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 $D_x=T_xT_x$ and $D_y=T_yT_y$ 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 $T_y$ , which 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 $A'$ , we will denote it as $p_{A',A}^B$ .

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) $$ \begin{align} \|Dh^{\mathcal F}_{D_x,D_y}(x)\!-\!p_{D_x,D_y}^{E_x}\| \leq C_{\mathcal F}d_{\mathcal F}(x,y)^{\theta}\quad \text{for all }y \in \mathcal F_{\delta}(x), \text{ for all } D_x \in \Delta_x, D_y \in \Delta_y. \end{align} $$

If instead of $\mathbb R^d$ we are in a smooth Riemannian manifold, the definition is similar, with the requirement that the condition in equation (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 $ :

$$ \begin{align*} t(E_x,\Delta,\delta)= \sup \bigg\{\frac 1{\sin(\angle(E_x,D_y))}:\ y\in{\mathcal W}_\delta(x), D_y\in\Delta_y\bigg\}. \end{align*} $$

In particular, we have

$$ \begin{align*} \|p_{D_y}^{E_x}\|\leq t(E_x,\Delta,\delta)\quad \text{for all } y \text{ such that } d(x,y)<\delta. \end{align*} $$

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

$$ \begin{align*} t({\mathcal F},\Delta)= \sup \bigg\{\frac 1{\sin(\angle(T_y{\mathcal F}(y),D_y))}:\ D_y\in\Delta_y\bigg\}. \end{align*} $$

We say that $\phi :\mathbb R^d\rightarrow \mathbb R^d$ is a linear parameterization of ( ${\mathcal W},{\mathcal F}$ ) if $\phi (\mathbb R^{\dim {\mathcal W}} \times \{0\}^{d-\dim {\mathcal W}})={\mathcal W}$ and $\phi (\mathbb R^{\dim {\mathcal F}}\times \{b\}\times \{0\}^{d-\dim {\mathcal W}})={\mathcal F}(\phi (0,b,0)) \text {for all } b\in \mathbb R^{\dim {\mathcal W}-\dim {\mathcal F}}$ ( $\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 parameterization of ( ${\mathcal W},{\mathcal F}$ ) at x.

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

$$ \begin{align*}Dh^{\tilde{\mathcal F}}_{\tilde D_x,\tilde D_y}=p^{\tilde E_x}_{\tilde D_x,\tilde D_y}\end{align*} $$

because $\tilde E_x$ contains the plane parallel to the linear foliation $\tilde {\mathcal F}$ . Denote $D'=D\phi (\tilde x)\tilde D_y$ and $\tilde D'=D\phi ^{-1}(x)D_y$ . Since $D\phi \tilde E_x=E_x$ , we have

$$ \begin{align*} p^{E_x}_{D_x,D_y}&=D\phi(\tilde x)|_{\tilde D'}\circ p^{\tilde E_x}_{\tilde D_x,\tilde D'}\circ D\phi^{-1}(x)|_{D_x}\\ &=D\phi(\tilde x)|_{\tilde D'}\circ p^{\tilde E_x}_{\tilde D_y,\tilde D'}\circ p^{\tilde E_x}_{\tilde D_x,\tilde D_y}\circ D\phi^{-1}(x)|_{D_x}\\ &=p^{E_x}_{D',D_y}\circ D\phi(\tilde x)|_{\tilde D_y}\circ p^{\tilde E_x}_{\tilde D_x,\tilde D_y}\circ D\phi^{-1}(x)|_{D_x}. \end{align*} $$

Then

$$ \begin{align*} \|Dh^{{\mathcal F}}_{D_x,D_y}-p^{E_x}_{D_x,D_y}\|&=\|D\phi(\tilde y)|_{\tilde D_y}\circ Dh^{\tilde{\mathcal F}}_{\tilde D_x,\tilde D_y}\circ D\phi^{-1}(x)|_{D_x}-p^{E_x}_{D_x,D_y}\kern-1.5pt\|\\&=\|(D\phi(\tilde y)|_{\tilde D_y}-p^{E_x}_{D',D_y}\circ D\phi(\tilde x)|_{\tilde D_y})\circ p^{\tilde E_x}_{\tilde D_x,\tilde D_y}\circ D\phi^{-1}(x)|_{D_x}\kern-1.5pt\|\\&\leq\|p^{E_x}_{D_y}\|\cdot\|D\phi(\tilde y)-D\phi(\tilde x)\|\cdot\|p^{\tilde E_x}_{\tilde D_y}\|\cdot\|D\phi^{-1}(x)\|\\&\leq\frac{t(E_x,\Delta,\delta)\cdot\|\phi\|_{C^{1+\theta}}\cdot\|\phi^{-1}\|_{C^1}^{1+\theta}\cdot \delta^{\theta}}{\sin(\angle(\tilde D_y,T\tilde F))}\\&\leq\|\phi\|_{C^{1+\theta}}^2\cdot\|\phi^{-1}\|_{C^1}^{2+\theta}\cdot t(E_x,\Delta,\delta)\cdot t({\mathcal F},\Delta)\cdot\delta^{\theta}. \end{align*} $$

We used the fact that

$$ \begin{align*} \sin(\angle(D_y,T_yF))\leq \sin(\angle(\tilde D_y,T\tilde F))\|D\phi\|\cdot\|D\phi^{-1}\kern-1.5pt\|.\\[-39pt] \end{align*} $$

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 notation for the bounds of $Df$ along $\Delta $ and $T\mathcal F$ :

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 2.3. 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 '\subset f_*\Delta $ and $\delta '<\unicode{x3bb} _{{\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 $\Delta ',f_*E_x$ and at scale $\delta '$ , where

(3)

Proof. Denote $E^{\prime }_x\hspace{-0.75pt}=\hspace{-0.75pt}Df(x)E_x, D^{\prime }_x\hspace{-0.75pt}=\hspace{-0.75pt}Df(x)D_x, D^{\prime }_y\hspace{-0.75pt}=\hspace{-0.75pt}Df(y)D_y, \tilde D\hspace{-0.75pt}=\hspace{-0.75pt}Df(x)D_y$ . Since $Df(x)$ takes the decomposition $E_x\oplus D_x$ to $E^{\prime }_x\oplus D^{\prime }_x$ , we have that

$$ \begin{align*} p_{D^{\prime}_x,D^{\prime}_y}^{E^{\prime}_x}\circ Df(x)|_{D_x}=p_{\tilde D,D^{\prime}_y}^{E^{\prime}_x}\circ Df(x)|_{D_y}\circ p_{D_x,D_y}^{E_x}. \end{align*} $$

We also have

For simplicity, we will use the notation $\unicode{x3bb} _{\Delta }^{\pm }, \unicode{x3bb} _{{\mathcal F}}$ . We have

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 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 a 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.

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

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 $\Delta ^{*'}_{2\epsilon _0}(y)$ are uniformly transverse for $d(x,y)<\delta _0$ : $t(\Delta ^*_{2\epsilon _0},\Delta ^{*'}_{2\epsilon _0},\delta _0)<2$ , where

  $$ \begin{align*} t(\Delta^*_{2\epsilon_0},\Delta^{*'}_{2\epsilon_0},\delta_0)= \sup \bigg\{\frac 1{\sin(\angle(D_1,D_2))}: d(x,y)\!<\!\delta_0, D_1\!\in\!\Delta^*_{2\epsilon_0}(x), D_2\!\in\!\Delta^{*'}_{2\epsilon_0}(y)\bigg\}; \end{align*} $$

• • the bunching condition holds at the ( $2\epsilon _0,\delta _0$ )-scale for all $g\in \mathcal U(f)$ , meaning that

  (4) $$ \begin{align} \frac{\unicode{x3bb}^+_{\Delta^c_{2\epsilon_0}}(g,x,\delta_0)}{\unicode{x3bb}^-_{\Delta^c_{2\epsilon_0}}(g,x,\delta_0) \unicode{x3bb}^-_{\Delta^u_{2\epsilon_0}}(g,x,\delta_0)^{\theta}}<\mu<1\quad \text{for all } g\in\mathcal U(f), \text{ for all } x\in M; \end{align} $$

• • 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 $T_yW\in \Delta (y)$ for all $y\in W$ .

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 2.7. 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 $U_1,U_2,\ldots ,U_k$ . Then $W^c_R(p)$ is covered by finitely many plaques ${\mathcal W}^c_{\delta _0}(x_i), 1\leq i\leq K$ from these foliation charts. Let $B_i={\mathcal W}^c_{3\delta _0}(x_i)$ and ${\mathcal W}^c_R(p)\subset {\mathcal W}_0=\bigcup _{i=1}^KB_i$ .

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 $\gamma '$ 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 $\gamma '$ 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;

• • $\|\tilde \gamma |_{B_{2\delta _0}E\cup V}\|_{C^2}\leq C(\epsilon )\max \{\|\gamma \|_{C^{1+\alpha }},\|\gamma |_V\|_{C^2}\}$ ;

• • $\tilde \gamma =\gamma $ 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 than $2\epsilon ^{\alpha } C_{\alpha }$ (in the exponential chart at $x_i$ ) is well defined between $B_i$ and $B_i^1$ . Let

$$ \begin{align*}{\mathcal W}_1={\mathcal W}_0\setminus\bigg(\bigcup_{i\in I_1}B_i\bigg)\cup \bigg(\bigcup_{i\in I_1}B_i^1\bigg). \end{align*} $$

Then ${\mathcal W}_1$ is an immersed submanifold of M, possible with self-intersections. Observe that we can extend $\tilde h^{su}_1$ as the identity outside $\bigcup _{i\in I_1}B_i^1$ and obtain a local diffeomorphism between ${\mathcal W}_0$ and ${\mathcal W}_1$ . Let $A_1=\bigcup _{i\in I_1}Gr(\gamma _i^1|_{B_{2\delta _0}E^c(x_i)})$ (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:

• • $B_{4\delta _0-\epsilon _1}E^c(x_i)\subset \pi _i(B_i^1)\subset B_{4\delta _0+\epsilon _1}E^c(x_i)$ ;

• • $\|\gamma _i-\gamma _i^1\|_{C^1}\leq \epsilon _1/2$ ;

• • $\|\gamma _i^1\|_{C^{1+\alpha }}\leq 2C_MC_{\alpha }$ ;

• • $\|\gamma _i^1|_{\pi _i(B_i^1\cap A_1)}\|_{C^2}\leq C(\epsilon )C_MC_{\alpha }$ ,

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 $\epsilon _1=2\epsilon ^{\alpha }C_MC_{\alpha }$ . Furthermore, each $B_i$ is diffeomorphic to $B_i^1$ by the holonomy $\tilde h^{su}_1$ of length less than $\epsilon _1$ . 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 $\bigcup _{i\in I_1}\tilde h^{su}_1({\mathcal W}^c_{2\delta _0-2\epsilon _1}(x_i))$ .

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 $\epsilon _2=4\epsilon ^{\alpha }C_M^2C_{\alpha }$ , and containing a smooth part $A_2\subset {\mathcal W}_2$ . Here, ${\mathcal W}_2$ 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:

• • $B_{4\delta _0-\epsilon _1-\epsilon _2}E^c(x_i)\subset \pi _i(B_i^2)\subset B_{4\delta _0+\epsilon _1+\epsilon _2}E^c(x_i)$ ;

• • $\|\gamma _i^1-\gamma _i^2\|_{C^1}\leq \epsilon _2/2$ ;

• • $\|\gamma _i^2\|_{C^{1+\alpha }}\leq 4C_M^2C_{\alpha }$ ;

• • $\|\gamma _i^2|_{\pi _i(B_i^1\cap A_2)}\|_{C^2}\leq C(\epsilon )^2C_M^2C_{\alpha }$ .

If $\epsilon $ is small enough so that ${\mathcal W}_2$ stays $C^1$ close to ${\mathcal W}^c$ , then

$$ \begin{align*} \bigcup_{i\in I_1\cup I_2}\tilde h^{su}_2\circ\tilde h^{su}_1({\mathcal W}^c_{2\delta_0-2\epsilon_1-2\epsilon_2}(x_i))\subset A_2. \end{align*} $$

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 $\epsilon _k=2^k\epsilon ^{\alpha }C_M^kC_{\alpha }$ , 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 $\|\gamma _i-\gamma _i^k\|_{C^1}\leq \tilde \epsilon /2$ ;

• • $\|\gamma _i^k\|_{C^{1+\alpha }}\leq 2^kC_M^kC_{\alpha }$ ;

• • $\|\gamma _i^k|_{\pi _i(B_i^1\cap A_k)}\|_{C^2}\leq C(\epsilon )^kC_M^kC_{\alpha }$ .

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

$$ \begin{align*}\bigcup_{i\in I}\tilde h^{su}_k\circ\cdots\circ\tilde h^{su}_1({\mathcal W}^c_{2\delta_0-2\tilde\epsilon}(x_i))\subset A_k. \end{align*} $$

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 $\overline \epsilon =\tilde \epsilon +\tilde \epsilon ^{\alpha }C_{\alpha }$ , $\tilde {\mathcal W}^c_{R,\overline \epsilon }(p)={\mathcal W}_k$ , $\tilde h^{su}_{\overline \epsilon }=\tilde h^{su}_k\circ \cdots \circ \tilde h^{su}_1$ , $\tilde C_{\alpha }=2^kC_M^{k+1}C_{\alpha }$ , and $\tilde C(\overline \epsilon )=C(\epsilon )^kC_M^{k+1}C_{\alpha }$ .

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 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 $\Delta ^u_{{\epsilon _0}/{10}}$ . 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 $\epsilon < ({\epsilon _0}/{10})$ 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 2.8. 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 parameterization $\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 notation, we will work in an exponential chart at x, and we will make an abuse of notation using the same notation for the objects in M and in the exponential chart.

Choose a decomposition $E_1\oplus E_2\oplus E_3=\mathbb R^d(=T_xM)$ 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 parameterization 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 parameterization of $\exp (B_{\delta _{\mathcal F}}\tilde E^{u}(y))$ . We can assume that $\alpha (y,0)=y$ , so $D_y\alpha (y,0)=\mathrm {id}_{\mathbb R^d}$ , 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 $\phi :B_{\delta _{{\mathcal F}}}\mathbb R^d\rightarrow \mathbb R^d$ ,

$$ \begin{align*} \phi(y_1,y_2,y_3)=(\alpha((y_1,\gamma(y_1)),y_2)+y_3. \end{align*} $$

It is clear from the definition that $\phi $ is a $\delta _{{\mathcal F}}$ -linear parameterization $\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

$$ \begin{align*} D\phi(0)=\begin{bmatrix} \mathrm{id}_{E_1} & 0 & 0\\ 0 & D_{y_2}\alpha(0,0) & 0\\ 0 & 0 & \mathrm{id}_{E_3} \end{bmatrix}\!. \end{align*} $$

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 2.9. Suppose that $\epsilon $ is small enough and $r-2\epsilon>\delta _{{\mathcal F}}$ , $\delta +\epsilon < {\delta _{{\mathcal F}}}/4$ . 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 $x'=\tilde h^{su}_{\epsilon }(x)\in \tilde {\mathcal W}^c_{R,\epsilon }(p)$ , so $d(x,x')<\epsilon $ . Then $\tilde {\mathcal W}^c_{R,\epsilon }(p)$ has size at $x'$ at least $r-2\epsilon>\delta _{{\mathcal F}}$ . This implies that $\tilde {\mathcal W}^{cu}_{R,\epsilon }(p)$ has size at $x'$ at least ${\delta _{{\mathcal F}}}/2$ . However, $d(y,x')\leq \delta +\epsilon < {\delta _{{\mathcal F}}}/4$ . The local product structure from Lemma 2.6 implies that ${\mathcal W}^s_{2(\delta +\epsilon )}(y)$ intersects transversely the disk centered at $x'$ of size $2(\delta +\epsilon )< {\delta _{{\mathcal F}}}/2$ 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 $\tilde {\mathcal W}^{cu}_{R,\epsilon }(p)$ .

3.1. 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 $0<\epsilon =\delta < ({\delta _{{\mathcal F}}}/{10})$ small enough so that all the conclusions from §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

$$ \begin{align*} {\mathcal W}_n={\mathcal W}^u_{\delta}{\mathcal W}^{c}_{R_n-r}(f^{-n}(p)). \end{align*} $$

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

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 2.9 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 $*_n=f^n(\tilde *_n)=h^s_{4\unicode{x3bb} _s^n\delta }(*)$ for $*\in \{p,q,x,z\}$ (the stable holonomy commutes with f and is uniformly contracted). Also denote $T_{p_n}=f^n(T_{\tilde p_n})=h^s_{4\unicode{x3bb} _s^n\delta }({\mathcal W}^c_{2\delta }(p))$ and $T_{q_n}=f^n(T_{\tilde q_n})=h^s_{4\unicode{x3bb} _s^n\delta }({\mathcal W}^c_{2\delta }(q))$ , 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;

• • $h^s_{4\unicode{x3bb} _s^n\delta }({\mathcal W}^{cu}_{\mathrm {loc}}(p))\subset f^n\tilde {\mathcal W}_n$ converges to ${\mathcal W}^{cu}_{\mathrm {loc}}(p)$ in the $C^1$ topology;

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

Let $T_{p_n}'=h^s_{4\unicode{x3bb} _s^n\delta }({\mathcal W}^c_{\delta }(p))\subset T_{p_n}$ . Then for n sufficiently large, there exists a well-defined holonomy of the foliation $f^n_*{\mathcal F}_n$ between the transversals $T_{p_n}'$ 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)

In other words, to obtain $h^n_{p,q}(x)$ for $x\in {\mathcal W}^c_{\delta }(p)$ , we move with the stable holonomy of size $4\unicode{x3bb} _s^n\delta $ to $T_{p_n}'\subset f^n\tilde {\mathcal W}_n$ , 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 $T_{p_n}'$ and $T_{q_n}$ , and then we move back by the stable holonomy of size $4\unicode{x3bb} _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 Lemmas 2.8 and 2.2, 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 $\tilde \Delta _n^k=\Delta ^{cs}_{\epsilon _0}\cap Tf^k\tilde {\mathcal W}_n\subset \Delta ^c_{2\epsilon _0}$ be a cone field tangent to $f^k\tilde {\mathcal W}_n$ . 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 $f_*^kE_{\tilde x_n}$ at scale $\delta $ : $t(f_*^kE_{\tilde x_n},\tilde \Delta _n^k,\delta )<2$ 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 2.3 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 $\tilde \Delta _n^n$ , $f_*^nE_{\tilde x_n}$ , and at scale $\delta $ for the constant $C_0=C_{{\mathcal F}}+ ({4\|Df\|_{C^{\theta }}}/{(1-\mu )\|Df^{-1}\|})$ 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 $d(h^{f^n_*{\mathcal F}_n}_{p_n,q_n}(x_n),h^{f^n_*{\mathcal F}_n}_{p_n,q_n}(x_n'))\leq L_0d(x_n,x_n')$ uniformly in n. Since $x_n$ converges to x, $x_n'$ converges to $x'$ , $h^{f^n_*{\mathcal F}_n}_{p_n,q_n}(x_n)$ converges to $h^u_{p,q}(x)$ , and $h^{f^n_*{\mathcal F}_n}_{p_n,q_n}(x_n')$ converges to $h^u_{p,q}(x')$ , it follows that $d(h^u_{p,q}(x),h^u_{p,q}(x'))\leq L_0d(x,x')$ .

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

Let us remember 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 $\limsup _{x\rightarrow p} ({d(f(x),g(x))}/{d(x,p)})=0$ . The equivalence classes form the space $J(M,p,N,q)$ of Lipschitz jets at $p,q$ . The distance between two Lipschitz jets is $d(J(f),J(g))=\limsup _{x\rightarrow p} ({d(f(x),g(x))}/{d(x,p)})$ , 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. Here, $J^d(M,p,N,q)$ 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 $y_n=h^{f^n_*{\mathcal F}_n}_{p_n,q_n}(x_n)$ , $E_{x_n}=f_*^nE_{\tilde x_n}$ , $\overline h^n=h^{f^n_*{\mathcal F}_n}_{p_n,q_n}$ . We have

(6) $$ \begin{align} \|D\overline h^n(x_n)-p^{E_{x_n}}_{D_{x_n},D_{y_n}}\kern-1.5pt\|\leq C_0d(x_n,y_n)^{\theta}. \end{align} $$

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:

• • $p_n+x'-\sigma _{p_n}(x')\in E_{p_n}\ \text { for all } x'\in B_{p_n}$ ;

• • $q_n+y'-\sigma _{q_n}(y')\in E_{p_n}\ \text { for all } y'\in B_{q_n}$ .

In other words, they are parameterizations 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 $g_n:T_{p_n}'\rightarrow T_{q_n}$ , $g_n=\sigma _{q_n}\circ p^{E_{p_n}}_{D_{p_n},D_{q_n}}\circ \sigma _{p_n}^{-1}$ . 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 notation $x_n=\sigma _{p_n}(x_n'), y_n=\sigma _{q_n}(y_n')$ . We can see that:

• • $\sigma _{p_n}(0)=p_n,\ \sigma _{q_n}(0)=q_n$ ;

• • $D\sigma _{p_n}(0)=\mathrm {id}_{D_{p_n}},\ D\sigma _{q_n}(0)=\mathrm {id}_{D_{q_n}}$ ;

• • $D\sigma _{p_n}(x_n')=p^{E_{p_n}}_{D_{p_n},D_{x_n}},\ D\sigma _{q_n}(y_n')=p^{E_{p_n}}_{D_{q_n},D_{y_n}}$ ;

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 $\|DG_n(0)\|\leq C_0d(p_n,q_n)^{\theta }$ . We have

$$ \begin{align*} &\|DG_n(x_n')\|=\|D\sigma_{q_n}^{-1}\circ D\overline h^n(x_n)\circ D\sigma_{p_n}(x_n')-p^{E_{p_n}}_{D_{p_n},D_{q_n}}\kern-1.5pt\|\\&\quad =\|p^{E_{p_n}}_{D_{q_n}}\circ(D\overline h^n(x_n)-p^{E_{x_n}}_{D_{x_n},D_{y_n}})\circ p^{E_{p_n}}_{D_{x_n}}+p^{E_{p_n}}_{D_{q_n}}\circ(p^{E_{x_n}}_{D_{x_n},D_{y_n}}-p^{E_{p_n}}_{D_{x_n},D_{y_n}})\circ p^{E_{p_n}}_{D_{x_n}}\|\\&\quad \leq\|p^{E_{p_n}}_{D_{q_n}}\|\cdot \|p^{E_{p_n}}_{D_{x_n}}\|\cdot(C_0d(x_n,y_n)^{\theta}+2d(E_{p_n},E_{x_n}))\\&\quad \leq4(C_0d(x_n,y_n)^{\theta}+2d(E_{p_n},E_{x_n})). \end{align*} $$

There exists $\gamma>0$ depending on $d(p,q)$ such that for all n sufficiently large and all $x_n\in T_{p_n}'$ with $d(x_n,p_n)<\gamma $ , we have:

• • $d(x_n,y_n)<2d(p,q)$ ;

• • $8d(E_{p_n},E_{x_n}))<C_0d(p,q)^{\theta }$ .

We deduce that if $d(x_n,p_n)<\gamma $ , then $\|DG(x_n')\|<5C_0d(p,q)^{\theta }$ , or G is Lipschitz with constant $5C_0d(p,q)^{\theta }$ . Then

$$ \begin{align*} &d(\sigma_{q_n}^{-1}\circ\overline h^n\circ\sigma_{p_n}(x_n'),p^{E_{p_n}}_{D_{p_n},D_{q_n}}(x_n'))\\ &\quad =d(G(x_n'),G(0))\leq 5C_0d(p,q)^{\theta}d(x_n',0)\leq 10C_0d(p,q)^{\theta}d(x_n,p_n) \end{align*} $$

and furthermore

$$ \begin{align*} \sup_{d(x_n,p_n)<\gamma}\frac{d(\overline h^n(x_n),g_n(x_n))}{d(x_n,p_n)}&\leq \mathrm{Lip}(\sigma_{q_n})\sup_{d(x_n,p_n)<\gamma}\frac{d(\sigma_{q_n}^{-1}\circ \overline h^n(x_n),\sigma_{q_n}^{-1}\circ g_n(x_n))}{d(x_n,p_n)}\\ &\leq 20C_0d(p,q)^{\theta}. \end{align*} $$

In other words, $d(J(\overline h^n),J(g_n))\leq 20C_0d(p,q)^{\theta }$ 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

$$ \begin{align*} \sup_{d(x,p)<\gamma}\frac{d(h^u_{p,q}(x),g_{p,q}(x))}{d(x,p)}\leq 20C_0d(p,q)^{\theta}, \end{align*} $$

where $g_{p,q}=\sigma _q\circ p^{E^{su}(p)}_{E^c(p),E^c(q)}\circ \sigma _p^{-1}$ . This means that $d(J(h^u_{p,q}),J(g_{p,q}))\leq 20 C_0d(p,q)^{\theta }$ for all $p\in M$ and $q\in {\mathcal W}^u_{\delta }(x)$ .

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

$$ \begin{align*} B_n=B(J(g_n), C_1d(f^n(p),f^n(q))^{\theta})\subset J^b({\mathcal W}^c_{\delta}(f^n(p)),f^n(p),{\mathcal W}^c_{\delta}(f^n(q)),f^n(q)) \end{align*} $$

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) $$ \begin{align} \mu C_1+\frac{4\mu\|Df\|_{C^{\theta}}}{\|Df^{-1}\|}<C_1. \end{align} $$

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

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

We claim that F is well defined. For this, we have to prove that if $d(J(h),J(g_n))\leq C_1d(f^n(p),{\kern-1pt}f^n(q))$ , then $d(J(f{\kern-1pt}\circ{\kern-1pt} h{\kern-1pt}\circ{\kern-1pt} f^{-1}),J(g_{n+1})){\kern-1pt}\leq{\kern-1pt} C_1d(f^{n+1}(p),{\kern-1pt}f^{n+1}(q))$ . Observe that

$$ \begin{align*} &d(J(f\circ h\circ f^{-1}),J(g_{n+1}))\\ &\quad\leq d(J(f\circ h\circ f^{-1}),J(f\circ g_n\circ f^{-1}))+d(J(f\circ g_n\circ f^{-1}),J(g_{n+1})). \end{align*} $$

On one hand, we have

$$ \begin{align*} &d(J(f\circ h\circ f^{-1}),J(f\circ g_n\circ f^{-1}))\\ &\quad\leq \mathrm{Lip} (f, f^{n}(q))\cdot d(J(h),J(g_n))\cdot \mathrm{Lip} (f^{-1},f^{n+1}(p))\\ &\quad\leq\frac{ \unicode{x3bb}^+_{\Delta^c_{2\epsilon_0}}(f,p,\delta_0)}{\unicode{x3bb}^-_{\Delta^c_{2\epsilon_0}}(f,p,\delta_0)\unicode{x3bb}^-_{\Delta^u_{2\epsilon_0}}(f,p,\delta_0)^{\theta}}C_1d(f^{n+1}(p),f^{n+1}(q))^{\theta}\\ &\quad\leq\mu C_1d(f^{n+1}(p),f^{n+1}(q))^{\theta}. \end{align*} $$

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

$$ \begin{align*} &d(J(f\circ g_n\circ f^{-1}),J(g_{n+1}))=\|D(f\circ g_n\circ f^{-1})-Dg_{n+1}\|\\&\quad=\|p^{E^{su}_{f^{n+1}(p)}}_{E^c_{f^{n+1}(q)}}\cdot(Df(f^n(q))-Df(f^n(p)))\cdot p^{E^{su}_{f^n(p)}}_{E^c_{f^n(q)}}\cdot Df^{-1}|_{E^c_{f^{n+1}(p)}}\kern-1.5pt\|\\&\quad\leq\frac{4\mu\|Df\|_{C^{\theta}}}{\unicode{x3bb}^-_{\Delta^c_{2\epsilon_0}}(f,p,\delta_0)\unicode{x3bb}^-_{\Delta^u_{2\epsilon_0}}(f,p,\delta_0)^{\theta}}d(f^{n+1}(p),f^{n+1}(q))^{\theta}\\&\quad\leq\frac{4\mu\|Df\|_{C^{\theta}}}{\|Df^{-1}\|}d(f^{n+1}(p),f^{n+1}(q))^{\theta}. \end{align*} $$

The estimates above together with the condition in equation (7) imply that F is indeed well defined.

Next, we modify the distance inside each fiber $B_n$ , we let $d_n= d/d(f^n(p), f^n(q))^{\theta }$ . 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 $J(\sigma ), J(\sigma ')\in B_n$ . Then

$$ \begin{align*} &d_{n+1}(J(f\circ\sigma\circ f^{-1}),J(f\circ\sigma'\circ f^{-1}))=\frac{d_{n+1}(J(f\circ\sigma\circ f^{-1}),J(f\circ\sigma'\circ f^{-1}))}{d(f^{n+1}(p),f^{n+1}(q))^{\theta}}\\&\quad\leq \mathrm{Lip} (f, f^{n}(q))\cdot \mathrm{Lip} (f^{-1},f^{n+1}(p))\cdot \frac{d(f^n(p),f^n(q))^{\theta}}{d(f^{n+1}(p),f^{n+1}(q))^{\theta}} \\&\qquad \cdot \frac{d(J(\sigma),J(\sigma'))}{d(f^n(p),f^n(q))^{\theta}}\\&\quad\leq\frac{ \unicode{x3bb}^+_{\Delta^c_{2\epsilon_0}}(f,p,\delta_0)}{\unicode{x3bb}^-_{\Delta^c_{2\epsilon_0}}(f,p,\delta_0)\unicode{x3bb}^-_{\Delta^u_{2\epsilon_0}}(f,p,\delta_0)^{\theta}}d_n(J(\sigma),J(\sigma'))\\&\quad\leq\mu d_n(J(\sigma),J(\sigma')). \end{align*} $$

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

Here, $B_n\cap J^d({\mathcal W}^c_{\delta }(f^n(p)),f^n(p),{\mathcal W}^c_{\delta }(f^n(q)),f^n(q))$ is a closed non-empty 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

$$ \begin{align*} \|Dh^u_{p,q}(p)-p^{E^{su}_p}_{E^c_p,E^c_q}\|\leq C_1d(p,q)^{\theta}. \end{align*} $$

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 $D_p,D_q\in \Delta ^c_{{\epsilon _0}/4}$ 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 $\Delta ^c_{{\epsilon _0}/2}$ . 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

$$ \begin{align*} \|Dh^u_{T_p,T_q}(p)-p^{E^{su}_p}_{D_p,D_q}\kern-1.5pt\|\leq C_1d(p,q)^{\theta}. \end{align*} $$

In other words, $Dh^u$ is ( $C_1,\theta $ )-Hölder along ${\mathcal W}^u$ at p with respect to $E^{su}_p,\Delta ^c_{{\epsilon _0}/4}$ 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 equation (4). Choose $\mu <\mu '<1$ . Since $E^c$ is uniformly $C^{\alpha }$ in a neighborhood of f, there exists $0<\delta '<\delta $ such that for all $g\in \mathcal U(f), p,q\in M, d(p,q)\leq \delta '$ , we have

$$ \begin{align*} \|p^{E^{su}_{p,g}}_{E^c_{p,g},E^c_{q,g}}\kern-1.2pt\|, \|p^{E^{su}_{p,g}}_{E^c_{q,g},E^c_{p,g}}\kern-1.2pt\|< \sqrt{\frac{\mu'}{\mu}}. \end{align*} $$

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 $N'=\{(p,q,g)\in N:\ q\in {\mathcal W}^u_{\delta '}(p,g)\}$ . Clearly, $N'$ is overflowed by G.

Let

$$ \begin{align*} \|\sigma\|_b=\sup_{(p,q,g)\in N'}\frac{\|\sigma(p,q,g)\|}{d(p,q)^{\theta}} \end{align*} $$

and let $\Sigma ^b$ be the space of sections in $\mathcal E$ over $N'$ bounded in $\|\cdot \|_b$ (in particular, $\sigma \in \Sigma ^b$ implies $\sigma (p,p,g)=0$ ). This is a complete metric space. Here, $\Sigma ^c\cap \Sigma ^b$ is the space of the sections which are both continuous and bounded in $\|\cdot \|_b$ , and is a closed non-empty subset of $\Sigma ^b$ (it contains the zero section).

The bundle map is $T:\mathcal E\rightarrow \mathcal E$ ,

$$ \begin{align*} T(p,q,g,L)=p^{E^{su}_{g(p),g}}_{E^c_{g(q),g}E^c_{g(p),g}}\circ Dg(q)|_{E^c_{q,g}}\circ p^{E^{su}_{p,g}}_{E^c_{p,g},E^c_{q,g}}\circ(\mathrm{id}+L)\circ Dg(p)|_{E^c_{p,g}}^{-1}-\mathrm{id}. \end{align*} $$

This is continuous in $(p,q,g)\in N',L\in \mathcal L(E^c_{p,g})$ .

The corresponding graph transform acts on sections $\sigma \in \Sigma ^b$ and it is the following:

$$ \begin{align*} &(T\sigma)(g(p),g(q),g)\\&\quad=p^{E^{su}_{g(p),g}}_{E^c_{g(q),g}E^c_{g(p),g}}\circ Dg(q)|_{E^c_{q,g}}\circ p^{E^{su}_{p,g}}_{E^c_{p,g},E^c_{q,g}} \circ(\mathrm{id}+\sigma(p,q,g))\circ Dg(p)|_{E^c_{p,g}}^{-1}-\mathrm{id}. \end{align*} $$

The connection with the holonomies is the following. If

$$ \begin{align*} \mathrm{id}+\sigma(p,q,g)=p^{E^{su}_{p,g}}_{E^c_{q,g},E^c_{p,g}}\circ H^u_{p,q,g}, \end{align*} $$

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

$$ \begin{align*} \mathrm{id}+(T\sigma)(g(p),g(q),g)&=p^{E^{su}_{g(p),g}}_{E^c_{g(q),g},E^c_{g(p),g}}\circ g_*H^u_{p,q,g}\\ &=p^{E^{su}_{g(p),g}}_{E^c_{g(q),g},E^c_{g(p),g}}\circ Dg(q)|_{E^c_{q,g}}\circ H^u_{p,q,g}\circ Dg(p)|_{E^c_{p,g}}^{-1}. \end{align*} $$

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

$$ \begin{align*} \mathrm{id}=p^{E^{su}_{g(p),g}}_{E^c_{g(q),g}E^c_{g(p),g}}\circ Dg(p)|_{E^c_{q,g}}\circ p^{E^{su}_{p,g}}_{E^c_{p,g},E^c_{q,g}}\circ Dg(p)|_{E^c_{p,g}}^{-1}. \end{align*} $$

Then,

$$ \begin{align*} \|T0\|_b&=\sup_{(g(p),g(q),g)\in N'}\frac{\|p^{E^{su}_{g(p),g}}_{E^c_{g(q),g},E^c_{g(p),g}}\circ Dg(q)|_{E^c_{q,g}}\circ p^{E^{su}_{p,g}}_{E^c_{p,g},E^c_{q,g}}\circ Dg(p)|_{E^c_{p,g}}^{-1}-\mathrm{id}\|}{d(g(p),g(q))^{\theta}}\\&\leq\sup_{(p,q,g)\in N'}\frac{\|p^{E^{su}_{g(p),g}}_{E^c_{g(p),g}}\circ (Dg(q)-Dg(p))|_{E^c_{q,g}}\circ p^{E^{su}_{p,g}}_{E^c_{p,g},E^c_{q,g}}\circ Dg(p)|_{E^c_{p,g}}^{-1}\kern-1.5pt\|}{d(g(p),g(q))^{\theta}}\\&\leq\sup_{(p,q,g)\in N'}\frac{4\|Dg\|_{C^{\theta}}}{\unicode{x3bb}^-_{\Delta^c_{2\epsilon_0}}(g,p,\delta')\unicode{x3bb}^-_{\Delta^u_{2\epsilon_0}}(g,p,\delta')^{\theta}}\\&\leq\frac{4\mu\|Dg\|_{C^{\theta}}}{\|Dg^{-1}\|}. \end{align*} $$

Now let us check that T is a contraction in $\Sigma ^b$ :

$$ \begin{align*} &\|T\sigma_1-T\sigma_2\|_b\\&\quad =\sup_{N'}\frac{\|p^{E^{su}_{g(p),g}}_{E^c_{g(q),g},E^c_{g(p),g}}\circ Dg(q)|_{E^c_{q,g}}\circ p^{E^{su}_{p,g}}_{E^c_{p,g},E^c_{q,g}}\circ(\sigma_1-\sigma_2)(p,q,g))\circ Dg(p)|_{E^c_{p,g}}^{-1}\kern-1.5pt\|}{d(g(p),g(q))^{\theta}}\\&\quad \leq\frac{\mu' \unicode{x3bb}^+_{\Delta^c_{2\epsilon_0}}(g,p,\delta')}{\mu \unicode{x3bb}^-_{\Delta^c_{2\epsilon_0}}(g,p,\delta')\unicode{x3bb}^-_{\Delta^u_{2\epsilon_0}}(g,p,\delta')^{\theta}}\|\sigma_1-\sigma_2\|_b\\&\quad \leq\mu'\|\sigma_1-\sigma_2\|_b. \end{align*} $$

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 $\Sigma ^b\cap \Sigma ^c$ , and we obtain that the unique invariant section in $\Sigma ^c$ is in fact continuous. However, the section

$$ \begin{align*} \sigma^u(p,q,g)=p^{E^{su}_{p,g}}_{E^c_{q,g},E^c_{p,g}}\circ Dh^u_{p,q,g}-\mathrm{id} \end{align*} $$

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 $p,q,g$ .

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 $p,q$ . This finishes the proof of Theorem 1.3.

3.7. Proof of Corollary 1.5

The proof is similar to the proof of Theorem 1.3. 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 direct the reader to [Reference Burns and Wilkinson4].

3.8. Proof of Theorem 1.7

The proof is actually contained in §3.6. Even if we do not know that there exists a (differentiable) holonomy between center leaves, we still obtain a continuous invariant section $\sigma ^u$ of T, and then $H^u_{p,q,g}=p^{E^{su}_{p,g}}_{E^c_{p,g},E^c_{q,g}}\circ (\sigma ^u_{p,q,g}+\mathrm {id})$ is the invariant continuous holonomy we are looking for, at least at the scale $\delta '$ . 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 $\sigma '$ corresponding to $H^u_{q,r}\circ H^u_{p,q}$ . Then the invariant section $\sigma $ is not unique, which is a contradiction.

The invariant holonomy which we obtain is Hölder because of the norm we use in the application of the invariant section theorem and the fact that the center bundle is Hölder, and by results in [Reference Kalinin and Sadovskaya7, Reference Kalinin and Sadovskaya8], it is the unique Hölder invariant holonomy. If f is dynamically coherent, then this unique holonomy has to be exactly the derivative of the holonomy between the center leaves.