta)$ size of ${\mathcal{W}}$ is
$\|{\mathcal{W}}\|_{C^{r},x,\delta}=\|g\|_{C^{r}}$.

Eventually modifying the Riemannian metric, we can assume that the invariant
subspaces are close to orthogonal.

######  Definition 2.3.

The continuous cone field $\Delta^{*}_{\epsilon}$ over $M$ is defined in the
following way: $\Delta^{*}_{\epsilon}(x)$ contains the subspaces of $T_{x}M$
which have the same dimension as $E^{*}(x)$ and are $\epsilon$-close to
$E^{*}(x)$, $*\in\\{s,c,u,cs,su,cu\\}$.

For $\epsilon$ small we have that $\Delta^{u}_{\epsilon}$ and
$\Delta^{cu}_{\epsilon}$ are forward invariant while $\Delta^{s}_{\epsilon}$
and $\Delta^{cs}_{\epsilon}$ are backward invariant.

Fix $\epsilon_{0},\delta_{0}>0$ and a $C^{1+\theta}$ neighborhood
$\mathcal{U}(f)$ of $f$ such that

  * •

the $s,sc$-cones of size $2\epsilon_{0}$ are backwards invariant while the
$u,cu$-cones are forward invariant, for all $g\in\mathcal{U}(f)$;

  * •

the cone fields $\Delta^{*}_{2\epsilon_{0}}$ are uniformly transverse for all
$*\in\\{s,c,u,cs,su,cu\\}$ at the scale $\delta_{0}$, meaning that for every
$x\in M$, within the exponential chart at $x$, $\Delta^{*}_{2\epsilon_{0}}(x)$
and $\Delta^{*^{\prime}}_{2\epsilon_{0}}(y)$ are uniformly transverse for
$d(x,y)<\delta_{0}$:
$t(\Delta^{*}_{2\epsilon_{0}},\Delta^{*^{\prime}}_{2\epsilon_{0}},\delta_{0})<2$;

  * •

The bunching condition holds at the ($2\epsilon_{0},\delta_{0}$)- scale for
all $g\in\mathcal{U}(f)$, meaning that

(4)
$\frac{\lambda^{+}_{\Delta^{c}_{2\epsilon_{0}}}(g,x,\delta_{0})}{\lambda^{-}_{\Delta^{c}_{2\epsilon_{0}}}(g,x,\delta_{0})\lambda^{-}_{\Delta^{u}_{2\epsilon_{0}}}(g,x,\delta_{0})^{\theta}}<\mu<1,\
\ \forall g\in\mathcal{U}(f),\forall x\in M.$

  * •

The center bundle is uniformly $C^{\alpha}$ and the local center manifolds are
uniformly $C^{1+\alpha}$, meaning that there exists $C_{\alpha}>0$ such that
for every $g\in\mathcal{U}(f)$ and every $x,y\in M$,
$d(E^{c}(x,g),E^{c}(y,g))\leq C_{\alpha}d(x,y)^{\alpha}$ and
$\|{\mathcal{W}}^{c}_{4\delta_{0}}(x,g)\|_{C^{1+\alpha},x,4\delta_{0}}<C_{\alpha}$.

The following lemma is an immediate consequence of the transversality. We say
that the submanifold $W$ is tangent to the cone field $\Delta$ if
$T_{y}W\in\Delta(y)$, for all $y\in W$.

######  Lemma 2.3 (Local product structure).

There exist $\delta_{p}>0$ such that for any $0<\delta\leq\delta_{p}$, any
$x,y\in M$ with $d(x,y)<\delta$, any $W_{2\delta}(x)$ $C^{1}$ manifold of size
$2\delta$ at $x$ tangent to $\Delta^{*}_{\epsilon_{0}}$, and any
$W_{2\delta}(y)$ $C^{1}$ manifold of size $2\delta$ at $y$ tangent to
$\Delta^{*^{\prime}}_{\epsilon_{0}}$, where $*$ and $*^{\prime}$ are
complementary combinations of $\\{s,c,u\\}$, then $W_{2\delta}(x)$ and
$W_{2\delta}(y)$ intersect transversally in a unique point.

Now we are ready to construct the smooth approximations..

####  2.2.1. Smooth uniform approximation of center manifolds

The first step is to approximate large pieces of center manifolds with smooth
ones, while keeping control on the smoothness of the approximations.

Fix a smooth approximation $\tilde{E}^{su}$ inside
$\Delta^{su}_{\epsilon_{0}}$. There exists $0<\epsilon_{1}<\epsilon_{0}$ such
that for every $p\in M$, the family
$\\{\exp_{x}(B_{\epsilon_{1}}\tilde{E}^{su}(x)):\ x\in{\mathcal{W}}^{c}(p)\\}$
subfoliate a tubular neighborhood of ${\mathcal{W}}^{c}(p)$. Let
$\tilde{h}^{su}$ be the holonomy generated by this subfoliation.

######  Lemma 2.4.

For any $\epsilon>0$ small enough, any $p\in M$ and any $R>0$, there exists a
smooth approximation of size $\epsilon$ of ${\mathcal{W}}^{c}_{R}(p)$, meaning
the following. There exists a smooth immersed manifold (possible with self-
intersections) $\tilde{\mathcal{W}}^{c}_{R,\epsilon}(p)$ tangent to
$\Delta^{c}_{\epsilon}$, together with a local diffeomorphism
$\tilde{h}^{su}_{\epsilon}$ given by the local $\tilde{h}^{su}$-holonomy.
Furthermore the approximations are uniform in the following sense. For every
$x\in\tilde{\mathcal{W}}^{c}_{R,\epsilon}(p)$ we have
$\|\tilde{\mathcal{W}}^{c}_{R,\epsilon}(p)\|_{C^{1+\alpha},x,\delta_{0}}\leq\tilde{C}_{\alpha}$
for some $\tilde{C}_{\alpha}$ independent of $p,R,\epsilon,f$, and
$\|\tilde{\mathcal{W}}^{c}_{R,\epsilon}(p)\|_{C^{2},x,\delta_{0}}\leq\tilde{C}(\epsilon)$
for some $\tilde{C}(\epsilon)$ independent of $p,R,f$ (but depends on
$\epsilon$).

###### Proof.

Cover $M$ by a finite number of foliation charts of ${\mathcal{W}}^{c}$ with
center leaves of size $\delta_{0}$, say $U_{1},U_{2},\dots,U_{k}$. Then
$W^{c}_{R}(p)$ is covered by finitely many leaves
${\mathcal{W}}^{c}_{\delta_{0}}(x_{i}),1\leq i\leq K$ from these foliation
charts. Let $B_{i}={\mathcal{W}}^{c}_{3\delta_{0}}(x_{i})$ and
${\mathcal{W}}^{c}_{R}(p)\subset{\mathcal{W}}_{0}=\cup_{i=1}^{K}B_{i}$.

Each $B_{i}$ is (contained in) the graph of a function
$\gamma_{i}:B_{4\delta_{0}}E^{c}(x_{i})\rightarrow E^{c^{\perp}}(x_{i})$ with
uniform $C^{1+\alpha}$ bounds. We will use the following standard
regularization procedure.

Suppose that $\gamma:U\rightarrow E^{\perp}$ is $C^{1+\alpha}$,
$B_{3\delta_{0}}E\subset U\subset E$, and it is also $C^{\infty}$ on some
subset $V\subset U$. For any $\epsilon>0$ sufficiently small we can use the
standard regularization and obtain $\gamma^{\prime}$ which is $C^{\infty}$ and
$C^{1}$ close to $\gamma$. Let $\rho$ be a smooth bump function which is one
on $B_{2\delta_{0}}E$ and zero outside $B_{3\delta_{0}}E$. Use $\rho$ to
interpolate between $\gamma^{\prime}$ and $\gamma$ and obtain a new function
$\tilde{\gamma}$ which is $C^{1+\alpha}$ on $B_{4\delta_{0}}E$, $C^{\infty}$
on $B_{2\delta_{0}}E\cup V$ and satisfies

  * •

$\|\tilde{\gamma}-\gamma\|_{C^{1}}\leq\epsilon^{\alpha}\|\gamma\|_{C^{1+\alpha}}$
on $U$,

  * •

$\|\tilde{\gamma}\|_{C^{1+\alpha}}\leq 2\|\gamma\|_{C^{1+\alpha}}$ on $U$,

  * •

$\|\tilde{\phi}|_{B_{2\delta_{0}}E\cup V}\|_{C^{2}}\leq
C(\epsilon)\max\\{\|\phi\|_{C^{1+\alpha}},\|\phi|_{V}\|_{C^{2}}\\}$,

  * •

$\tilde{\phi}=\phi$ outside $B_{3\delta_{0}}E$,

where $C(\epsilon)>2$ depends only on $\epsilon$ (and $\rho$).

We proceed with perturbing the leaves in $U_{1}$. Let $I_{j}=\\{1\leq i\leq
k:{\mathcal{W}}^{c}_{\delta_{0}}(x_{i})\in U_{j}\\}$. By performing the
perturbation described above to each $B_{i},i\in I_{1}$ we obtain new
submanifolds $B_{i}^{1}$ which are graphs of the functions $\gamma_{i}^{1}$ in
exponential charts at $x_{i}$. The holonomy $\tilde{h}^{su}_{1}$ of length
smaller that $2\epsilon^{\alpha}C_{\alpha}$ (in the exponential chart at
$x_{i}$) is well defined between $B_{i}$ and $B_{i}^{1}$. Let

${\mathcal{W}}_{1}={\mathcal{W}}_{0}\setminus(\cup_{i\in
I_{1}}B_{i})\cup(\cup_{i\in I_{1}}B_{i}^{1}).$

The ${\mathcal{W}}_{1}$ is an immersed submanifold of $M$, possible with self-
intersections. Observe that we can extend $\tilde{h}^{su}_{1}$ as the identity
outside $\cup_{i\in I_{1}}B_{i}^{1}$ and obtain a local diffeomorphism between
${\mathcal{W}}_{0}$ and ${\mathcal{W}}_{1}$. Let $A_{1}=\cup_{i\in
I_{1}}Gr\left(\gamma_{i}^{1}|_{B_{2\delta_{0}}E^{c}(x_{i})}\right)$ (the
smooth part of ${\mathcal{W}}_{1}$). For $i\notin I_{1}$ define
$B_{i}^{1}=\tilde{h}^{su}_{1}(B_{i})\subset{\mathcal{W}}_{1}$. Then
${\mathcal{W}}_{1}$ is the union of $B_{i}^{1}$, and each $B_{i}^{1}$ is
related to $B_{i}$ by $\tilde{h}^{su}_{1}$. If $\pi_{i}$ is the projection on
the first coordinate in the exponential chart at $x_{i}$, then $B_{i}^{1}$ is
the graph of a function $\gamma_{i}^{1}:\pi_{i}(B_{i}^{1})\rightarrow
E^{c^{\perp}}(x_{i})$ satisfying

  * •

$B_{4\delta_{0}-\epsilon_{1}}E^{c}(x_{i})\subset\pi_{i}(B_{i}^{1})\subset
B_{4\delta_{0}+\epsilon_{1}}E^{c}(x_{i})$,

  * •

$\|\gamma_{i}-\gamma_{i}^{1}\|_{C^{1}}\leq\epsilon_{1}/2$,

  * •

$\|\gamma_{i}^{1}\|_{C^{1+\alpha}}\leq 2C_{M}C_{\alpha}$,

  * •

$\|\gamma_{i}^{1}|_{\pi_{i}(B_{i}^{1}\cap A_{1})}\|_{C^{2}}\leq
C(\epsilon)C_{M}C_{\alpha}$,

where $C_{M}$ depends on the Riemannian manifold $M$ and $\delta_{0}$
(measures the size of the change of coordinates between nearby exponential
charts) and $\epsilon_{1}=2\epsilon^{\alpha}C_{M}C_{\alpha}$. Furthermore each
$B_{i}$ is diffeomorphic to $B_{i}^{1}$ by the holonomy $\tilde{h}^{su}_{1}$
of length less that $\epsilon_{1}$. Observe that if $\epsilon$ is small enough
so ${\mathcal{W}}_{1}$ stays $C^{1}$ close to ${\mathcal{W}}^{c}$, then we
have that $A_{1}$ contains $\cup_{i\in
I_{1}}\tilde{h}^{su}_{1}\left({\mathcal{W}}^{c}_{2\delta_{0}-2\epsilon_{1}}(x_{i})\right)$.

Now we proceed with perturbing the leaves corresponding to $U_{2}$. In a
similar manner we obtain a submanifold ${\mathcal{W}}_{2}$ related to
${\mathcal{W}}_{1}$ by the holonomy $\tilde{h}^{su}_{2}$ of length
$\epsilon_{2}=4\epsilon^{\alpha}C_{M}^{2}C_{\alpha}$, and containing a smooth
part $A_{2}\subset{\mathcal{W}}_{2}$. ${\mathcal{W}}_{2}$ is the union of
$B_{i}^{2}=\tilde{h}^{su}_{2}(B_{i}^{1})$, and each $B_{i}^{2}$ is the graph
of a function $\gamma_{i}^{2}:\pi_{i}(B_{i}^{2})\rightarrow
E^{c^{\perp}}(x_{i})$ satisfying

  * •

$B_{4\delta_{0}-\epsilon_{1}-\epsilon_{2}}E^{c}(x_{i})\subset\pi_{i}(B_{i}^{2})\subset
B_{4\delta_{0}+\epsilon_{1}+\epsilon_{2}}E^{c}(x_{i})$,

  * •

$\|\gamma_{i}^{1}-\gamma_{i}^{2}\|_{C^{1}}\leq\epsilon_{2}/2$,

  * •

$\|\gamma_{i}^{2}\|_{C^{1+\alpha}}\leq 4C_{M}^{2}C_{\alpha}$,

  * •

$\|\gamma_{i}^{2}|_{\pi_{i}(B_{i}^{1}\cap A_{2})}\|_{C^{2}}\leq
C(\epsilon)^{2}C_{M}^{2}C_{\alpha}$.

If $\epsilon$ is small enough so that ${\mathcal{W}}_{2}$ stays $C^{1}$ close
to ${\mathcal{W}}^{c}$, then

$\cup_{i\in I_{1}\cup
I_{2}}\tilde{h}^{su}_{2}\circ\tilde{h}^{su}_{1}\left({\mathcal{W}}^{c}_{2\delta_{0}-2\epsilon_{1}-2\epsilon_{2}}(x_{i})\right)\subset
A_{2}.$

Continue by induction perturbing on each $U_{i}$ until we reach $U_{k}$. We
get a submanifold ${\mathcal{W}}_{k}$ related to ${\mathcal{W}}_{k-1}$ by the
holonomy $\tilde{h}^{su}_{k}$ of length
$\epsilon_{k}=2^{k}\epsilon^{\alpha}C_{M}^{k}C_{\alpha}$, and containing a
smooth part $A_{k}\subset{\mathcal{W}}_{k}$. In particular ${\mathcal{W}}_{k}$
is related to ${\mathcal{W}}_{0}$ by the holonomy $\tilde{h}^{su}$ of length
$\tilde{\epsilon}=\sum_{i=1}^{k}\epsilon_{i}$. Furthermore ${\mathcal{W}}_{k}$
is the union of $B_{i}^{k}=\tilde{h}^{su}_{k}(B_{i}^{k-1})$, and each
$B_{i}^{k}$ is the graph of a function
$\gamma_{i}^{k}:\pi_{i}(B_{i}^{k})\rightarrow E^{c^{\perp}}(x_{i})$ satisfying

  * •

$B_{4\delta_{0}-\tilde{\epsilon}}E^{c}(x_{i})\subset\pi_{i}(B_{i}^{k})\subset
B_{4\delta_{0}+\tilde{\epsilon}}E^{c}(x_{i})$,

  * •

$\|\gamma_{i}^{k-1}-\gamma_{i}^{k}\|_{C^{1}}\leq\epsilon_{k}/2$, so
$\|\gamma_{i}-\gamma_{i}^{k}\|_{C^{1}}\leq\tilde{\epsilon}/2$,

  * •

$\|\gamma_{i}^{k}\|_{C^{1+\alpha}}\leq 2^{k}C_{M}^{k}C_{\alpha}$,

  * •

$\|\gamma_{i}^{k}|_{\pi_{i}(B_{i}^{1}\cap A_{k})}\|_{C^{2}}\leq
C(\epsilon)^{k}C_{M}^{k}C_{\alpha}$.

If $\epsilon$ is small enough then we also have

$\cup_{i\in
I}\tilde{h}^{su}_{k}\circ\dots\circ\tilde{h}^{su}_{1}\left({\mathcal{W}}^{c}_{2\delta_{0}-2\tilde{\epsilon}}(x_{i})\right)\subset
A_{k}.$

We can make this construction until the end for any $\epsilon$ small enough
such that $\tilde{\epsilon}<\delta_{0}/2$ (and ${\mathcal{W}}_{k}$ is close to
${\mathcal{W}}^{c}$ so the estimates on the smooth part hold).

If $x=(a,\gamma_{i}^{k}(a))\in{\mathcal{W}}^{k}$ (in a chart at $x_{i}$), let
$x_{0}=(a,\gamma_{i}(a))\in{\mathcal{W}}^{c}(x-i)$. We have
$d(E^{c}(x),E^{c}(x_{0}))\leq C_{\alpha}\tilde{\epsilon}^{\alpha}$ and
$d(E^{c}(x_{0}),Gr(D\gamma_{i}^{k}(a)))\leq\tilde{\epsilon}$, so
${\mathcal{W}}_{k}$ is tangent to
$\Delta^{c}_{\tilde{\epsilon}+\tilde{\epsilon}^{\alpha}C_{\alpha}}$.

Since $\lim_{\epsilon\rightarrow
0}\tilde{\epsilon}+\tilde{\epsilon}^{\alpha}C_{\alpha}=0$, the conclusions of
the lemma hold with
$\overline{\epsilon}=\tilde{\epsilon}+\tilde{\epsilon}^{\alpha}C_{\alpha}$,
$\tilde{\mathcal{W}}^{c}_{R,\overline{\epsilon}}(p)={\mathcal{W}}_{k}$,
$\tilde{h}^{su}_{\overline{\epsilon}}=\tilde{h}^{su}_{k}\circ\dots\circ\tilde{h}^{su}_{1}$,
$\tilde{C}_{\alpha}=2^{k}C_{M}^{k+1}C_{\alpha}$ and
$\tilde{C}(\overline{\epsilon})=C(\epsilon)^{k}C_{M}^{k+1}C_{\alpha}$.

∎

####  2.2.2. Smooth uniform approximation of center-unstable manifolds and of
unstable foliation

The second step is to use the smooth approximation of the center in order to
construct smooth approximations of local center-unstable pieces together with
a subfoliation close to the unstable one.

Fix a smooth global approximation $\tilde{E}^{u}$ of $E^{u}$, say within
$\Delta^{u}_{\frac{\epsilon_{0}}{1}0}$. We know from the previous step that
$\tilde{\mathcal{W}}^{c}_{R,\epsilon}(p)$ are uniformly $C^{1+\alpha}$ for all
$p,R,\epsilon,f$. Then there exists
$0<\delta_{{\mathcal{F}}}<\min\\{\delta_{0},\delta_{p}\\}$ such that, for
every $p,R,\epsilon,f$, the family
$\\{\exp(B_{\delta_{\mathcal{F}}}\tilde{E}^{u}(x)):\
x\in\tilde{\mathcal{W}}^{c}_{R,\epsilon}(p)\\}$ foliates a smooth submanifold
inside a tubular neighborhood of $\tilde{\mathcal{W}}^{c}_{R,\epsilon}(p)$; we
denote this submanifold $\tilde{\mathcal{W}}^{cu}_{R,\epsilon}(p)$, and the
foliation ${\mathcal{F}}^{u}_{R,\epsilon,p}$. By assuming that
$\epsilon<\frac{\epsilon_{0}}{10}$ and eventually making
$\delta_{{\mathcal{F}}}$ smaller we have that
$\tilde{\mathcal{W}}^{cu}_{R,\epsilon}(p)$ is tangent to
$\Delta^{cu}_{\epsilon_{0}}$ and ${\mathcal{F}}^{u}_{R,\epsilon,p}$ is tangent
to $\Delta^{u}_{\epsilon_{0}}$. We also have that
$\tilde{\mathcal{W}}^{cu}_{R,\epsilon}(p)$ and
${\mathcal{F}}^{u}_{R,\epsilon,p}$ are uniformly $C^{r}$, $r\geq 2$ with
respect to $p,R,f$ (the $C^{r}$ bounds do however depend on $\epsilon$).

######  Lemma 2.5.

For any $\epsilon>0$ small enough there exists a constant
$C_{\phi}(\epsilon)>0$ such that for every $p,R,f$ and any
$x\in\tilde{\mathcal{W}}^{c}_{R,\epsilon}(p)$, there exists a
$\delta_{{\mathcal{F}}}$-linear parametrization $\phi$ of
($\tilde{\mathcal{W}}^{c}_{R,\epsilon}(p),{\mathcal{F}}^{u}_{R,\epsilon,p}$)
at $x$ with $\|\phi\|_{C^{2}},\|\phi^{-1}\|<C_{\phi}(\epsilon)$.

###### Proof.

For simplicity of the notations we will work in an exponential chart at $x$,
and we will make an abuse of notations using the same notation for the objects
in $M$ and in the exponential chart.

Choose a decomposition $E_{1}\oplus E_{2}\oplus E_{3}=\mathbb{R}^{d}(=T_{x}M)$
with $E_{1}=T_{x}\tilde{\mathcal{W}}^{c}_{R,\epsilon}(p)$,
$E^{2}=\tilde{E}^{u}_{x}$ and $E_{3}$ orthogonal on $E_{1},E_{2}$. Let
$\alpha:B_{\delta_{{\mathcal{F}}}}\mathbb{R}^{d}\times
B_{\delta_{{\mathcal{F}}}}E_{2}\rightarrow\mathbb{R}^{d}$ be a smooth
parametrization of the family
$\\{\exp(B_{\delta_{\mathcal{F}}}\tilde{E}^{u}(y)):\ y\in
B_{\delta_{{\mathcal{F}}}}\mathbb{R}^{d}\\}$, in other words $\alpha(y,\cdot)$
is a parametrization of $exp(B_{\delta_{\mathcal{F}}}\tilde{E}^{u}(y))$. We
can assume that $\alpha(y,0)=y$, so $D_{y}\alpha(y,0)=Id_{\mathbb{R}^{d}}$,
and $D_{y_{2}}\alpha(y,0)$ is a linear map from $E_{2}$ to
$\tilde{E}^{u}_{y}$, uniformly bounded from zero and infinity. In particular
$D_{y_{2}}\alpha(0,0)$ is an automorphism of $E_{2}$. The map $\alpha$ is
$C^{\infty}$ and its size depends only on the Riemannian structure on $M$ and
the choice of $\tilde{E}^{u}$.

Let $\gamma:B_{\delta_{{\mathcal{F}}}}E_{1}\rightarrow E_{2}\oplus E_{3}$ be a
smooth function such that its graph is the local manifold
$\tilde{\mathcal{W}}^{c}_{R,\epsilon}(p)$ in a neighborhood of $x$. Then
$\gamma$ has the $C^{1+\alpha}$ size bounded by $2\tilde{C}_{\alpha}$ and its
$C^{2}$ size bounded by $2\tilde{C}(\epsilon)$.

Let $\phi:B_{\delta_{{\mathcal{F}}}}\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$,

$\phi(y_{1},y_{2},y_{3})=(\alpha((y_{1},\gamma(y_{1})),y_{2})+y_{3}.$

It is clear from the definition that $\phi$ is a
$\delta_{{\mathcal{F}}}$-linear parametrization $\phi$ of
($\tilde{\mathcal{W}}^{c}_{R,\epsilon}(p),{\mathcal{F}}^{u}_{R,\epsilon,p}$)
at $x$. The $C^{2}$ size of $\phi$ is bounded by some $C_{\phi}(\epsilon)$
which depends on $\tilde{C}(\epsilon)$, the Riemannian structure of $M$ and
the choice of $\tilde{E}^{u}$. The $C^{1+\alpha}$ size of $\phi$ is bounded by
some constant which depends on $\tilde{C}(\alpha)$, the Riemannian structure
of $M$ and the choice of $\tilde{E}^{u}$.

We have

$D\phi(0)=\begin{bmatrix}Id_{E_{1}}&0&0\\\ 0&D_{y_{2}}\alpha(0,0)&0\\\
0&0&Id_{E_{3}}\end{bmatrix}.$

The determinant is uniformly bounded away from zero, so eventually readjusting
$\delta_{F}$ we have that the $C^{1}$ size of $\phi^{-1}$ is uniformly
bounded. This finishes the proof. ∎

We claim that ${\mathcal{W}}^{u}_{\delta}{\mathcal{W}}^{c}_{R-r}(p)$ and
$\tilde{\mathcal{W}}^{cu}_{R,\epsilon}(p)$ are related by local stable
holonomy for some $r>0$ and $\epsilon,\delta$ sufficiently small.

######  Lemma 2.6.

Suppose that $\epsilon$ is small enough and
$r-2\epsilon>\delta_{{\mathcal{F}}}$,
$\delta+\epsilon<\frac{\delta_{{\mathcal{F}}}}{4}$. Then for any $p,R,f$, with
$R>r$, the stable holonomy of size $2(\delta+\epsilon)$ gives a local
homeomorphism from ${\mathcal{W}}^{u}_{\delta}{\mathcal{W}}^{c}_{R-r}(p)$ to
(a subset of) $\tilde{\mathcal{W}}^{cu}_{R,\epsilon}(p)$.

###### Proof.

Let $x\in{\mathcal{W}}^{c}_{R-r}(p)$ and $y\in{\mathcal{W}}^{u}_{\delta}(x)$.
Let
$x^{\prime}=\tilde{h}^{su}_{\epsilon}(x)\in\tilde{\mathcal{W}}^{c}_{R,\epsilon}(p)$,
so $d(x,x^{\prime})<\epsilon$. Then $\tilde{\mathcal{W}}^{c}_{R,\epsilon}(p)$
has size at $x^{\prime}$ at least $r-2\epsilon>\delta_{{\mathcal{F}}}$. This
implies that $\tilde{\mathcal{W}}^{cu}_{R,\epsilon}(p)$ has size at
$x^{\prime}$ at least $\frac{\delta_{{\mathcal{F}}}}{2}$. On the other hand
$d(y,x^{\prime})\leq\delta+\epsilon<\frac{\delta_{{\mathcal{F}}}}{4}$. The
local product structure from Lemma 2.3 implies that
${\mathcal{W}}^{s}_{2(\delta+\epsilon)}(y)$ intersects transversely the disk
centered at $x^{\prime}$ of size
$2(\delta+\epsilon)<\frac{\delta_{{\mathcal{F}}}}{2}$ in
$\tilde{\mathcal{W}}^{cu}_{R,\epsilon}(p)$ in a point
$h^{s}_{2(\delta+\epsilon)}(y)$. Then $h^{s}_{2(\delta+\epsilon)}(y)$ is a
local homeomorphism from
${\mathcal{W}}^{u}_{\delta}{\mathcal{W}}^{c}_{R-r}(p)$ to
$\tilde{\mathcal{W}}^{cu}_{R,\epsilon}(p)$. ∎

##  3\. Proofs

We divide the proof of Theorem 1.1 in several steps.

###  3.1. Approximation of the unstable holonomies: construction of
$h^{n}_{p,q}$

We start with the construction of an approximation of the unstable holonomy
inside center-unstable leaves. From now on we fix $r=2\delta_{{\mathcal{F}}}$
and $0<\epsilon=\delta<\frac{\delta_{{\mathcal{F}}}}{10}$ small enough so that
all the conclusions from sub-section 2.2 hold.

Let $p\in M$, $q\in{\mathcal{W}}^{u}_{\delta}(p)$,
$x\in{\mathcal{W}}^{c}_{\delta}(p)$,
$z=h^{u}_{p,q}(x)\in{\mathcal{W}}^{c}_{2\delta}(q)$ and
$R_{n}=3{\delta}\sup_{x\in M}\|Df^{-n}|_{E^{c}}\|+r$. We start iterating back
by $f^{-n}$. Denote

${\mathcal{W}}_{n}={\mathcal{W}}^{u}_{\delta}{\mathcal{W}}^{c}_{R_{n}-r}(f^{-n}(p)).$

Observe that
$f^{-n}({\mathcal{W}}^{c}_{2\delta}(p),f^{-n}({\mathcal{W}}^{c}_{2\delta}(p)\subset{\mathcal{W}}_{n}$.

As in the previous section we consider the approximation
$\tilde{\mathcal{W}}^{cu}_{R_{n},\epsilon}(f^{-n}(p)):=\tilde{\mathcal{W}}_{n}$
and its subfoliation
${\mathcal{F}}_{R_{n},\epsilon,f^{-n}(p)}:={\mathcal{F}}_{n}$. Lemma 2.6
implies that the stable holonomy of size (smaller than) $4\delta$,
$h^{s}_{4\delta}:{\mathcal{W}}_{n}\rightarrow\tilde{\mathcal{W}}_{n}$, is a
local homeomorphism. Denote
$\tilde{*}_{n}=h^{s}_{4\delta}(f^{-n}(*))\in{\mathcal{W}}^{n}$ for
$*\in\\{p,q,x,z\\}$.

Let
$T_{\tilde{p}_{n}}=h^{s}_{4\delta}(f^{-n}({\mathcal{W}}^{c}_{2\delta}(p)))$
and
$T_{\tilde{q}_{n}}=h^{s}_{4\delta}(f^{-n}({\mathcal{W}}^{c}_{2\delta}(q)))$,
they are $C^{1+\alpha}$ transversals to the foliation ${\mathcal{F}}_{n}$ in
$\tilde{\mathcal{W}}_{n}$ (they are in fact tangent to
$\Delta^{c}_{\epsilon}$).

Now we iterate $\tilde{\mathcal{W}}_{n}$ and ${\mathcal{F}}_{n}$ forward by
$f^{n}$. Denote
$*_{n}=f^{n}(\tilde{*}_{n})=h^{s}_{4\lambda_{s}^{n}\delta}(*)$, for
$*\in\\{p,q,x,z\\}$ (the stable holonomy commutes with $f$ and is uniformly
contracted). Also denote
$T_{p_{n}}=f^{n}(T_{\tilde{p}_{n}})=h^{s}_{4\lambda_{s}^{n}\delta}({\mathcal{W}}^{c}_{2\delta}(p))$
and
$T_{q_{n}}=f^{n}(T_{\tilde{q}_{n}})=h^{s}_{4\lambda_{s}^{n}\delta}({\mathcal{W}}^{c}_{2\delta}(q))$,
they are again $C^{1+\alpha}$ transversals to $f^{n}_{*}{\mathcal{F}}_{n}$
inside $f^{n}\tilde{\mathcal{W}}_{n}$.

The partial hyperbolicity implies that

  * •

$p_{n},q_{n}$ and $x_{n}$ converge exponentially to $p,q,x$;

  * •

$T_{p_{n}}$ and $T_{q_{n}}$ converge to ${\mathcal{W}}^{c}_{2\delta}(p)$ and
${\mathcal{W}}^{c}_{2\delta}(q)$ in the $C^{1}$ topology;

  * •

$h^{s}_{4\lambda_{s}^{n}\delta}({\mathcal{W}}^{cu}_{loc}(p))\subset
f^{n}\tilde{\mathcal{W}}_{n}$ converges to ${\mathcal{W}}^{cu}_{loc}(p)$ in
the $C^{1}$ topology;

  * •

$f^{n}_{*}{\mathcal{F}}_{n,loc}$ converges to ${\mathcal{W}}^{u}_{loc}$ in the
following sense: if $a_{n}$ converges to $a$ then
$f^{n}_{*}{\mathcal{F}}_{n,loc}(a_{n})$ converges to
${\mathcal{W}}^{u}_{loc}(a)$ in the $C^{1}$ topology.

Let
$T_{p_{n}}^{\prime}=h^{s}_{4\lambda_{s}^{n}\delta}({\mathcal{W}}^{c}_{\delta}(p))\subset
T_{p_{n}}$. Then for $n$ sufficiently large there exists a well defined
holonomy of the foliation $f^{n}_{*}{\mathcal{F}}_{n}$ between the
transversals $T_{p_{n}}^{\prime}$ and $T_{q_{n}}$. We denote this holonomy
$h_{p_{n},q_{n}}^{f^{n}_{*}{\mathcal{F}}_{n}}$ and observe that it is
$C^{1+\theta}$.

Define
$h^{n}_{p,q}:{\mathcal{W}}^{c}_{\delta}(p)\rightarrow{\mathcal{W}}^{c}_{2\delta}(q)$,

(5)
$h^{n}_{p,q}=h^{s^{-1}}_{4\lambda_{s}^{n}\delta}\circ(h^{f^{n}_{*}{\mathcal{F}}_{n}}_{p_{n},q_{n}})\circ
h^{s}_{4\lambda_{s}^{n}\delta},$

In other words, in order to obtain $h^{n}_{p,q}(x)$ for
$x\in{\mathcal{W}}^{c}_{\delta}(p)$, we move with the stable holonomy of size
$4\lambda_{s}^{n}\delta$ to $T_{p_{n}}^{\prime}\subset
f^{n}\tilde{\mathcal{W}}_{n}$, then we move with the holonomy given by the
foliation $f_{*}^{n}{\mathcal{F}}_{n}$ of $f^{n}(\tilde{\mathcal{W}}_{n})$
between the $C^{1}$ transversals $T_{p_{n}}^{\prime}$ and $T_{q_{n}}$, and
then we move back by the stable holonomy of size $4\lambda_{s}^{n}\delta$ to
${\mathcal{W}}^{c}_{2\delta}(q)$.

Clearly $h^{n}_{p,q}$ is continuous, since the stable holonomies are Hölder
continuous, while the holonomy $h^{f^{n}_{*}{\mathcal{F}}_{n}}_{p_{n},q_{n}}$
is $C^{1+\theta}$.

###  3.2. $h^{n}_{p,q}$ converges uniformly to $h^{u}_{p,q}$.

This follows immediately from the remarks in the previous section.

###  3.3. $h^{u}_{p,q}$ is Lipschitz.

We first show that $Dh^{f^{n}_{*}{\mathcal{F}}_{n}}_{p_{n},q_{n}}(x_{n})$ is
bounded uniformly in $n,x_{n}$.

Let $\tilde{\Delta}_{n}=\Delta^{cs}_{\epsilon_{0}}\cap
T\tilde{\mathcal{W}}_{n}\subset\Delta^{c}_{2\epsilon_{0}}$ be a cone field
tangent to ${\mathcal{W}}_{n}$. Let
$E_{\tilde{x}_{n}}=E^{s}_{\tilde{x}_{n}}\oplus
T_{\tilde{x}_{n}}{\mathcal{F}}_{n}$. Since the cone fields
$\Delta^{*}_{2\epsilon_{0}}$ are uniformly transverse at the scale
$\delta_{0}$, we have $t(E_{\tilde{x}_{n}},\tilde{\Delta}_{n},\delta)<2$ and
$t({\mathcal{F}}_{n},\tilde{\Delta}_{n})<2$. In view of the Lemma 2.5 and
Lemma 2.1 we have that $Dh^{{\mathcal{F}}_{n}}$ is
($C_{{\mathcal{F}}},\theta$)-Hölder along ${\mathcal{F}}_{n}$ at
$\tilde{x}_{n}$ with respect to $\tilde{\Delta}_{n}$, $E_{\tilde{x}_{n}}$ and
at scale $\delta$, for some constant $C_{{\mathcal{F}}}$ independent on
$p,n,\tilde{x}_{n}$ and $g$ in $\mathcal{U}(f)$.

Let $\tilde{\Delta}_{n}^{k}=\Delta^{cs}_{\epsilon_{0}}\cup
Tf^{k}\tilde{\mathcal{W}}_{n}\subset\Delta^{c}_{2\epsilon_{0}}$ be a cone
field tangent to $f^{k}{\mathcal{W}}_{n}$. Observe that
$\tilde{\Delta}_{n}^{k+1}\subset f_{*}\tilde{\Delta}_{n}^{k}$ because of the
backward invariance of $\Delta^{cs}_{\epsilon_{0}}$. Since
$Tf_{*}^{k}{\mathcal{F}}_{n}$ stays tangent to $\Delta^{u}_{\epsilon_{0}}$, we
also have uniform transversality between $\tilde{\Delta}_{n}^{k}$ and both
$f_{*}^{k}{\mathcal{F}}_{n}$ and $f_{*}^{k}E_{\tilde{x}_{n}}$ at scale
$\delta$: $t(f_{*}^{k}E_{\tilde{x}_{n}},\tilde{\Delta}_{n}^{k},\delta)<2$ and
$t(f_{*}^{k}{\mathcal{F}}_{n},\tilde{\Delta}_{n}^{k})<2$. Due to the fact that
${\mathcal{F}}_{n}$ is uniformly expanding, we can apply successively Lemma
2.2 and using the bunching condition we conclude that
$Dh^{f_{*}^{n}{\mathcal{F}}_{n}}$ is ($C_{0},\theta$)-Hölder along
$f_{*}^{n}{\mathcal{F}}_{n}$ at $f^{n}(\tilde{x}_{n})=x_{n}$ with respect to
$\tilde{\Delta}_{n}^{n}$, $f_{*}^{n}E_{\tilde{x}_{n}}$ and at scale $\delta$,
for the constant
$C_{0}=C_{{\mathcal{F}}}+\frac{4\|Df\|_{C^{\theta}}}{(1-\mu)\|Df\|}$
independent of $p,n,\tilde{x}_{n}$. The constant also works for $g$ within the
neighborhood $\mathcal{U}(f)$ of $f$ (eventually readjusting $\mathcal{U}(f)$
or $C_{0}$). Then $Dh^{f^{n}_{*}{\mathcal{F}}_{n}}_{p_{n},q_{n}}(x_{n})$ is
bounded uniformly by some constant $L_{0}$, so
$h^{f^{n}_{*}{\mathcal{F}}_{n}}_{p_{n},q_{n}}$ is Lipschitz with constant
$L_{0}$ uniformly in $n$.

Now the fact that $h^{u}_{p,g}$ is Lipschitz is just a simple consequence of
the fact that $h^{f^{n}_{*}{\mathcal{F}}_{n}}_{p_{n},q_{n}}$ are uniformly
Lipschitz. We have that
$d(h^{f^{n}_{*}{\mathcal{F}}_{n}}_{p_{n},q_{n}}(x_{n}),h^{f^{n}_{*}{\mathcal{F}}_{n}}_{p_{n},q_{n}}(x_{n}^{\prime}))\leq
L_{0}d(x_{n},x_{n}^{\prime})$ uniformly in $n$. Since $x_{n}$ converges to
$x$, $x_{n}^{\prime}$ converges to $x^{\prime}$,
$h^{f^{n}_{*}{\mathcal{F}}_{n}}_{p_{n},q_{n}}(x_{n})$ converges to
$h^{u}_{p,q}(x)$ and
$h^{f^{n}_{*}{\mathcal{F}}_{n}}_{p_{n},q_{n}}(x_{n}^{\prime})$ converges to
$h^{u}_{p,q}(x^{\prime})$, it follows that
$d(h^{u}_{p,q}(x),h^{u}_{p,q}(x^{\prime}))\leq L_{0}d(x,x^{\prime})$.

###  3.4. Estimate on the Lipschitz jet of $h^{u}_{p,q}$.

Let us remind the definition of Lipschitz jets. Let $M,N$ be two metric
spaces, $p\in M$, $q\in N$. Two functions $f,g:M\rightarrow N$ such that
$f(p)=g(p)=q$ are equivalent if $\limsup_{x\rightarrow
p}\frac{d(f(x),g(x))}{d(x,p)}=0$. The equivalence classes form the space
$J(M,p,N,q)$ of Lipschitz jets at $p,q$. The distance between two Lipschitz
jets is $d(J(f),J(g))=\limsup_{x\rightarrow p}\frac{d(f(x),g(x))}{d(x,p)}$, it
can be infinite and is independent of the representatives $f$ and $g$. A
Lipschitz jet is bounded if the distance to the jet of the constant function
is finite. The space of bounded Lipschitz jets at $p,q$, $J^{b}(M,p,N,q)$, is
a complete metric space. If $M,N$ are differentiable manifolds, then the space
of differentiable Lipschitz jets at $p,q$, $J^{d}(M,p,N,q)$, is formed by the
jets which have a representative which is differentiable. $J^{d}(M,p,N,q)$ is
a closed subspace of $J^{b}(M,p,N,q)$.

For simplicity let us denote $D_{x_{n}}=T_{x_{n}}T_{p_{n}}$,
$D_{y_{n}}=T_{y_{n}}T_{q_{n}}$, where
$y_{n}=h^{f^{n}_{*}{\mathcal{F}}_{n}}_{p_{n},q_{n}}(x_{n})$,
$E_{x_{n}}=f_{*}^{n}E_{\tilde{x}_{n}}$,
$\overline{h}^{n}=h^{f^{n}_{*}{\mathcal{F}}_{n}}_{p_{n},q_{n}}$. We have

(6) $\|D\overline{h}^{n}(x_{n})-p^{E_{x_{n}}}_{D_{x_{n}},D_{y_{n}}}\|\leq
C_{0}d(x_{n},y_{n})^{\theta}$

In particular we have that $D_{x_{n}}$ and $D_{y_{n}}$ converge exponentially
to $E^{c}(x),E^{c}(z)$ when $n$ goes to infinity, while $E_{x_{n}}$ converges
exponentially to $E^{s}(x)\oplus E^{u}(x)$.

We will work in an exponential chart at $p_{n}$, and we will make an abuse of
notation keeping the notation of the points. Let $B_{p_{n}},B_{q_{n}}$ be the
balls or radius $\delta$ in $D_{p_{n}},D_{q_{n}}$. We can choose
$C^{1+\alpha}$ maps $\sigma_{p_{n}}:B_{p_{n}}\rightarrow T_{p_{n}}$ and
$\sigma_{q_{n}}:B_{q_{n}}\rightarrow T_{q_{n}}$ such that

  * •

$p_{n}+x^{\prime}-\sigma_{p_{n}}(x^{\prime})\in E_{p_{n}},\ \forall
x^{\prime}\in B_{p_{n}}$;

  * •

$q_{n}+y^{\prime}-\sigma_{q_{n}}(y^{\prime})\in E_{p_{n}},\ \forall
y^{\prime}\in B_{q_{n}}$.

In other words they are parametrisations of $T_{p_{n}},T_{q_{n}}$ given by the
projection from $B_{p_{n}},B_{q_{n}}$ parallel to $E_{p_{n}}$. Using them we
can define $g_{n}:T_{p_{n}}^{\prime}\rightarrow T_{q_{n}}$,
$g_{n}=\sigma_{q_{n}}\circ
p^{E_{p_{n}}}_{D_{p_{n}},D_{q_{n}}}\circ\sigma_{p_{n}}^{-1}$. This means that
$g_{n}$ has $Dg_{n}(p_{n})=p^{E_{p_{n}}}_{D_{p_{n}},D_{q_{n}}}$. We will
analyze the Lipschitz jets of $\overline{h}_{n}$ and $g_{n}$ at $p_{n}$.

We will use the notations
$x_{n}=\sigma_{p_{n}}(x_{n}^{\prime}),y_{n}=\sigma_{q_{n}}(y_{n}^{\prime})$.
We can see that

  * •

$\sigma_{p_{n}}(0)=p_{n},\ \sigma_{q_{n}}(0)=q_{n}$;

  * •

$D\sigma_{p_{n}}(0)=Id_{D_{p_{n}}},\ D\sigma_{q_{n}}(0)=Id_{D_{q_{n}}}$;

  * •

$D\sigma_{p_{n}}(x_{n}^{\prime})=p^{E_{p_{n}}}_{D_{p_{n}},D_{x_{n}}},\
D\sigma_{q_{n}}(y_{n}^{\prime})=p^{E_{p_{n}}}_{D_{q_{n}},D_{y_{n}}}$;

Let
$G_{n}=\sigma_{q_{n}}^{-1}\circ\overline{h}^{n}\circ\sigma_{p_{n}}-p^{E_{p_{n}}}_{D_{p_{n}},D_{q_{n}}}:B_{p_{n}}\rightarrow
D_{q_{n}}$. We have that $G_{n}(0)=0$ and $\|DG_{n}(0)\|\leq
C_{0}d(p_{n},q_{n})^{\theta}$. We have

$\displaystyle\|DG_{n}(x_{n}^{\prime})\|$
$\displaystyle=\|D\sigma_{q_{n}}^{-1}\circ D\overline{h}^{n}(x_{n})\circ
D\sigma_{p_{n}}(x_{n}^{\prime})-p^{E_{p_{n}}}_{D_{p_{n}},D_{q_{n}}}\|$
$\displaystyle=\|p^{E_{p_{n}}}_{D_{q_{n}}}\circ(D\overline{h}^{n}(x_{n})-p^{E_{x_{n}}}_{D_{x_{n}},D_{y_{n}}})\circ
p^{E_{p_{n}}}_{D_{x_{n}}}+p^{E_{p_{n}}}_{D_{q_{n}}}\circ(p^{E_{x_{n}}}_{D_{x_{n}},D_{y_{n}}}-p^{E_{p_{n}}}_{D_{x_{n}},D_{y_{n}}})\circ
p^{E_{p_{n}}}_{D_{x_{n}}}\|$
$\displaystyle\leq\|p^{E_{p_{n}}}_{D_{q_{n}}}\|\cdot\|p^{E_{p_{n}}}_{D_{x_{n}}}\|\cdot(C_{0}d(x_{n},y_{n})^{\theta}+2d(E_{p_{n}},E_{x_{n}}))$
$\displaystyle\leq 4(C_{0}d(x_{n},y_{n})^{\theta}+2d(E_{p_{n}},E_{x_{n}})).$

There exists $\gamma>0$ depending on $d(p,q)$ such that for all $n$
sufficiently large and all $x_{n}\in T_{p_{n}}^{\prime}$ with
$d(x_{n},p_{n})<\gamma$, we have

  * •

$d(x_{n},y_{n})<2d(p,q)$;

  * •

$8d(E_{p_{n}},E_{x_{n}}))<C_{0}d(p,q)^{\theta}$.

We deduce that if $d(x_{n},p_{n})<\gamma$ then
$\|DG(x_{n}^{\prime})\|<5C_{0}d(p,q)^{\theta}$, or $G$ is Lipschitz with
constant $5C_{0}d(p,q)^{\theta}$. Then

$d(\sigma_{q_{n}}^{-1}\circ\overline{h}^{n}\circ\sigma_{p_{n}}(x_{n}^{\prime}),p^{E_{p_{n}}}_{D_{p_{n}},D_{q_{n}}}(x_{n}^{\prime}))=d(G(x_{n}^{\prime}),G(0))\leq
5C_{0}d(p,q)^{\theta}d(x_{n}^{\prime},0)\leq
10C_{0}d(p,q)^{\theta}d(x_{n},p_{n}).$

and furthermore

$\displaystyle\sup_{d(x_{n},p_{n})<\gamma}\frac{d(\overline{h}^{n}(x_{n}),g_{n}(x_{n}))}{d(x_{n},p_{n})}$
$\displaystyle\leq$ $\displaystyle
Lip(\sigma_{q_{n}})\sup_{d(x_{n},p_{n})<\gamma}\frac{d(\sigma_{q_{n}}^{-1}\circ\overline{h}^{n}(x_{n}),\sigma_{q_{n}}^{-1}\circ
g_{n}(x_{n}))}{d(x_{n},p_{n})}$ $\displaystyle\leq$ $\displaystyle
20C_{0}d(p,q)^{\theta}$

In other words $d(J(\overline{h}^{n}),J(g_{n}))\leq 20C_{0}d(p,q)^{\theta}$ in
$J^{b}(T_{p_{n}},p_{n},T_{q_{n}},q_{n})$ (in fact in
$J^{d}(T_{p_{n}},p_{n},T_{q_{n}},q_{n})$). Since $\gamma$ is independent of
$n$ this relation can be passed to the limit when $n$ goes to infinity and we
get

$\sup_{d(x,p)<\gamma}\frac{d(h^{u}_{p,q}(x),g_{p,q}(x))}{d(x,p)}\leq
20C_{0}d(p,q)^{\theta},$

where $g_{p,q}=\sigma_{q}\circ
p^{E^{su}(p)}_{E^{c}(p),E^{c}(q)}\circ\sigma_{p}^{-1}$. This means that
$d(J(h^{u}_{p,q}),J(g_{p,q}))\leq 20C_{0}d(p,q)^{\theta}$, for all $p\in M$
and $q\in{\mathcal{W}}^{u}_{\delta}(x)$.

######  Remark 3.1.

$g_{p,q}$ is differentiable and the derivative is
$p^{E^{su}(p)}_{E^{c}(p),E^{c}(q)}$. The bound obtained also works for the
neighborhood $\mathcal{U}(f)$.

###  3.5. $h^{u}_{p,q}$ is differentiable.

We will use the invariant section theorem. Let
$q\in{\mathcal{W}}^{u}_{\delta}(p)$, $q\neq p$. For simplicity let us denote
$g_{f^{n}(p),f^{n}(q)}=g_{n},p^{E^{su}(p)}_{E^{c}(p),E^{c}(q)}=\pi_{n}$. The
base is $\mathbb{Z}$ with the discrete topology, and the base map is $T$, the
translation by one. The fiber over $n$ is

$B_{n}=B(J(g_{n}),C_{1}d(f^{n}(p),f^{n}(q))^{\theta})\subset
J^{b}({\mathcal{W}}^{c}_{\delta}(f^{n}(p)),f^{n}(p),{\mathcal{W}}^{c}_{\delta}(f^{n}(q)),f^{n}(q))$

if $n\leq 0$, where $C_{1}=20C_{0}$. In particular we have
$J(h^{u}_{f^{n}(p),f^{n}(q)})\in B_{n}$. Observe that since $C_{1}>C_{0}$ we
have

(7) $\mu C_{1}+\frac{4\mu\|Df\|_{C^{\theta}}}{\|Df\|}<C_{1}.$

The subset $\mathbb{Z}^{-}=\mathbb{Z}\setminus\mathbb{N}$ is overflowed by
$T$. The bundle map is

$F(n,J(h))=(n+1,J(f\circ h\circ f^{-1})).$

We claim that $F$ is well defined. For this we have to prove that if
$d(J(h),J(g_{n}))\leq C_{1}d(f^{n}(p),f^{n}(q))$ then $d(J(f\circ h\circ
f^{-1}),J(g_{n+1}))\leq C_{1}d(f^{n+1}(p),f^{n+1}(q))$. Observe that

$d(J(f\circ h\circ f^{-1}),J(g_{n+1}))\leq d(J(f\circ h\circ f^{-1}),J(f\circ
g_{n}\circ f^{-1}))+d(J(f\circ g_{n}\circ f^{-1}),J(g_{n+1})).$

On one hand we have

$\displaystyle d(J(f\circ h\circ f^{-1}),J(f\circ g_{n}\circ f^{-1}))$
$\displaystyle\leq Lip(f,f^{n}(q))\cdot d(J(h),J(g_{n}))\cdot
Lip(f^{-1},f^{n+1}(p))$
$\displaystyle\leq\frac{\lambda^{+}_{\Delta^{c}_{2\epsilon_{0}}}(f,p,\delta_{0})}{\lambda^{-}_{\Delta^{c}_{2\epsilon_{0}}}(f,p,\delta_{0})\lambda^{-}_{\Delta^{u}_{2\epsilon_{0}}}(f,p,\delta_{0})^{\theta}}C_{1}d(f^{n+1}(p),f^{n+1}(q))^{\theta}$
$\displaystyle\leq\mu C_{1}d(f^{n+1}(p),f^{n+1}(q))^{\theta}.$

On another hand, since $g_{n}$ and $g_{n+1}$ are differentiable, we have

$\displaystyle d(J(f\circ g_{n}\circ f^{-1}),J(g_{n+1}))$
$\displaystyle=\|D(f\circ g_{n}\circ f^{-1})-Dg_{n+1}\|$
$\displaystyle=\|p^{E^{su}_{f^{n+1}(p)}}_{E^{c}_{f^{n+1}(q)}}\cdot(Df(f^{n}(q))-Df(f^{n}(p)))\cdot
p^{E^{su}_{f^{n}(p)}}_{E^{c}_{f^{n}(q)}}\cdot Df^{-1}|_{E^{c}_{f^{n+1}(p)}}\|$
$\displaystyle\leq\frac{4\mu\|Df\|_{C^{\theta}}}{\lambda^{-}_{\Delta^{c}_{2\epsilon_{0}}}(f,p,\delta_{0})\lambda^{-}_{\Delta^{u}_{2\epsilon_{0}}}(f,p,\delta_{0})^{\theta}}d(f^{n+1}(p),f^{n+1}(q))^{\theta}$
$\displaystyle\leq\frac{4\mu\|Df\|_{C^{\theta}}}{\|Df\|}d(f^{n+1}(p),f^{n+1}(q))^{\theta}.$

The estimates above together with the condition 7 imply that $F$ is indeed
well defined.

Next we modify the distance inside each fiber $B_{n}$, we let
$d_{n}=\frac{d}{d(f^{n}(p),f^{n}(q))}$. Let $\Sigma^{b}$ be the space of
sections over $\mathbb{Z}^{-}$, with the supremum distance
$d_{\sup}=\sup_{n\in\mathbb{Z}^{-}}d_{n}$. It is clear that
$(\Sigma^{b},d_{\sup})$ is a complete metric space. We claim that $F$ is a
uniform bundle contraction over $\mathbb{Z}^{-}$.

Let $J(\sigma),J(\sigma^{\prime})\in B_{n}$. Then

$\displaystyle d_{n+1}(J(f\circ\sigma\circ
f^{-1}),J(f\circ\sigma^{\prime}\circ f^{-1}))$
$\displaystyle=\frac{d_{n+1}(J(f\circ\sigma\circ
f^{-1}),J(f\circ\sigma^{\prime}\circ f^{-1}))}{d(f^{n+1}(p),f^{n+1}(q))}$
$\displaystyle\leq Lip(f,f^{n}(q))\cdot
Lip(f^{-1},f^{n+1}(p))\cdot\frac{d(f^{n}(p),f^{n}(q))}{d(f^{n+1}(p),f^{n+1}(q))}\cdot$
$\displaystyle\ \ \ \ \ \ \
\cdot\frac{d(J(\sigma),J(\sigma^{\prime}))}{d(f^{n}(p),f^{n}(q))}$
$\displaystyle\leq\frac{\lambda^{+}_{\Delta^{c}_{2\epsilon_{0}}}(f,p,\delta_{0})}{\lambda^{-}_{\Delta^{c}_{2\epsilon_{0}}}(f,p,\delta_{0})\lambda^{-}_{\Delta^{u}_{2\epsilon_{0}}}(f,p,\delta_{0})^{\theta}}d_{n}(\sigma,\sigma^{\prime})$
$\displaystyle\leq\mu d_{n}(\sigma,\sigma^{\prime}).$

This shows that $F$ induces a contraction on $\Sigma^{b}$, so there exists a
unique invariant bounded section $\sigma(n)\in B_{n}$.

$B_{n}\cap
J^{d}({\mathcal{W}}^{c}_{\delta}(f^{n}(p)),f^{n}(p),{\mathcal{W}}^{c}_{\delta}(f^{n}(q)),f^{n}(q))$
is a closed nonempty subset of $B_{n}$, so we can apply again the invariant
section theorem to this closed sub-bundle, which is clearly preserved by $F$,
and we get that the unique invariant section must contain actually
differentiable jets at all points.

We can check that the jet of the holonomy is also an invariant bounded section
of $F$. Uniqueness of the invariant section implies then that the holonomy is
differentiable at every points $p,q\in{\mathcal{W}}^{u}_{\delta}(p)$, and
satisfies

$\|Dh^{u}_{p,q}(p)-p^{E^{su}_{p}}_{E^{c}_{p},E^{c}_{q}}\|\leq
C_{1}d(p,q)^{\theta}.$

######  Remark 3.2.

We proved the differentiability of the unstable holonomy between (nearby)
center leaves. However we can adapt the proof for any two transversals to
${\mathcal{W}}^{u}$ inside a center-unstable leaf. A sketch of the proof is
the following.

Let $T_{p},T_{q}$ be two $C^{1}$ transversals to ${\mathcal{W}}^{u}$
restricted to ${\mathcal{W}}^{cu}(p)$, and denote $D_{p}$ and $D_{q}$ their
tangent planes in $p,q$. Assume that
$D_{p},D_{q}\in\Delta^{c}_{\frac{\epsilon_{0}}{4}}$ and $d(p,q)<\delta/4$
(otherwise iterate back a finite number of times). Choose
$\tilde{\mathcal{W}}^{s}$ a smooth approximation of ${\mathcal{W}}^{s}$ in a
tubular neighborhood of ${\mathcal{W}}^{cu}_{\delta}(p)$. The local
$\tilde{\mathcal{W}}^{s}$ holonomy takes $T_{p},T_{q}$ to the $C^{1}$
transversals $T_{p}^{n},T_{q}^{n}$ to $f_{*}^{n}{\mathcal{F}}_{n}$ inside
$f^{n}{\mathcal{W}}_{n}$. If $n$ is sufficiently large,
$f^{n}{\mathcal{W}}_{n}$ is close to ${\mathcal{W}}^{cu}(p)$, and the local
$\tilde{\mathcal{W}}^{s}$ holonomy takes $D_{p},D_{q}$ to subspaces
$D_{p}^{n},D_{q}^{n}$ inside $\Delta^{c}_{\frac{\epsilon_{0}}{2}}$. We do have
again the uniform control of the regularity of the
$f_{*}^{n}{\mathcal{F}}_{n}$ holonomy between $T_{p}^{n}$ and $T_{q}^{n}$, so
we can pass it to the limit as before and obtain that the unstable holonomy
between $T_{p}$ and $T_{q}$ is differentiable, with

$\|Dh^{u}_{T_{p},T_{q}}(p)-p^{E^{su}_{p}}_{D_{p},D_{q}}\|\leq
C_{1}d(p,q)^{\theta}.$

In other words $Dh^{u}$ is ($C_{1},\theta$)-Hölder along ${\mathcal{W}}^{u}$
at $p$ with respect to $E^{su}_{p},\Delta^{c}_{\frac{\epsilon_{0}}{4}}$ and at
scale $\delta/4$ for all $p\in M$. The result holds for the neighborhood
$\mathcal{U}(f)$.

###  3.6. $Dh^{u}_{p,q}$ is continuous in $p,q,f$.

We will apply again the invariant section theorem in yet another space. First
let us refine the bunching bound from 4. Choose $\mu<\mu^{\prime}<1$. Since
$E^{c}$ is uniformly $C^{\alpha}$ in a neighborhood of $f$, there exists
$0<\delta^{\prime}<\delta$ such that for all $g\in\mathcal{U}(f),p,q\in
M,d(p,q)\leq\delta^{\prime}$, we have

$\|p^{E^{su}_{p,g}}_{E^{c}_{p,g},E^{c}_{q,g}}\|,\|p^{E^{su}_{p,g}}_{E^{c}_{q,g},E^{c}_{p,g}}\|<\sqrt{\frac{\mu^{\prime}}{\mu}}.$

The base space is $N=M^{2}\times\mathcal{U}(f)$, with the $C^{1}$ topology on
$\mathcal{U}(f)$. The base map is $G(p,q,g)=(g(p),g(q),g)$, which is
continuous. At each $(p,q,g)\in N$ we consider the fiber
$\mathcal{E}_{p,q,g}=\mathcal{L}(E^{c}_{p,g})$, the linear maps from
$E^{c}_{p,g}$ to itself, with the usual norm given by the Riemannian metric.
Since the center bundle is continuous with respect to the point and the map,
we obtain a continuous Banach bundle $\mathcal{E}$ over $N$. Let
$N^{\prime}=\\{(p,q,g)\in N:\
q\in{\mathcal{W}}^{u}_{\delta^{\prime}}(p,g)\\}$. Clearly $N^{\prime}$ is
overflowed by $G$.

Let

$\|\sigma\|_{b}=\sup_{(p,q,g)\in
N^{\prime}}\frac{\|\sigma(p,q,g)\|}{d(p,q)^{\theta}}.$

and let $\Sigma^{b}$ the space of sections in $\mathcal{E}$ over $N^{\prime}$
bounded in $\|\cdot\|_{b}$ (in particular $\sigma\in\Sigma^{b}$ implies
$\sigma(p,p,g)=0$). This is a complete metric space.
$\Sigma^{c}\cap\Sigma^{b}$ is the space of the sections which are both
continuous and bounded in $\|\cdot\|_{b}$, this is a closed nonempty subset of
$\Sigma^{b}$ (it contains the zero section).

The bundle map is

$(T\sigma)(g(p),g(q),g)=p^{E^{su}_{g(p),g}}_{E^{c}_{g(q),g}E^{c}_{g(p),g}}\circ
Dg(q)|_{E^{c}_{q,g}}\circ
p^{E^{su}_{p,g}}_{E^{c}_{p,g},E^{c}_{q,g}}\circ(Id+\sigma(p,q,g))\circ
Dg(p)|_{E^{c}_{p,g}}^{-1}-Id.$

This is continuous in $p,q,g,\sigma$.

The connection with the holonomies is the following. If

$Id+\sigma(p,q,g)=p^{E^{su}_{p,g}}_{E^{c}_{q,g},E^{c}_{p,g}}\circ
H^{u}_{p,q,g},$

where $H^{u}_{p,q,g}:E^{c}_{p,g}\rightarrow E^{c}_{q,g}$ is the candidate for
the derivative of the holonomy, then

$Id+(T\sigma)(g(p),g(q),g)=p^{E^{su}_{g(p),g}}_{E^{c}_{g(q),g},E^{c}_{g(p),g}}\circ
g_{*}H^{u}_{p,q,g}=p^{E^{su}_{g(p),g}}_{E^{c}_{g(q),g},E^{c}_{g(p),g}}\circ
Dg(q)|_{E^{c}_{q,g}}\circ H^{u}_{p,q,g}\circ Dg(p)|_{E^{c}_{p,g}}^{-1}.$

Let us check that $T$ applied to the zero section is in $\Sigma^{b}$. We
remark first that

$Id=p^{E^{su}_{g(p),g}}_{E^{c}_{g(q),g}E^{c}_{g(p),g}}\circ
Dg(p)|_{E^{c}_{q,g}}\circ p^{E^{su}_{p,g}}_{E^{c}_{p,g},E^{c}_{q,g}}\circ
Dg(p)|_{E^{c}_{p,g}}^{-1}.$

Then

$\displaystyle\|T0\|_{b}$ $\displaystyle=\sup_{(g(p),g(q),g)\in
N^{\prime}}\frac{\|p^{E^{su}_{g(p),g}}_{E^{c}_{g(q),g},E^{c}_{g(p),g}}\circ
Dg(q)|_{E^{c}_{q,g}}\circ p^{E^{su}_{p,g}}_{E^{c}_{p,g},E^{c}_{q,g}}\circ
Dg(p)|_{E^{c}_{p,g}}^{-1}-Id\|}{d(g(p),g(q))^{\theta}}$
$\displaystyle\leq\sup_{(p,q,g)\in
N^{\prime}}\frac{\|p^{E^{su}_{g(p),g}}_{E^{c}_{g(p),g}}\circ(Dg(q)-Dg(p))|_{E^{c}_{q,g}}\circ
p^{E^{su}_{p,g}}_{E^{c}_{p,g},E^{c}_{q,g}}\circ
Dg(p)|_{E^{c}_{p,g}}^{-1}\|}{d(g(p),g(q))^{\theta}}$
$\displaystyle\leq\sup_{(p,q,g)\in
N^{\prime}}\frac{4\|Dg\|_{C^{\theta}}}{\lambda^{-}_{\Delta^{c}_{2\epsilon_{0}}}(g,p,\delta^{\prime})\lambda^{-}_{\Delta^{u}_{2\epsilon_{0}}}(g,p,\delta^{\prime})^{\theta}}$
$\displaystyle\leq\frac{4\mu\|Dg\|_{C^{\theta}}}{\|Dg\|}.$

Now let us check that $T$ is a contraction in $\Sigma^{b}$.

$\displaystyle\|T\sigma_{1}-T\sigma_{2}\|_{b}$
$\displaystyle=\sup_{N^{\prime}}\frac{\|p^{E^{su}_{g(p),g}}_{E^{c}_{g(q),g},E^{c}_{g(p),g}}\circ
Dg(q)|_{E^{c}_{q,g}}\circ
p^{E^{su}_{p,g}}_{E^{c}_{p,g},E^{c}_{q,g}}\circ(\sigma_{1}-\sigma_{2})(p,q,g))\circ
Dg(p)|_{E^{c}_{p,g}}^{-1}\|}{d(g(p),g(q))^{\theta}}$
$\displaystyle\leq\frac{\mu^{\prime}\lambda^{+}_{\Delta^{c}_{2\epsilon_{0}}}(g,p,\delta^{\prime})}{\mu\lambda^{-}_{\Delta^{c}_{2\epsilon_{0}}}(g,p,\delta^{\prime})\lambda^{-}_{\Delta^{u}_{2\epsilon_{0}}}(g,p,\delta^{\prime})^{\theta}}\|\sigma_{1}-\sigma_{2}\|_{b}$
$\displaystyle\leq\mu^{\prime}\|\sigma_{1}-\sigma_{2}\|_{b}.$

Since $\Sigma^{b}$ is a complete metric space, we obtain that there is a
unique invariant section in $\Sigma^{b}$. Continuous sections are preserved by
$T$, so we can also apply the Banach fixed point Theorem in
$\Sigma^{b}\cap\Sigma^{d}$, and we obtain that the unique invariant section in
$\Sigma^{d}$ is in fact continuous. On the other hand the section

$\sigma^{u}(p,q,g)=p^{E^{su}_{p,g}}_{E^{c}_{q,g},E^{c}_{p,g}}\circ
Dh^{u}_{p,q,g}-Id$

is an invariant section of $T$ inside $\Sigma^{b}$, so it must be the unique
invariant section. Since $p^{E^{su}_{p,g}}_{E^{c}_{q,g},E^{c}_{p,g}}$ is
continuous in $p,q,g$, we obtain that $Dh^{u}$ is also continuous in $p,q,g$,
which finishes the proof of Theorem 1.1.

######  Remark 3.3.

If we consider the restriction to the base space $M^{2}\times\\{f\\}$, then we
have a Hölder map in a Hölder bundle, so the invariant section theorem will
provide us with a Hölder continuous invariant section, which means that
$Dh^{u}_{p,q,f}$ is actually Hölder in $p,q$.

###  3.7. Proof of Corollary 1.1

The proof is similar to the proof of Theorem 1.1. The space is not compact (it
is a disjoint union of $\mathbb{R}^{d}$), but the bounds are uniform. The
invariant foliations are globally defined graphs so in this case the
approximation of the pair (${\mathcal{W}}^{cu},{\mathcal{W}}^{u}$) is actually
much easier. We can take $\tilde{\mathcal{W}}^{cu}$ to be the $cu$-subspace
passing through the origin, and the subfoliation ${\mathcal{F}}$ to be the
subfoliation by $u$-subspaces. For more details on fake foliations we send the
reader to [4].

###  3.8. Proof of Theorem 1.2

The proof is actually contained in the section 3.6. Even if we don’t know that
there exists a (differentiable) holonomy between center leaves, we still
obtain a continuous invariant section $\sigma^{u}$ of $T$, and then
$H^{u}_{p,q,g}=p^{E^{su}_{p,g}}_{E^{c}_{p,g},E^{c}_{q,g}}\circ(\sigma^{u}_{p,q,g}+Id)$
is the invariant continuous holonomy we are looking for, at least at the scale
$\delta^{\prime}$. In order to define it for all $q\in{\mathcal{W}}^{u}(p)$ we
iterate forward and use invariance under $f$. Doing this we have automatically
the invariance under $f$ and the continuity with respect to the points. To
prove that $H^{u}_{q,r}\circ H^{u}_{p,q}=H^{u}_{p,r}$ we can use the
uniqueness of the invariant section. If the relation does not hold, we can
modify the invariant section $\sigma$ along the orbit of $(p,r)$, replacing it
with the $\sigma^{\prime}$ corresponding to $H^{u}_{q,r}\circ H^{u}_{p,q}$.
Then the invariant section $\sigma$ is not unique, which is a contradiction.

## References

  * [1] A. Avila, J. Santamaria and M. Viana, Holonomy invariance: rough regularity and applications to Lyapunov exponents. Astérisque 358 (2013), 13–74.
  * [2] A. Brown, Smoothness of stable holonomies inside center-stable manifolds. Erg. Th. Dynam. Syst. 42 (2022), no. 12, 3593–3618.
  * [3] K. Burns and A. Wilkinson, A note on stable holonomy between centers. (preliminary version.) Notes on A. Wilkinson’s personal webpage, 2005.
  * [4] K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems. Annals of Mathematics (2) 171 (2010), no. 1, 451–489.
  * [5] S. Crovisier, D. Obata and M. Poletti, Uniqueness of $u$-Gibbs measures for hyperbolic skew products on $\mathbb{T}^{4}$. preprint arXiv:2209.09151
  * [6] M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds. Springer Lecture Notes in Mathematics, 583 (1977).
  * [7] B. Kalinin and V. Sadovskaya, Cocycles with one exponent over partially hyperbolic systems. Geom. Dedicata 167 (2013), 167-188.
  * [8] M. Leguil and L. Piñeyrúa, Accesibility for dynamically coherent partially hyperbolic diffeomorphisms with 2D center. preprint arXiv:2112.12762
  * [9] C. Liang, K. Marin, and J. Yang, Lyapunov exponents of partially hyperbolic volume preserving maps with 2-dimensional center bundle. Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), no. 6, 1687–1706.
  * [10] C. Liang, K. Marin and J. Yang, $C^{1}$-openness of non-uniform hyperbolic diffeomorphisms with bounded $C^{2}$ norm. Erg. Th Dynam. Syst., 40 (2020), no. 11, 3078–3104.
  * [11] C. Liang, R. Saghin, F. Yang and J. Yang, Shub’s example revisited. preprint arXiv:2303.17775
  * [12] K. Marin, $C^{r}$-density of (non-uniform) hyperbolicity in partially hyperbolic symplectic diffeomorphisms. Comment. Math. Helv. 91 (2016), no. 2, 357–396.
  * [13] D. Obata, On the holonomies of strong stable foliations. Notes on D. Obata’s personal webpage, 2018.
  * [14] D. Obata, Open sets of partially hyperbolic skew products having a unique SRB measure. preprint arXiv:2107.12916v2 .
  * [15] C. Pugh, M. Shub and A. Wilkinson, Partial differentiability of invariant splittings. Journal of Statistical Physics 114 (2004), no. 3-4, 891–921.
  * [16] F. Rodriguez Hertz, M. A. Rodriguez Hertz, and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle. Invent. Math., 172 no. 2 (2008), 353–381.