2.1 Setup

Fix $k\in \mathbb {N}$ and let $\mathbb {R}^k$ be decomposed into subvector spaces

$$ \begin{align*} \mathbb{R}^k= \mathbb {R}^s\oplus \mathbb {R}^c\oplus \mathbb {R}^u. \end{align*} $$

For each $n\in \mathbb {Z}$ , let $A_n\colon \mathbb {R}^s\to \mathbb {R}^s$ , $B_n\colon \mathbb {R}^c\to \mathbb {R}^c$ , and $C_n\colon \mathbb {R}^u\to \mathbb {R}^u$ be invertible linear maps and let

$$ \begin{align*} L_n = \left(\begin{array}{ccc}A_n & 0 & 0 \\0 & B_n & 0 \\0 & 0 & C_n\end{array}\right) \end{align*} $$

be the associated invertible linear map preserving the decomposition $\mathbb {R}^k= \mathbb {R}^s\oplus \mathbb {R}^c\oplus \mathbb {R}^u$ . We assume each component of the decomposition $\mathbb {R}^k= \mathbb {R}^s\oplus \mathbb {R}^c\oplus \mathbb {R}^u$ is non-trivial, though the results can be formulated (with fewer conditions) in the case where $\mathbb {R}^u$ is degenerate.

We assume there are constants

$$ \begin{align*} -\mu< \eta^{\prime}_n< \kappa^{\prime}_n< \gamma^{\prime}_n\le \hat \gamma^{\prime}_n< \hat \kappa^{\prime}_n<\hat \eta^{\prime}_n <\mu \end{align*} $$

such that for every $n\in \mathbb {Z}$ ,

1. (1) $e^{\eta ^{\prime }_n}\le m(A_n) \le \|A_n\| \le e^{\kappa ^{\prime }_n}$ ,

2. (2) $e^{ \gamma ^{\prime }_n}\le m(B_n) \le \|B_n\| \le e^{ \hat \gamma ^{\prime }_n}$ ,

3. (3) $e^{\hat \kappa ^{\prime }_n}\le m(C_n) \le \|C_n\| \le e^{ \hat \eta ^{\prime }_n}.$

Here $\|\cdot \|$ is the operator norm induced by the standard norm on the corresponding Euclidean spaces and $m(A) := \|A{}^{-1}\|^{-1}$ denotes the associated conorm of A. Throughout, we will further assume that $\sup \{ \kappa ^{\prime }_n\}<0$ . We do not impose any assumptions on the signsFootnote ¹ of $\gamma ^{\prime }_n$ , $\hat \gamma ^{\prime }_n, \hat \kappa ^{\prime }_n$ and $\hat \eta ^{\prime }_n$ .

We assume, moreover, that

$$ \begin{align*} \inf \{ |\kappa^{\prime}_n-\gamma^{\prime}_n|, \hat \kappa^{\prime}_n-\hat \gamma^{\prime}_n \}>0. \end{align*} $$

Fix some

$$ \begin{align*} 0<\epsilon_0\le \inf \{|\kappa^{\prime}_n-\gamma^{\prime}_n|,\hat \kappa^{\prime}_n-\hat \gamma^{\prime}_n , |\kappa^{\prime}_n|, 1 \}/10. \end{align*} $$

Anticipating perturbing the linear maps $L_n$ below, we set

$$ \begin{align*} \kappa_n= \kappa^{\prime}_n+\epsilon_0, \quad \hat \kappa_n=\hat \kappa^{\prime}_n- \epsilon_0, \quad \eta_n= \eta^{\prime}_n- \epsilon_0, \quad\hat\eta_n= \hat \eta^{\prime}_n+ \epsilon_0, \end{align*} $$

$$ \begin{align*} \gamma_n= \gamma^{\prime}_n- \epsilon_0,\quad \hat \gamma_n= \hat \gamma^{\prime}_n+ \epsilon_0. \end{align*} $$

2.2 Bunching criteria

Fix $0<\beta <1$ , which will be the Hölder regularity of the derivatives of perturbations of $L_n$ below. We assume that $ \hat \gamma ' _n-\gamma ^{\prime }_n$ and $\epsilon _0$ are sufficiently small so that there exists $0<\bar \theta <\beta $ satisfying

(5) $$ \begin{align} \sup (\kappa_n - \eta_n \bar \theta - \gamma_n) <0 \quad\text{ and }\quad \sup({-\hat \kappa_n} + \hat \eta_n \bar \theta +\hat \gamma_n)<0 \end{align} $$

and $\theta < \bar \theta $ with

(6) $$ \begin{align} \sup(\kappa_n\beta- \kappa_n \theta - \gamma_n\beta) <0 \end{align} $$

and

(7) $$ \begin{align} \sup ( \hat \gamma _n- \gamma_n +\kappa_n \theta)< 0. \end{align} $$

These are the analogues of (1), (2), and (3) above.

Condition (5) ensures that certain invariant distributions defined below are uniformly $\bar \theta $ -Hölder. Condition (7) is a standard bunching condition. Note that with $\theta =\bar \theta $ , (7) is the bunching condition stated in [Reference Burns and WilkinsonBW1, Theorem 0.3]. Our proof, however, requires a stronger bunching criteria imposed by (6). In particular, we use heavily (6) in our proof of Lemma 3.1 below. Note from (5) that $\theta = \beta \bar \theta $ satisfies (6).

2.3 Family of perturbations

We introduce the dynamics $f_n$ we study for the remainder as $C^1$ small perturbations of the linear maps $L_n$ . We begin with some notational conventions used throughout the paper.

2.3.1 Notational conventions

We let $\|\cdot \|$ denote the standard Euclidean norm on $\mathbb {R}^k$ and write d for the induced distance. Given a subspace $U\subset \mathbb {R}^k$ , we write $S U$ for the unit sphere in U relative to the Euclidean norm $\|\cdot \|$ . If $T\colon U\to V$ is linear we write $ T_*\colon S U\to S V$ for the induced map. We recall that if $T\colon U\to V$ is a linear isomorphism with $a\le m(T)\le \|T\|\le b$ then $ T_* $ is bi-Lipschitz with constants $b{}^{-1} a$ and $ba{}^{-1}$ . Finally, if $N\subset \mathbb {R}^k$ is an embedded submanifold we write $SN:= STN$ for the unit sphere bundle over N. Given a diffeomorphism $g\colon N_1\to N_2$ , we write $g_*\colon SN_1 \to S N_2$ for the renormalized derivative map

$$ \begin{align*} g_*(x,v) = \bigg(g(x), \frac{1}{ \|D_xg (v)\| }D_xg(v)\bigg). \end{align*} $$

In what follows, we consider $C^{1+\beta }$ diffeomorphisms $f\colon \mathbb {R}^k\to \mathbb {R}^k$ with uniform estimates on the $(1+\beta )$ -norms: namely, viewing $x\mapsto D_xf$ as a map from $\mathbb {R}^k$ to the space of linear maps, we assume that $ \sup _{x\in \mathbb {R}^k } \|D_x f\|<\infty $ and that $Df$ is $\beta $ -Hölder continuous with

$$ \begin{align*} \mathrm{H}\ddot{\rm o}\mathrm{l}^{{\beta}}(Df) := \sup_{x \neq y } \bigg\{\dfrac{ \| D_xf- D_y f \|} {d(x,y) ^{\beta}}\bigg\} <\infty. \end{align*} $$

Given submanifolds $N_1$ and $N_2$ and a diffeomorphism $h\colon N_1\to N_2$ then, as the linear maps $D_{x} h$ and $D_y h$ have different domains for $x\neq y\in N_1$ , we define the Hölder variation of $Dh$ and $h_*$ as functions between metric spaces: Assuming $N_1$ has bounded diameter, define the $\beta $ -Hölder variation of $Dh \colon TN_1\to TN_2$ to be

$$ \begin{align*}\mathrm{H}\ddot{\rm o}\mathrm{l}^{{\beta}} (Dh) := \sup_{\substack{(x,v)\neq (y,u) \in SN_1 } }\bigg\{ \dfrac{ d(Dh(x,v),Dh (y,u)) }{d((x,v), (y,u)) ^{\beta}}\bigg\}\end{align*} $$

where, given $(x,v)$ and $(y,u)$ in $T\mathbb {R}^k$ , we write

$$ \begin{align*}d((x,v), (y,u)) = \max\{d(x,y), d(v,u)\}. \end{align*} $$

Similarly define ${\mathrm{H}\ddot{\rm o}\mathrm{l}}^{{\beta }} ( h_*)$ . The $C^{1+\beta }$ -norm of h is $\max \{ \|h\|_{C^1}, {\mathrm{H}\ddot{\rm o}\mathrm{l}}^{{\beta }} ( Dh ) \}$ .

2.3.2 Families of perturbations

For the remainder of §2 and throughout §3, we fix $f_n\colon \mathbb {R}^k\to \mathbb {R}^k$ to be a sequence of $C^{1+\beta }$ diffeomorphisms with $f_n(0) = 0$ for each n. Fix $\epsilon '>0$ sufficiently small satisfying Proposition 2.1 below. We assume there is a $C_0>1$ such that for each $n\in \mathbb {Z}$ :

1. (1) $\|f_n-L_n\|_{{C^1}}\le \epsilon '$ , and $\|f_n{}^{-1} -L_n{}^{-1} \|_{{C^1}}\le \epsilon '$ ;

2. (2) ${\mathrm{H}\ddot{\rm o}\mathrm{l}}^{{\beta }} (Df_n)<C_0$ , and ${\mathrm{H}\ddot{\rm o}\mathrm{l}}^{{\beta }} (Df_n{}^{-1})<C_0$ .

Note then that for some $C_1\ge C_0>1$ we have:

1. (3) ${\mathrm{H}\ddot{\rm o}\mathrm{l}}^{{\beta }}_{\mathrm {loc}} ( ( f_n)_*)\le C_1 $ , and ${\mathrm{H}\ddot{\rm o}\mathrm{l}}^{{\beta }}_{\mathrm {loc}} (( f_n{}^{-1})_* )\le C_1 $ ;

2. (4) $\| (D_x f_n^{\pm 1})_*\|_{C^1}\le C_1$ and $\|D_x f_n ^{\pm 1 } \| \le C_1$ for every x.

Here, ${\mathrm{H}\ddot{\rm o}\mathrm{l}}^{{\beta }}_{\mathrm {loc}} ( f_*) $ is the local Hölder variation of $f_*\colon S\mathbb {R}^k\to S\mathbb {R}^k$ defined as

$$ \begin{align*} \mathrm{H}\ddot{\rm o}\mathrm{l}^{{\beta}}_{\mathrm{loc}} ( f_*) := \sup_{ 0<d((x,v), (y,u)) \le 1 }\bigg\{ \dfrac{ d(f_*(x,v),f_* (y,u)) }{d((x,v), (y,u)) ^{\beta}}\bigg\}. \end{align*} $$

Moreover, as it holds in all applications we have in mind, one may assume that $f_n(y) = L_n(y)$ for all y with $\|y\|\ge 1$ .

From the graph transform method, given $\epsilon '>0$ sufficiently small and a sequence $f_n$ of diffeomorphisms as above, we may construct foliations of $\mathbb {R}^k$ by pseudo-stable and pseudo-unstable manifolds. (See [Reference Hirsch, Pugh and ShubHPS, Theorem 5.1], [Reference Burns and WilkinsonBW1, Proposition 3.1], or [Reference Pugh and ShubPS1, Theorem 3.16] for more details.) To summarize, we have the following proposition.

Proposition 2.1. There exist $\beta '>\bar \theta $ and $\beta ">0$ so that for every sufficiently small $\epsilon '>0$ and every $C_0>1$ as above there is a $\hat C>0$ such that for every $n\in \mathbb {Z}$ , $\star = \{u,c,s,cu,cs\}$ , and $x\in \mathbb {R}^k$ there are manifolds $W^{\star }_n(x)$ containing x with:

1. (1) $e^{\hat \kappa _n} d(x,y) \le d(f_n(x), f_n(y)) \le e^{\hat \eta _n} d(x,y) $ for $y\in W^u_n(x)$ ;

2. (2) $e^{ \gamma _n} d(x,y) \le d(f_n(x), f_n(y)) \le e^{\hat \eta _n} d(x,y) $ for $y\in W^{cu}_n(x)$ ;

3. (3) $e^{ \gamma _n} d(x,y) \le d(f_n(x), f_n(y)) \le e^{\hat \gamma _n} d(x,y)$ for $y\in W^c_n(x)$ ;

4. (4) $e^{ \eta _n } d(x,y) \le d(f_n(x), f_n(y)) \le e^{\hat \gamma _n } d(x,y)$ for $y\in W^{cs}_n(x)$ ;

5. (5) $e^{ \eta _n} d(x,y) \le d(f_n(x), f_n(y)) \le e^{ \kappa _n} d(x,y)$ for $y\in W^s_n(x)$ ;

6. (6) $f_n (W^{\star }_n(x)) = W^{\star }_{n+1} (f_n(x))$ for every $x\in \mathbb {R}^k$ and $n\in \mathbb {Z}$ .

7. (7) If $y\in W^{\star }_n(x)$ then $W^{\star }_n(x) = W^{\star }_n(y)$ . In particular, the partition into $W^{\star }_n$ -manifolds foliates $\mathbb {R}^k$ ; moreover, the partition into $W^s_n$ -manifolds subfoliates each $W^{cs}_n(x)$ .

8. (8) Each $W^{\star }_n(x)$ is the graph of a $C^{1+\mathrm {H}\ddot{\rm o}\mathrm {lder}}$ function $G^{\star } _n(x)\colon \mathbb {R}^{\star }\to (\mathbb {R}^{\star })^{\perp }$ with $\| D_uG^{\star } _n(x)\|\le \tfrac 1 3$ for all $u\in \mathbb {R}^{\star }$ and

   1. (a) ${\mathrm{H}\ddot{\rm o}\mathrm{l}}^{{\beta }}( D G^{\star } _n(x)) \le \hat C$ for $\star = s$ ;

   2. (b) ${\mathrm{H}\ddot{\rm o}\mathrm{l}}^{{\beta '}}( D G^{\star } _n(x)) \le \hat C$ for $\star = cs, c, cu$ ;

   3. (c) ${\mathrm{H}\ddot{\rm o}\mathrm{l}}^{{\beta "}}( D G^{\star } _n(x)) \le \hat C$ for $\star = u$ .

   Moreover, the functions $G_n^{\star } (x)$ depend continuously (in the $C^{1+\mathrm {H}\ddot{\rm o}\mathrm {lder}}$ topology) on x.

For a discussion of the $C^{1+\mathrm {H}\ddot{\rm o}\mathrm {lder}}$ -regularity of individual leaves in property (8) (of Proposition 2.1), see for instance [Reference Pugh and ShubPS1, §6]. Note, in particular, that while the foliation of $\mathbb {R}^k$ into $W^{\star }_n$ -leaves is only continuous, each leaf $W^{\star }_n(x)$ is a uniformly $C^{1+\mathrm {H}\ddot{\rm o}\mathrm {lder}}$ -embedded submanifold.

Write $E^{\star }_n(x):= T_xW^{\star }_n(x)$ . From our choice of $\bar \theta>0$ satisfying (5), it follows (for example, from the $C^r$ -section theorem [Reference Hirsch, Pugh and ShubHPS, p. 30]; see also discussion following [Reference Burns and WilkinsonBW1]) that the tangent spaces $E^{\star }_n(x)$ are Hölder continuous with exponent $\bar \theta $ and constant uniform in $x\in \mathbb {R}$ and $n\in \mathbb {Z}$ for $\star = cu, cs$ , and hence for $\star = c$ . For $\star = s,u$ , the distributions $E^{\star }_n(x)$ are Hölder continuous with exponent satisfying analogues of (5). As discussed in [Reference Pugh and ShubPS1, Theorem 6.6], each $W^{cu}_n(x)$ is $C^{1+\beta '}$ whenever $0<\beta '\le \beta $ satisfies

$$ \begin{align*}\sup \{\kappa_n -(1+\beta') \gamma_n\} <0.\end{align*} $$

Since $\eta _n<\gamma _n$ , we have

$$ \begin{align*}\sup \{\kappa_n -(1+\beta') \gamma_n\} < \sup\{\kappa_n -\gamma_n - \beta' \eta_n\}\end{align*} $$

and equation (5) implies each $W^{cu}_n(x)$ is $C^{1+\beta '}$ for some $\beta '>\bar \theta $ . Similarly, equation (5) implies each $W^{cs}_n(x)$ is $C^{1+\beta '}$ for some $\beta '>\bar \theta $ ; thus each $W^{c}_n(x)$ is $C^{1+\beta '}$ . We also note in the case where $\inf \{\hat \kappa _n\}>0$ , we may take $\beta "=\beta $ .

We write

$$ \begin{align*}W^{\star}_n(x,R) := \{ y\in W_n^{\star} (x): d(x,y)<R\}.\end{align*} $$

2.4 $C^{1+\mathrm {H}\ddot{\rm o}\mathrm {lder}}$ holonomies inside center-stable manifolds

Fix $R>0$ . Fix $p\in \mathbb {R}^k$ , $n\in \mathbb {Z}$ , and $q\in W^s_n(p,R)$ . Let $\hat D_1 $ and $\hat D_2$ be two uniformly $C^{1+\beta '}$ embedded, $\dim (\mathbb {R}^{cu})$ -dimensional manifolds without boundary and with $p\in \hat D_1$ and $q\in \hat D_2$ . We assume the diameter of each $D_i$ is less than $1$ and that each subspace $T_x\hat D_i$ is sufficiently transverse to $\mathbb {R}^s$ : Given $v\in T_x \hat D_i$ , write $v= v^s + v^{cu}$ where $v^s\in \mathbb {R}^s$ and $v^{cu}\in \mathbb {R}^{cu}$ ; we then assume $\|v^{cu}\|\ge 3 \|v^s\|$ . Let $D_1= W^{cs}_n(p)\cap \hat D_1$ and $D_2= W^{cs}_n(q)\cap \hat D_2$ . Given $x\in D_1$ , let $h_{D_1, D_2}(x)$ denote the unique point y in $D_2$ with $y\in W^s_n(x)$ if such a point exists. Note that the domain and codomain of $h_{D_1, D_2}$ are open subsets of $D_1$ and $D_2$ . By restriction of domain and codomain we may assume $h_{D_1, D_2}\colon D_1 \to D_2$ is a homeomorphism.

Our main result is the following theorem.

In particular, we have the following corollary.

2.5 Main theorem

As discussed below, it is sufficient to prove a special case of Theorem 2.2.

Recall that each $W^c_n(x)$ is a uniformly $C^{1+\beta '}$ -embedded manifold and intersects $W^s_n(y)$ for every $y\in W^{cs}_n(x)$ . Moreover, each $E^{cu}_n(y)$ is uniformly transverse to both $\mathbb {R}^s$ and $E^s_n(y)$ . It suffices to prove Theorem 2.2 for the distinguished family of transversals to $W^s_n$ given by the family of center manifolds. Given $n\in \mathbb {Z}$ , $p\in \mathbb {R}^k$ , and $q\in W^s_n(p)$ , we write $h^s_{p,q,n}\colon W^c_n(p)\to W^c_n(q)$ for the stable holonomy map between center manifolds. More precisely, given $z\in W^c_n(p)$ , let

$$ \begin{align*}h^s_{p,q,n}(z) = W^{cu}_n(q) \cap W^{s}_n(z).\end{align*} $$

As both $\{W _n^{s}(x): x\in W^{cs}_n(p)\}$ and $\{W _n^{c}(x): x\in W^{cs}_n(p)\}$ subfoliate $ W^{cs}_n(p)$ , it follows that $h^s_{p,q,n}(z) \in W^{c}_n(q).$ Moreover, by the global transverseness of the manifolds, the maps $h^s_{p,q,n}$ have domain all of $W^c_n(p)$ and map onto $W^c_n(q).$

The main result of this paper is the following theorem.

Theorem 2.4. There exist $0<R_0<1$ and $\hat \alpha>0$ with the following property. Let $p\in \mathbb {R}^k$ and $q\in W^s_n(p, R_0)$ . Then the holonomy map

$$ \begin{align*}h^s_{p,q,n}\colon W^c_n(p,1) \to W^c_n(q)\end{align*} $$

is a $C^{1+ \hat \alpha }$ -diffeomorphism onto its image. Moreover, the $C^{1+\hat \alpha }$ -norm of $h^s_{p,q,n}$ is uniform across the choice of p, q and n.

The $R_0>0$ for which the theorem holds is given by criteria in §3.1.2 below.

We recall that the composition of finitely many $C^{1+ \hat \alpha }$ diffeomorphisms is again a $C^{1+ \hat \alpha }$ diffeomorphism. Since $R_0>0$ , for any fixed $R>0$ and any $q\in W^s_n(p, R)$ , the holonomy map

$$ \begin{align*}h^s_{p,q,n}\colon W^c_n(p,1) \to W^c_n(q)\end{align*} $$

is the composition of finitely many $C^{1+ \hat \alpha }$ -diffeomorphisms and the $C^{1+\hat \alpha }$ -norm of $h^s_{p,q,n}$ is uniform across the choice of p, q and n.

Taking $\hat \alpha <\beta '$ and using that holonomies are uniformly $C^{1+\hat \alpha }$ , we use Journé’s theorem [Reference JournéJou] or related discussions in [Reference Pugh, Shub and WilkinsonPSW, §6] to conclude that leaves of the partition $\{W^s_n(x), x\in \mathbb {R}^k\}$ restrict to a $C^{1+\hat \alpha }$ -foliation inside each $W^{cs}_n(p)$ . The smoothness of holonomies for arbitrary transversals in Theorem 2.2 then follows by considering foliation charts. In particular, given arbitrary transversals $D_1 $ and $D_2$ to $\{W^s_n(x), x\in W^{cs}_n(p) \}$ inside $W^{cs}_n(p)$ as above, it follows that the holonomy map $h^s_{D_1, D_2}$ is uniformly $C^{1+\hat \alpha }$ on its domain.

3.1 Initial approximations, additional notation, and sequence of approximate holonomies

3.1.1 Initial approximations

Given $n\in \mathbb {Z}$ and arbitrary $p,q\in \mathbb {R}^k$ with $q\in W^s_n(p, 1)$ , we assume there exists a uniformly $C^{1+\beta '} $ initial approximation

$$ \begin{align*}\pi_{p,q,n} \colon W^c_n(p, 1)\to W^c_n(q)\end{align*} $$

to the stable holonomy map $h_{p,q,n}^s$ with the following properties. There is a constant $C_2>1$ such that for every $n\in \mathbb {Z}$ , $ p\in \mathbb {R}^k$ , and $q\in W^s(p,1)$ we have:

1. (1) $d(\pi _{p,q,n} (p), q)\le C_2 d(p,q)$ and $d(p, \pi _{p,q,n} (p))\le C_2d(p,q)$ ;

2. (2) $d ((\pi _{p,q,n} )_*(v), v)\le C_2d(p,q)^{\bar \theta }$ for all $v\in SW^c_n(p)$ ;

3. (3) $|\| D\pi _{p,q,n} \| - 1|\le C_2d(p,q)^{\bar \theta }$ ;

4. (4) if $p'\in W^c_n(p)$ and $q'\in W^s_n(p', 1)\cap W^c_n(q)$ then $\pi _{p,q,n} $ and $\pi _{p',q',n} $ coincide on $W^c_n(p,1)\cap W^c_n(p',1)$ .

For instance, we may define such a system of approximating maps $\{\pi _{p,q,n}\}$ by linear projection: for $z\in W^c_n(p, 1)$ , define $\pi _{p,q,n} ( z)$ to be the unique point of intersection of $W^c_n(q)$ and $ z + \mathbb {R}^u \oplus \mathbb {R}^s$ . One may verify that the above properties hold for this choice of $\pi _{p,q,n}$ .

3.1.2 Additional constants

Fix $\alpha>0$ and $0<\hat \theta < \theta < \bar \theta < \hat \beta <\beta $ for the remainder with

(8) $$ \begin{align} \sup \bigg\{ \frac{ (1+\alpha)(\hat\gamma _n- \gamma_n)}{-\kappa_n }\bigg\}<\hat \theta \end{align} $$

and

(9) $$ \begin{align} \sup\bigg\{\dfrac{ \kappa_n}{\kappa_n- \gamma_n}\theta\bigg\} <\hat \beta. \end{align} $$

The existence of such $\alpha $ and $\hat \theta $ follow from (7); the existence of such a $\hat \beta $ follows from (6). Set $\bar \kappa = \sup \{\kappa _n\} <0 $ . Set

(10) $$ \begin{align} \omega = \sup\{\kappa_n \theta + (1+\alpha)(\hat\gamma _n- \gamma_n)\}, \quad \hat \omega = \sup\{\kappa_n\hat \theta + (1+\alpha)(\hat\gamma _n- \gamma_n)\} .\end{align} $$

We have $\omega <\hat \omega <0$ from the choice of $\hat \theta < \theta $ in (8).

For the remainder of §3, fix $0<\delta <1 $ such that for all $n\in \mathbb {Z}$ we have

(11) $$ \begin{align} 1+ e^{-(\hat \gamma_n- \gamma_n)} C_1 \delta^{\beta-\hat \beta} \le e^{\alpha (\hat \gamma_n- \gamma_n)}.\end{align} $$

Such a $\delta>0$ exists since $\inf \{e^{\alpha (\hat \gamma _n- \gamma _n)}\}>1$ .

Given $p,q$ , and n with $q\in W^s_n(p)$ , define

$$ \begin{align*}\rho(p,q,n):= \sup \{d(x, h^s_{p,q,n}(x)): x\in W^c_n(p,1)\}.\end{align*} $$

Take $0<\rho _0<1$ such that

(12) $$ \begin{align} (3 C_2 C_1 +1)^{\bar \theta^{-1}} \rho_0\le \delta \end{align} $$

where $C_1\ge 1$ is as in §2.3 and $C_2\ge 1$ is as above. Fix $0<R_0<1$ so that for all $n\in \mathbb {Z}$ , $p\in \mathbb {R}^k$ , and $q\in W^s_n(p, R_0)$ we have

$$ \begin{align*}\rho(p,q,n) \le \rho_0.\end{align*} $$

With this $R_0$ we establish Theorem 2.4.

3.1.3 Additional notation

It is enough to prove Theorem 2.4 in the case where $n=0$ . For the remainder of §3, we fix p and q in $ \mathbb {R}^k$ with $q\in W^s_0(p, R_0)$ as in Theorem 2.4. Write $h:=h_{p,q,0}^s$ .

Given $n, j\in \mathbb {Z}$ , write:

• • $f^{(j)}_n:= \mathrm {id}$ , $j=0$ ;

• • $f^{(j)}_n:= f_{n+j-1} \circ \cdots \circ f_n$ , $j>0$ ;

• • $f^{(j)}_n:= f_{n+j}{}^{-1} \circ \cdots \circ f_{n-1}{}^{-1}$ , $j<0$ ;

• • for $z\in \mathbb {R}^k$ , write $z_n = f^{(n)}_0(z)$ ;

• • write $D_n\subset W^c_n(p_n):= f^{(n)}_0(W^c_0(p,1))$ ;

• • let $\displaystyle \kappa ^{(j)}_n =\begin {cases}\kappa _{n+j-1}+ \cdots + \kappa _n, & j>0,\\ 0, &j= 0,\\ -\kappa _{n+j}- \cdots - \kappa _{n-1}, & j<0; \end {cases}$

• • similarly, define $\hat \kappa ^{(j)}_n, \gamma ^{(j)}_n,$ and $ \hat \gamma ^{(j)}_n $ .

Note that if $x\in W^c_0(p,1) = D_0$ and $y= h(x)\in W^c_0(q)$ then for all $n\ge 0$ we have

$$ \begin{align*}d(x_{n},y_{n})\le e^{\kappa _0^{(n)}} d(x,y)\le e^{\kappa _0^{(n)}} { \rho(p,q,0)<\rho_0.}\end{align*} $$

Since $d(x_n,y_n)<1$ , we obtain initial approximations $\pi _{x_{n},y_{n},n}$ satisfying the properties in §3.1.1. By property (4) of the approximate holonomy maps $\pi _{p_{n}, q_{n}, n}$ , it follows that the collection of maps $\{\pi _{x_{n},y_{n},n}: x_n\in D_{n}\}$ coincide with the restriction of a single approximation which we denote by $\pi _{n} \colon D_{n}\to W^c_{n}(q_{n})$ for the remainder. Note that $\pi _{n} \colon D_{n}\to W^c_{n}(q_{n})$ has all the properties enumerated in §3.1.

3.1.4 Approximate holonomies

For $n\ge 0$ , we define $h_{n}\colon W^c_0(p,1) \to W^c_0(q)$ to be successive approximations to h given by

$$ \begin{align*}h_{n} \colon x\mapsto f^{(-n)}_{n} ( \pi_{n} (x_{n}))=f^{(-n)}_{n} ( \pi_{n} (f^{(n)}_0(x))).\end{align*} $$

Note that each $h_{n} $ is a $C^{1+\beta '}$ diffeomorphism onto its image. Although the $(1+\beta ')$ -norms of the sequence $h_{n}$ may not be controlled, Theorem 2.2 follows by showing that $h_{n} $ converges to $ h \colon W^c_0(p,1) \to W^c_0(q)$ in the $C^1$ topology. We then show $Dh\colon SW^c_0(p,1) \to TW^c_0(q)$ is Hölder continuous with uniform estimates for some Hölder exponent $0\,{<}\, \hat \alpha \,{<}\,\beta '$ .

3.2 An auxiliary lemma

Given $\xi = (x, v)$ and $\zeta = (y, w)$ in $S \mathbb {R}^k$ , recall that we write $d(\xi ,\zeta ) = \sup \{d(x,y), d(v,w)\}$ . Given $\xi =(x,v)\in S\mathbb {R}^k $ , we write

$$ \begin{align*}\xi_{n}=(x_{n}, v_{n}):= (f^{(n)}_0)_*(\xi) \in S\mathbb {R}^k.\end{align*} $$

Similarly, write $\zeta _{n}:= (y_{n}, w_{n})$ .

Recall the parameters $0<\delta <1$ satisfying (11) and $\alpha $ and $\hat \beta $ satisfying (8).

Lemma 3.1. Given $x\in \mathbb {R}^k$ , $\xi =(x,v)$ , $\zeta =(y,w) \in S W^c_0(x) $ , $0\le r\le \delta $ , and $n\ge 0$ , suppose that $d(x_{n}, y_{n})\le r e^{\kappa _0^{(n)}}$ , $d(\xi _{n}, \zeta _{n})\le r^{\bar \theta }e^{\kappa _0^{(n)} \theta } $ , and for all $0\le k\le n$ that

$$ \begin{align*}d(x_{k}, y_{k})\le \delta.\end{align*} $$

Then, for all $0\le k\le n $ ,

$$ \begin{align*}d(x_{k}, y_{k})\le r e^{ \kappa_0^{(n)} - \gamma_{k}^{(n-k)} } \quad\text{and}\quad d(\xi_{k}, \zeta_{k})\le r^{\bar \theta } e^{ \kappa_0^{(n)} \theta + (1+\alpha)(\hat\gamma_{k}^{(n-k)}- \gamma_{k}^{(n-k)}) } .\end{align*} $$

In particular,

$$ \begin{align*}d(\xi, \zeta) \le r^{\bar \theta } e^{ \kappa_0^{(n)} \theta + (1+\alpha) (\hat\gamma_0^{(n)}- \gamma_0^{(n)})} .\end{align*} $$

Proof For the final assertion, note that

$$ \begin{align*}r e^{\kappa_{0}^{(n)} - \gamma_{0}^{(n)}}\le r^{\bar \theta } e^{\kappa_{0}^{(n)} \theta + (1+\alpha)(\hat \gamma_{0}^{(n)}- \gamma_{0}^{(n)})}\end{align*} $$

follows from inequality (5) as $ \theta \le (({\kappa _n -\gamma _n})/{\eta _n})\le (({\kappa _n -\gamma _n})/{\kappa _n}) $ holds for all n.

We prove the first two assertions by backwards induction on k starting with $k=n$ . We clearly have

$$ \begin{align*}d(x_{(k-1)}, y_{(k-1)}) \le e^{-\gamma_{k-1}} d(x_{k}, y_{k}) \le r e^{\kappa _{0}^{(n)} - \gamma _{k-1}^{(n-(k-1))} } .\end{align*} $$

Moreover, we have

$$ \begin{align*} d(\xi_{(k-1) }, \zeta_{(k-1)}) &= d((f_{k}^{(-1)})_* (x_{k}, v_{k}) , (f_{k}^{(-1)})_* (y_{k}, w_{k})) \\ &\le d((f_{k}^{(-1)})_* (x_{k}, v_{k}) , (f_{k}^{(-1)})_* (x_{k}, w_{k})) \\ & \quad + d((f_{k}^{(-1)})_* (x_{k}, w_{k}) , (f_{k}^{(-1)})_* (y_{k}, w_{k})) \\ &\le e^{(\hat\gamma_{k-1}-\gamma_{k-1})}d(v_{k}, w_{k}) + C_1d(x_{k} , y_{k})^{\beta} \\ &\le e^{(\hat\gamma_{k-1}-\gamma_{k-1})} (d(v_{k}, w_{k}) + e^{-(\hat\gamma_{k-1}-\gamma_{k-1})}C_1 d(x_{k} , y_{k})^{\beta} )\\ &\le e^{(\hat \gamma_{k-1}-\gamma_{k-1})} (1 + e^{- (\hat \gamma_{k-1}-\gamma_{k-1})}C_1d(x_{k} , y_{k})^{\beta-\hat \beta} ) \\&\quad\cdot \max \{ d(x_{k} , y_{k})^{\hat \beta} ,d(v_{k}, w_{k})\} \\ &\le e^{ (1+\alpha) (\hat\gamma_{k-1}- \gamma_{k-1})} \max \{ r^{\hat \beta} e^{ \hat \beta \kappa _0^{(n)} - \hat \beta \gamma_{k}^{ (n- k)} } , r^{\bar \theta } e^{ \kappa _0^{(n)} \theta + (1+\alpha)(\hat \gamma_{k}^{ (n- k)}- \gamma_{k}^{ (n- k)}) } \}. \notag \end{align*} $$

The last line follows from the induction hypothesis and the choice of $\delta>0$ in (11).

From (9) we have

$$ \begin{align*} \hat \beta \kappa _0^{(n)} - \hat \beta \gamma_{k}^{ (n- k)} & = \hat \beta \kappa_{0}^{(k)} + \hat \beta \kappa_{k}^{(n-k)} - \hat \beta\gamma _{k}^{ (n- k)} \\ &\le \hat \beta \kappa_{0}^{(k)}+ \theta \kappa_{k}^{(n-k)} \\ &\le \theta \kappa_{0}^{(n)} \\ &\le \theta \kappa _0^{(n)} + (1+\alpha)(\hat \gamma_{k}^{ (n- k)}- \gamma_{k}^{ (n- k)}). \end{align*} $$

Hence

$$ \begin{align*}r^{\hat \beta} e^{ \hat \beta \kappa _0^{(n)} - \hat \beta \gamma_{k}^{ (n- k)} } \le r^{\bar \theta } e^{ \kappa _0^{(n)} \theta + (1+\alpha)(\hat \gamma_{k}^{ (n- k)}- \gamma_{k}^{ (n- k)}) } \end{align*} $$

and the result follows.

3.3 Step 1: $C^0$ convergence

Recall the ‘true’ holonomies $h^s_{p,q,n}$ . We fix $h= h^s_{p,q,0}$ and let $h_n$ be the approximate holonomies in §3.1.4. We have the following lemma.

Proof First (by equivariance of $W^s_n$ -manifolds) we have $f^{(-n)} _n\circ h^s_{p_{n}, q_{n},n} \circ f^{(n)} _0= h^s_{p,q,0}$ . For $x\in W^c_0(p,1)$ we have

$$ \begin{align*}d(x_{n}, f^{(n)}_0(h(x)))= d(x_{n}, h^s_{p_{n}, q_{n},n}(x_{{n}}))\le e^{\kappa_0^{(n)}} \rho(p,q,0)\le e^{\kappa_0^{(n)}}.\end{align*} $$

By property (1) of the maps $\pi _{n}$ ,

$$ \begin{align*} d(h_n(x), h(x)) &= d(f^{(-n)}_0( \pi_{n} (x_{n})), f^{(-n)}_n(h^s_{p_{n}, q_{n},n}(x_{n})))\\ &\le e^{ -\gamma_{0}^{(n)} } d(\pi_{n} (x_{n}), h^s_{p_{n}, q_{n},n}(x_{n}))\\ &\le e^{ -\gamma_{0}^{(n)} } C_2 d(x_n, h^s_{p_{n}, q_{n},n}(x_{n}))\\ &\le C_2 e^{{\kappa_0^{(n)}} -\gamma_{0}^{( n)} } \rho_0.\\[-33pt] \end{align*} $$

3.4 Step 2: Convergence of the projectivized derivative

Consider now the projectivized derivatives $(h_n)_*\colon S W^c_0(p,1) \to S W^c_0(q).$ We show that the sequence $(h_n)_*$ is Cauchy. Set

(13) $$ \begin{align}L_1= C_2 + \sum_{n=0}^{\infty} 3C_2 C_1 e^{\omega n} \end{align} $$

where $\omega <0$ is as in (10).

Lemma 3.3. The sequence of maps $(h_n)_*\colon S W^c_0(p,1) \to S W^c_0(q) $ is uniformly Cauchy. Moreover, defining $h_*\colon S W^c_0(p,1) \to S W^c_0(q)$ to be the limit $h_*= \lim _{n\to \infty } (h_n)_*$ , for $(x,v)\in SW^c_0(p,1)$ we have

$$ \begin{align*}d((x,v), h_* (x,v))\le L_1 d(x, h(x) ) ^{\bar \theta}.\end{align*} $$

Proof With $\xi = (x,v)$ , let $y = h(x)$ and let $\zeta ^n =(y^n, w^n)= (h_n)_*(\xi )$ .

We have

$$ \begin{align*}d(y^n, y^{n+1}) = d( f^{(-n)}_{n} (\pi_{n} (x_{n})), f^{(-n-1)}_{n+1}(\pi_{n+1}(x_{n+1})))\end{align*} $$

and, using property (1) of the maps $\pi _{n}$ ,

$$ \begin{align*} d( \pi_{n} (x_{n}), f_{n+1}^{(-1)}( \pi_{n+1}(x_{n+1}))) &\le d( x_{n}, \pi_{n} (x_{n}))+ ( f_{n+1}^{(-1)}(x_{n+1} ), f_{n+1}^{(-1)}(\pi_{n+1}(x_{n+1})))\\ &\le C_2 e^{\kappa _0^{(n)}}d(x,y) + C_1 C_2 e^{\kappa_0^{(n+1)}}d(x,y) \\ &\le 2C_1 C_2 e^{\kappa_0 ^{(n)} }d(x,y). \end{align*} $$

Similarly,

(14) $$ \begin{align} d(\zeta^n, \zeta^{n+1}) &= d( (f^{(-n)}_{n})_* (\pi_{n})_* (\xi_{n}), (f^{(-n-1)}_{n+1})_* (\pi_{n+1})_* (\xi_{n+1})) \end{align} $$

and

(15) $$ \begin{align} &d( (\pi_{n})_* (\xi_{n}), (f^{(-1)}_{n+1})_* (\pi_{n+1})_* (\xi_{n+1})) \notag \\ &\quad\le d( \xi_{n}, (\pi_{n})_* (\xi_{n}) ) + d ((f_{n+1}^{(-1)})_* \xi_{n+1}, ( f_{n+1}^{(-1)})_* (\pi_{n+1})_* (\xi_{n+1})) .\end{align} $$

With $(\pi _{n+1})_* (x_{n+1}, v_{n+1})= (y',w')$ we have

(16) $$ \begin{align} &d ((f_{n+1}^{(-1)})_* \xi_{n+1}, ( f_{n+1}^{(-1)})_* (\pi_{n+1})_* (\xi_{n+1}) )\notag\\ &\quad= d ((f_{n+1}^{(-1)})_* (x_{n+1}, v_{n+1}) , ( f_{n+1}^{(-1)})_* ( y',w') ) \notag\\ &\quad\le d ((f_{n+1}^{(-1)})_* (x_{n+1}, v_{n+1}) , ( f_{n+1}^{(-1)})_* ( x_{n+1},w') ) \notag \\ &\qquad+ d ((f_{n+1}^{(-1)})_* (x_{n+1}, w') , ( f_{n+1}^{(-1)})_*( y',w') )\notag\\ &\quad\le C_1 d( v_{n+1}, w') + C_1 d(x_{n+1}, y')^{\beta}\notag\\ &\quad\le C_1 C_2 d( x_{n+1},y_{n+1})^{\bar \theta} + C_1 (C_2 d(x_{n+1}, y_{n+1}))^{\beta} \\ &\quad\le C_1 C_2 e^{\kappa_0 ^{(n+1)} \bar \theta} d(x,y)^{\bar \theta} + C_1 C_2^{\beta} e^{\beta \kappa_0 ^{(n+1)} } d(x,y)^{\beta} \notag\\ &\quad\le 2 C_1 C_2 e^{\bar \theta \kappa_0 ^{(n+1)} }d(x,y)^{\bar \theta }\notag \end{align} $$

where the first term in (16) follows from property (2) of the approximate holonomies in §3.1.1 and the second term in (16) uses property (1) of the approximate holonomies. Combined with (15) and using property (2) of the approximate holonomies $\pi _n$ , we then have

$$ \begin{align*} &d( (\pi_{n})_* (\xi_{n}), (f^{(-1)}_{n+1})_* (\pi_{n+1})_* (\xi_{n+1})) \\ &\quad\le C_2 e^{\bar \theta \kappa_0^{( n)}}d(x,y)^{\bar \theta} + 2C_1 C_2 e^{ \bar \theta \kappa_0^{ (n+1)}} d(x,y)^{\bar \theta}\\ &\quad\le 3C_2 C_1 e^{\bar \theta \kappa _0^{(n)}} d(x,y)^{\bar \theta }. \end{align*} $$

Applying Lemma 3.1 (with $r= (3C_1C_2)^{\bar \theta ^{-1}}d(x,y) $ ) to (14) with the choice of $\rho _0$ satisfying (12), it follows that

(17) $$ \begin{align} d((h_n)_*\xi, (h_{n+1})_*\xi) =d(\zeta^n, \zeta^{n+1} ) \le e^{ \kappa_0^{(n)} \theta + (1+\alpha)(\hat \gamma_0^{(n)}- \gamma_0^{(n)}) } 3C_2 C_1 d(x,y)^{\bar \theta}. \end{align} $$

By (10),

$$ \begin{align*}\kappa_0^{(n)} \theta + (1+\alpha)(\hat \gamma_0^{(n)}- \gamma_0^{(n)}) \le n\omega,\end{align*} $$

and it follows that $(h_n)_* $ is uniformly Cauchy on $SW^c_0(p,1)$ .

Moreover, for any $\xi =(x,v)\in SW^c_0(p,1)$ we have

$$ \begin{align*} d(\xi, h_* (\xi) ) &\le d(\xi, (\pi_0)_* \xi) + \sum_{n=0}^{\infty} d((h_n)_*\xi, (h_{n+1})_*\xi)\\ &= d(\xi, (\pi_0)_* \xi) + \sum_{n=0}^{\infty} d(\zeta^n, \zeta^{n+1} ) \\ &\le C_2d(x,y)^{\bar \theta } + \sum_{n=0}^{\infty} e^{ \kappa_0^{(n)} \theta + (1+\alpha)(\hat \gamma_0^{(n)}- \gamma_0^{(n)}) } 3C_2 C_1 d(x,y)^{\bar \theta}\\ &\le C_2d(x,y)^{\bar \theta } + \sum_{n=0}^{\infty} e^{ n\omega } 3C_2 C_1 d(x,y)^{\bar \theta} \end{align*} $$

where we use properties (1) and (2) of the approximate holonomies $\pi _0$ to bound $d(\xi , (\pi _0)_* \xi )\le C_2d(x,y)^{\bar \theta } $ . Thus, with $L_1$ as above, for any $\xi = (x,v) \in SW^c_0(p,1)$ and with $y= h(x)$ we have

$$ \begin{align*} &d(\xi, h_* (\xi) ) \le L _1d(x,y) ^{\bar \theta }.\\[-42pt] \end{align*} $$

Note that the convergence of the projectivized derivative of the stable holonomies in Lemma 3.3 is independent of the choice of $p,q\in \mathbb {R}^k$ or $n\in \mathbb {Z}$ in Theorem 2.2. Thus for all $n\in \mathbb {Z}$ , $p'\in \mathbb {R}^k$ , and $q'\in W^s_{n} (p', R_0)$ , let $(h^s_{p',q',n})_*$ denote the projectivized derivative of stable holonomies constructed as above. We have for all $\xi '= (x',v')\in SW^c_n(p',1)$ that

$$ \begin{align*}d(\xi,(h^s_{p',q',n})_* (\xi) ) \le L _1d(x',h^s_{p',q',n}(x')) ^{\bar \theta }.\end{align*} $$

Moreover, from the definition of the limit in Lemma 3.3 defining $(h^s_{p',q',n})_*$ , we have that $(h^s_{p',q',n})_*$ is a holonomy for the projectivized derivative cocycle:

(18) $$ \begin{align}(f_n)_* \circ (h^s_{p',q',n})_* = (h^s_{f_n(p'),f_n(q'),n+1})_* \circ (f_n)_*.\end{align} $$

Indeed, return to the case $n=0$ , let $h = h^s_{p,q, 0}$ , and consider $\xi =(x,v)\in SW^c_0(p,1)$ . Then

$$ \begin{align*} (f_0)_* \circ h_* (\xi) &= \lim_{n\to \infty}(f_0)_*\circ (h_n)_*(\xi )\\ &= \lim_{n\to \infty}(f_0)_* (f_n^{(-n)}\circ \pi_n\circ f_0^{(n)})_*(\xi) \\ &= \lim_{n\to \infty} (f_n^{(-(n-1))}\circ \pi_n\circ f_1^{(n-1)})_*(f_0)_*(\xi )\\ &= (h^s_{f_0(p), f_0(q), 1})_*\circ (f_0)_*(\xi). \end{align*} $$

To show that the holonomies are $C^1$ , we next show that each $(h^s_{p',q',n})_*$ coincides with the projectivization of a continuous $D h^s_{p',q',n}\colon TW^c_n(p',1)\to TW^c_n(q')$ .

3.5 Step 3: The sequence of maps $ Dh_n$ is uniformly Cauchy

We return to the notation in Step 2. In particular, we recall our distinguished $p,q\in \mathbb {R}^k$ and the maps $h_n$ approximating $h= h^s_{p,q,0}$ .

We first derive two simple distortion estimates. Fix $\xi =(x,v)\in S W^c_0(p,1)$ . With $\xi _{n}:= (f^{(n)}_0)_*(\xi ) $ , let $y=h(x)$ , $\zeta =(y,w) := h_*(\xi )$ , $\zeta _{n}=(y_{n}, w_{n}) = (f_0 ^{(n)})_* (\zeta ) = (h^s_{p_{n},q_{n}, n})_* (\xi _{n}),$ and $\hat \zeta ^n= (h_n)_*(\xi )= (\hat y^n, \hat w^{n})$ . Write $\hat \zeta ^n_i = (f^{(i)}_0)_*\hat \zeta ^n$ . Then

$$ \begin{align*}\hat \zeta^n_{n }:= (f_0^{(n)})_*(\hat \zeta ^n)= (\hat y^n_{n}, \hat w^{n}_n) = (\pi_{n})_* (\xi_{n}).\end{align*} $$

From property (1) of the approximate holonomies $\pi _n$ ,

(19) $$ \begin{align} d(y_{n}, \hat y^n_{n}) \le C_2 d(x_n, y_n) \le C_2 e^{\kappa _0^{(n)} }d(x,y)\le e^{\kappa _0^{(n)} }\delta.\end{align} $$

From Lemma 3.3, we have

$$ \begin{align*}d(\xi_{n}, \zeta_{n})\le L _1e^{\bar \theta \kappa _0^{(n)}} d(x,y)^{\bar \theta }.\end{align*} $$

By properties (1) and (2) of the approximate holonomies $\pi _n$ ,

$$ \begin{align*}d(\xi_{n}, \hat \zeta^n_{n}) \le \max\{ d(x_n, \hat y_n ^n), d (v_n, \hat w^n _n)\} \le C_2 e^{\bar \theta \kappa _0^{(n)}} d(x,y)^{\bar \theta }.\end{align*} $$

Hence,

$$ \begin{align*}d(\zeta_{n}, \hat \zeta^n_{n})\le (C_2 +L_1 ) e^{ \bar \theta \kappa _0^{(n)} }.\end{align*} $$

Let $n_0$ be such that $(C_2 +L _1) e^{ \bar \theta \kappa _0^{(n_0)}}\le e^{ \theta \kappa _0^{(n_0)}} \delta ^{ \bar \theta }$ so that for $n\ge n_0$ we have

(20) $$ \begin{align}d(\zeta_{n}, \hat \zeta^n_{n}) \le e^{ \theta \kappa _0^{(n)} } \delta^{\bar \theta}. \end{align} $$

Given $\xi = (x,v) \in T\mathbb {R}^k$ , define $\|\xi \| = \|v\|$ . For each i, the map $S \mathbb {R}^k \to \mathbb {R}$ given by $\zeta \to \log \|Df_i(\zeta )\|$ is $\beta $ -Hölder on $S\mathbb {R}^k$ with uniform Hölder constant $C_3$ . Recall $\hat \theta <\theta $ satisfying (8) and $\hat \omega $ satisfying (10). Let

$$ \begin{align*}K_1:= \sum _{k=0}^{\infty} C_3 (\delta^{\bar \theta} e^{ \hat \omega k } ) ^{\beta} .\end{align*} $$

Lemma 3.4. For all $n\ge n_0$ ,

$$ \begin{align*}\exp (- K_1e^{ \beta (\theta -\hat \theta ) \kappa_0^{(n)} }) \le \dfrac{\prod_{i=0} ^{n-1} \|Df_{i} \hat \zeta^n_{i}\|}{\prod_{i=0} ^{n-1} \|Df_{i} \zeta_{i}\|} \le \exp(K_1e^{ \beta (\theta -\hat \theta ) \kappa_0^{(n)} }).\end{align*} $$

(Note, in particular, that the middle ratio goes to $1$ as $n\to \infty $ .)

Proof Recalling Lemma 3.1 (with $r=\delta $ and estimates (19) and (20)), for $n\ge n_0$ we have

$$ \begin{align*} \bigg| \log \dfrac{\prod_{i=0} ^{n-1} \|Df_{i} \hat \zeta^n_{i}\|}{\prod_{i=0} ^{n-1} \|Df_{i} \zeta_{i}\|} \bigg| &= \bigg | \sum _{i=0}^{n-1} \log \|Df_{i} \hat \zeta^n_{i}\| - \log \|Df_{i} \zeta_{i}\| \bigg |\\ &\le \sum _{i=0}^{n-1} C_3 d (\hat \zeta^n_{i},\zeta_{i})^{\beta} \\ &\le \sum _{i=0}^{n-1} C_3 (\delta^{\bar \theta} e^{ \kappa_{0}^{(n)} \theta + (1+\alpha)(\hat \gamma_i^{(n-i)}- \gamma_i^{(n-i)})} ) ^{\beta} \\ &\le e^{ \beta \kappa_0^{(n)}(\theta -\hat \theta ) } K_1 \end{align*} $$

where we use that

$$ \begin{align*}& \kappa_{0}^{(n)} \theta+ (1+\alpha)(\hat \gamma_i^{(n-i)}- \gamma_i^{(n-i)} ) \\ &\quad\le \kappa_{0}^{(n)} (\theta- \hat \theta) + \kappa_{i}^{(n-i)} \hat \theta + (1+\alpha)(\hat \gamma_i^{(n-i)}- \gamma_i^{(n-i)} )\\ &\quad\le \kappa_{0}^{(n)} (\theta- \hat \theta) + \hat \omega (n-i) \end{align*} $$

in the last inequality.

Similarly, letting

$$ \begin{align*}K_2 = \exp \bigg(\sum _{i=1}^{\infty } C_3 L_1^{\beta} e^{\bar \kappa \bar \theta\beta i} \bigg),\end{align*} $$

we have the following lemma.

Lemma 3.5. For all $n>0$ ,

$$ \begin{align*}K_2{}^{-1} \le \dfrac{\prod_{i=0} ^{n-1} \|Df_{i} \xi_{i}\|}{\prod_{i=0} ^{n-1} \|Df_{i} \zeta_{i}\|} \le K _2.\end{align*} $$

Proof From Lemma 3.3 we obtain

$$ \begin{align*} \bigg| \log \dfrac{\prod_{i=0} ^{n-1} \|Df_{i} \xi_{i}\|}{\prod_{i=0} ^{n-1} \|Df_{i} \zeta_{i}\|} \bigg| &= \bigg | \sum _{i=0}^{n-1} \log \|Df_{i} \xi_{i}\| - \log \|Df_{i} \zeta_{i}\| \bigg |\\ &\le \sum _{i=0}^{n-1} C_3 d( \xi_{i},\zeta_{i})^{\beta} \\ &\le \sum _{i=0}^{n-1} C_3 [L_1 d(x_i, y_i)^{\bar \theta } ]^{\beta} \\ &\le \sum _{i=0}^{n-1} C_3 L_1^{\beta} (e^{\kappa_0^{(i)} \bar \theta} ) ^{\beta} d(x,y)^{\beta} \\ &\le \log( K_2). \\[-34pt]\end{align*} $$

We now approximate the derivatives $D h\colon TW^c_0(p) \to TW^c_0(q)$ by the bundle maps

$$ \begin{align*}\Delta_{n} \colon TW^c_0(p,1)\to TW^c_0(q)\end{align*} $$

defined as follows: given $n\ge 0$ and $(x,v) \in TW^c_0(p,1)$ , let

$$ \begin{align*}\Delta_{n} \colon (x,v)\mapsto Df^{(-n)}_{n}\big(\big(\| D f^{(n)}_0 (v)\| (h^s_{p_{n},q_{n},n})_*((f^{(n)}_0)_*(x,\tfrac{v}{\|v\|}))\big)\big).\end{align*} $$

From (18) the projectivization of each $\Delta _n$ coincides with $(h^s_{p,q,0})_* = h_*$ . With $h = h^s_{p,q,0}$ and $(x,v) \in SW^c_0(0)$ , let $h_*(x,v) = (y,w)$ . We have

$$ \begin{align*}\|\Delta_{n} (x,v) \| = \frac {\|D_xf_0^{(n)}(v)\|}{\|D_yf_0^{(n)}(w)\|}\end{align*} $$

and $\Delta _{n} (x,v) = (y, \|\Delta _{n} (x,v) \| w)$ . From Lemma 3.5, $\|\Delta _{n} \| $ is uniformly bounded over the choice of $(x,v) \in SW^c_0(p,1)$ and n.

Given $\xi =(x,v)\in SW^c_0(p)$ , we have

$$ \begin{align*}\|Dh_n (\xi) \|= \|\Delta_n (\xi)\| \cdot \dfrac{\prod_{i=0} ^{n-1} \|D f_{i} \zeta_{i}\|}{\prod_{i=0} ^{n-1} \|D f_{i} \hat \zeta^n_{i}\|} \cdot \| D\pi_{p_{n},q_{n},n}((f^{(n)}_0)_*(x,v ))\|.\end{align*} $$

It then follows from Lemma 3.4 and property (3) of the approximate holonomies $\pi _n$ that

$$ \begin{align*}\|Dh_n (\xi)\| - \|\Delta_n (\xi)\| \to 0\end{align*} $$

uniformly in $ \xi \in SW^c_0(p)$ . Combined with Lemma 3.3, we thus obtain the following

It follows that the sequence $D h_n $ converges uniformly if and only if the sequence $\Delta _n$ converges uniformly.

Proof Given $\xi =(x,v) \in SW^c_0(p,1) $ with $\xi _{n}= (x_{n}, v_{n})\in S D_{n}$ , let $\zeta =(y,w) = h_*(\xi )\in SW^c_0(q) $ and $\zeta _{n}=(y_{n}, w_{n})= (h^s_{p_{n},q_{n},{n}})_{*}(\xi _{n}) $ . Observe that both $\Delta _{n} (x,v)$ and $ \Delta _{n+1} (x, v) $ have footprint y. Then

(21) $$ \begin{align} &\| \Delta_{n} (x,v) - \Delta_{n+1} (x, v) \| \notag \\ &\quad= \frac{\| D_x f_0^{(n)} (v)\| }{ \| D_{y}f^{(n)}_0 ( {w})\| } \|D_{y_{n}} f_{n} (w_{n})\|^{-1} \notag \\&\qquad \cdot \vert\kern-0.25ex\vert\kern-0.25ex\vert D_{y_{n}} (f_{n}) (w_{n})\| - \|D_{x_{n}} (f_{n}) (v_{n}) \vert\kern-0.25ex\vert\kern-0.25ex\vert\notag \\ & \quad\le K_2 C_1 ( \vert\kern-0.25ex\vert\kern-0.25ex\vert D_{x_{n}} (f_{n}) (v_{n}) \| - \|D_{x_{n}} (f_{n}) (w_{n})\vert\kern-0.25ex\vert\kern-0.25ex\vert \notag \\ &\qquad + \vert\kern-0.25ex\vert\kern-0.25ex\vert D_{x_{n}} (f_{n}) (w_{n})\| - \|D_{y_{n}} (f_{n}) (w_{n})\vert\kern-0.25ex\vert\kern-0.25ex\vert)\notag \\ & \quad\le K_2 C_1 ( C_1 L _1e^{\bar \theta \kappa _0^{(n)}} + C_0 e^{\kappa _0^{(n)} \beta }) \notag \\ &\quad\le K_2 C_1 2 C_1 L _1 e^{ \bar \theta \kappa _0^{(n)}} \\ &\quad\le K_2 C_1 2 C_1 L _1 e^{\bar \theta \bar \kappa n}\notag \end{align} $$

where we use Hölder continuity of $x\mapsto D_xf$ to bound $\|D_{x_{n}} (f_{n}) (w_{n})\| - \|D_{y_{n}} (f_{n}) (w_{n})\|$ and Lemma 3.3 to bound

$$\begin{align*}\vert\kern-0.25ex\vert\kern-0.25ex\vert D_{x_{n}} (f_{n}) (v_{n}) \| - \|D_{x_{n}} (f_{n}) (w_{n})\vert\kern-0.25ex\vert\kern-0.25ex\vert \le C_1 \|v_n -w_n\|\le C_1 L_1 d(x_n, y_n)^{\bar \theta}.\\[-42pt]\end{align*}$$

From Claim 3.6 and Lemma 3.7 it follows that the sequence of maps $D h_{ n} $ converges uniformly. As $h_{ n}$ converges to h we have that $h = h^s_{p,q,n}$ is differentiable and that $Dh_{ n}$ converges to $Dh$ . Furthermore, $\|Dh^s_{p,q,n}\|\le K_2$ . This completes the proof of the $C^1$ properties in Theorem 2.4.

3.6 Step 4: Hölder continuity of $Dh$

We now show that $Dh$ is Hölder continuous. We begin with the following estimate.

Claim 3.8. There is a $c_0>0$ such that if $d(x_k,y_k)\le 1$ for all $0\le k\le n-1$ and $d((x,v), (y,w))\le 1$ , then

$$ \begin{align*}d((f_0^{(n)})_*(x,v), (f_0^{(n)})_*(y,w))\le e^{c_0 n} d( (x,v), (y,w))^{\beta}.\end{align*} $$

Proof We have

$$ \begin{align*} &d((f_0^{(n)})_*(x,v), (f_0^{(n)})_*(y,w)) \\ &\quad\le d((f_0^{(n)})_*(x,v), (f_0^{(n)})_*(x,w))+ d((f_0^{(n)})_*(x,w), (f_0^{(n)})_*(y,w))\\ &\quad\le (C_1)^{2n} d(v,w) + d((f_0^{(n)})_*(x,w), (f_0^{(n)})_*(y,w)). \end{align*} $$

We have

$$ \begin{align*}d((f_0^{(1)})_*(x,w), (f_0^{(1)})_*(y,w)) \le C_1 d( x,y)^{\beta}. \end{align*} $$

Proceeding inductively, for $k\ge 2$ we claim that

$$ \begin{align*}d((f_0^{(k)})_*(x,w), (f_0^{(k)})_*(y,w)) \le k C_1 ^{ (1+\beta)k} d( x,y)^{\beta}.\end{align*} $$

Indeed,

(22) $$ \begin{align} &d((f_0^{(k)})_*(x,w), (f_0^{(k)})_*(y,w)) \notag \\ &\quad = d((D_{x_{k-1}}f_{k-1}) _* (D_{x}f_0^{(k-1)})_* (w) , (D_{y_{k-1}}f_{k-1}) _* (D_{y}f_0^{(k-1)})_* (w) ) \notag \\ &\quad\le d((D_{x_{k-1}}f_{k-1}) _* (D_{x}f_0^{(k-1)})_* (w) , (D_{y_{k-1}}f_{k-1}) _* (D_{x}f_0^{(k-1)})_* (w) ) \notag\\ &\qquad + d((D_{y_{k-1}}f_{k-1}) _* (D_{x}f_0^{(k-1)})_* (w) , (D_{y_{k-1}}f_{k-1}) _* (D_{y}f_0^{(k-1)})_* (w) ) \notag\\ & \quad\le C_1 d(x_{k-1}, y_{k-1}) ^{\beta}+ C_1 (k-1) C_1 ^{ (1+\beta)(k-1)} d( x,y)^{\beta} \\ &\quad\le C_1 (C_1)^{k\beta} d(x, y) ^{\beta}+ (k-1) C_1 ^{ (1+\beta)k} d( x,y)^{\beta} \notag \\&\quad \le k C_1 ^{ (1+\beta)k} d( x,y)^{\beta} \notag\end{align} $$

where the inductive hypothesis is used in (22). Take $c_0 = 2\log C_1 +1$ .

We now show that the maps $h_*\colon SW^c_0(p,1)\to SW^c_0(q)$ and $Dh\colon SW^c_0(p,1)\to TW^c_0(q)$ are Hölder continuous.

Fix any $\epsilon _1>0$ (to appear only in the proof of Claim 3.10 below).

• • Take $a_0= \max \{ \sup \{ ({\kappa _n - \hat \gamma _n })/{\kappa _n}\}, \sup \{ ({ { \kappa _n }- \beta \hat \gamma _n -\epsilon _1 })/{\beta \kappa _n}\} ,\sup \{( { { \kappa _n }- c_0 -\epsilon _1})/ { { \beta \kappa _n } }\} \}>0$ . Observe that

  $$ \begin{align*}a_0 \ge \sup \bigg\{\dfrac { 1- {c_0}( \kappa_n){}^{-1} - {\epsilon_1}({ \kappa_n}){}^{-1} }{ \beta }\bigg\} \ge \beta^{-1}>1.\end{align*} $$

• • Set $\bar \alpha = \min \{ \inf \{ ({\kappa _n - \gamma _n })/{a_0 \kappa _n}\}, \inf \{ ({ \theta \kappa _n + (1+\alpha )(\hat \gamma _n - \gamma _n)})/{\kappa _n a_0 } \} , {\bar \theta }/{a_0 } \}.$ We have $0<\bar \alpha <\bar \theta <1 $ .

• • Recall our fixed $0<\rho _0<1 $ satisfying (12) and $R_0>0$ such that $\rho (p,q,n)\le \rho _0$ for all p and $q\in W^s_n (p, R_0)$ .

• • Set $\rho _1= \rho _0^{1/\beta } <\rho _0.$

Proof It is enough to consider the case $n=0$ . Fix $p\in \mathbb {R}^k$ and $q\in W^s_0(p, R_0)$ .

Given $n\in \mathbb {N}$ , set

$$ \begin{align*} r_n : = \rho_1 e^{\kappa_0^{(n)} a_0 }. \end{align*} $$

Consider any pair $\xi := (x,v)$ and $ \xi ':= (x',v') $ in $SW^c_0(p, 1)$ . It is enough to consider $\xi $ and $\xi '$ sufficiently close so that for some $1\le n$ ,

$$ \begin{align*} d(x,x') \le r_n , \quad d (v, v') \le r_n, \end{align*} $$

and either

$$ \begin{align*} d(x,x')\ge r_{n+1}\quad \text{or} \quad d (v, v') \ge r_{n+1}. \end{align*} $$

Let

$$ \begin{align*} \zeta= (y, w) = h_*(\xi) \quad \text{and} \quad \zeta^n= (y^n, w^n) = (h_n)_*(\xi). \end{align*} $$

Similarly define $\zeta '=(y', w') = h_*(\xi ')$ and $\zeta ^{\prime n} = (y^{\prime n}, w^{\prime n}) = (h_n)_*(\xi ')$ . From (17) and (13) we have

$$ \begin{align*} d(\zeta, \zeta^n) &\le \sum _{j=n}^{\infty} d(\zeta^{j}, \zeta^{j+1}) \\ &\le \sum _{j=n}^{\infty} e^{ \kappa_0^{(j)} \theta + (1+\alpha)(\hat \gamma_0^{(j)}- \gamma_0^{(j)}) } 3C_2 C_1 d(x,y)^{\bar \theta} \\ &\le e^{ \kappa_0^{(n)} \theta + (1+\alpha)(\hat \gamma_0^{(n)}- \gamma_0^{(n)}) } \sum _{j=0}^{\infty} e^{\omega j} 3C_2 C_1 d(x,y)^{\bar \theta} \\ &\le e^{\kappa_0^{(n)} \theta + (1+\alpha)(\hat \gamma_0^{(n)}- \gamma_0^{(n)})} L_1 d(x,y)^{\bar \theta} \\ &\le e^{\bar \alpha a_0 \kappa_0^{(n)} } L_1 \rho^{\bar \theta}_0\\ & = L_1 \rho_1^{\beta \bar \theta-\alpha} r_n ^{\bar \alpha } \\ & \le L_1 \rho_1^{\beta {\bar \theta} -\bar \alpha} e^{\mu\bar \alpha a_0} r_{n+1}^{ \bar \alpha }, \end{align*} $$

and similarly,

$$ \begin{align*} d(\zeta', \zeta^{\prime n}) \le L_1 \rho_1^{\beta {\bar \theta} -\bar \alpha} e^{\mu\bar \alpha a_0} r_{n+1}^{ \bar \alpha }. \end{align*} $$

Note that for all $0\le k\le n$ we have

$$ \begin{align*} d(x_{k}, x^{\prime}_{k} ) \le e^{\hat \gamma _0^{(k)}} r_n = \rho_1 e^{a_0\kappa _0^{(n)} + \hat \gamma _0^{(k)}} \le \rho_1 e^{\kappa_0^{( k)} }\le \rho_0 e^{\kappa_0^{( k)} } <\delta <1. \end{align*} $$

From Claim 3.8 and the choice of $a_0$ , for all $0\le k\le n$ we have

$$ \begin{align*} d(v_k, v_k')\le d(\xi_{k}, \xi^{\prime}_{k})\le e^{c_0 k} (r_n)^{\beta} = e^{c_0 k} \rho_1 ^{\beta} e^{ \kappa_0^{(n)} a_0 \beta} \le \rho_0 e^{\kappa_0^{(n)} }. \end{align*} $$

From properties (1) and (2) of the maps $\pi _n$ we have

• • $d(x_{n},y^n_{n}) \le C_2 e^{\kappa _0^{(n)}} \rho _0 $ ,

• • $d(x^{\prime }_{ n},y^{\prime n}_{n}) \le C_2 e^{ \kappa _0^{(n)} } \rho _0 $ ,

• • $d (v_{ n}, w^n_ {n}) \le C_2 e^{\kappa _0^{(n)} \bar \theta } \rho ^{\bar \theta }_0 $ , and

• • $d (v^{\prime }_{n}, w^{\prime n}_{n}) \le C_2 e^{\kappa _0^{(n)} \bar \theta } \rho ^{\bar \theta } _0.$

Thus

$$ \begin{align*} d(y_{n}^n,y^{\prime n}_{n}) \le (1+2C_2) e^{\kappa _0^{(n)} }\rho _0 <\delta \end{align*} $$

and

$$ \begin{align*} d (w_{ n}^n, w^{\prime n}_{ n}) \le ( \rho_0 e^{\kappa _0^{(n)}}+2C_2 \rho^{\bar \theta }_0e^{\kappa _0^{(n)} \bar \theta }) \le (1+2C_2) e^{\kappa _0^{(n)} \theta } \rho^{\bar \theta }_0. \end{align*} $$

From Lemma 3.1 (with $r= (1+2C_2) \rho _0$ ) we have

$$ \begin{align*} d(y^n,y^{\prime n}) & \le (1+2C_2) e^{\kappa_0^{(n)} - \gamma_0^{(n)}}\rho_0\\ & \le (1+2C_2) e^{\bar \alpha a_0 \kappa_0^{(n)} }\rho_0\\ & \le (1+2C_2) e^{\mu\bar \alpha a_0} \rho_1^{\beta-\bar \alpha} (r_{n+1})^{\bar \alpha } \end{align*} $$

and (from Lemma 3.1 with $r= (1+2C_2) ^{\bar \theta {}^{-1}} \rho _0<\delta $ )

$$ \begin{align*} d(w^n,w^{\prime n}) &\le (1+2C_2) \rho^{\bar \theta }_0 e^{\kappa_0^{(n)} \theta + (1+\alpha) (\hat \gamma _0^{(n)} - \gamma_0^{(n)}) } \\ & \le (1+2C_2) \rho^{\bar \theta }_0 e^{\bar \alpha a_0 \kappa_0^{(n)} } \\ & \le (1+2C_2) e^{\mu a_0 \bar \alpha} \rho_1^{\bar \theta \beta -\bar \alpha} ( r_{n+1} )^{\bar \alpha}. \end{align*} $$

It follows that there is some uniform $K_3>0$ so that

$$ \begin{align*}d(\zeta, \zeta') \le d(\zeta, \zeta^n) + d(\zeta^n, \zeta^{\prime n}) + d(\zeta', \zeta^{\prime n}) \le K_3 r_{n+1}^{\bar \alpha }\end{align*} $$

whence $d(\zeta , \zeta ') \le K_3 d(\xi , \xi ')^{\bar \alpha }$ .

Claim 3.10. The function $SW^c_n(p,1)\to SW^c_n(q)$ given by

$$ \begin{align*}(x,v)\mapsto \|D_xh^s_{p,q,n} (v)\|\end{align*} $$

is $ ( \bar \alpha a_0{}^{-1}) $ -Hölder with Hölder constant uniform in all choices of p, $q\in W^s_n(p, R_0)$ , and $n\in \mathbb {Z}$ .

Proof Again, it is enough to consider the case $n=0$ . Fix $p\in \mathbb {R}^k$ , $q\in W^s_0(p, R_0)$ , and set $h = h^s_{p,q,0}.$

We retain the previous notation: given $n\in \mathbb {N}$ , set

$$ \begin{align*}r_n : = \rho_1 e^{\kappa_0^{(n)} a_0 }\end{align*} $$

and consider $\xi := (x,v)$ and $ \xi ':= (x',v') $ in $SW^c_0(p, 1)$ with $d(\xi ,\xi ') \le r_n $ and $d(\xi ,\xi ') \ge r_{n+1}$ . Write $\zeta = (y, w) = h_*(\xi )$ and $\zeta '= (y', w') = h_*(\xi ')$ .

Recall that for all $(x,v)\in SW^c_0(p)$ we have $\|\Delta _n(x,v)\| - \|D_x h (v) \| \to 0$ as $n\to \infty $ . Moreover, from (21) and using that $\bar \alpha a_0 \le \bar \theta $ , we have for some uniform $ K_4$ and $K_5$ that

(23) $$ \begin{align} \vert\kern-0.25ex\vert\kern-0.25ex\vert\Delta_n(x,v)\| - \|D_x h (v)\vert\kern-0.25ex\vert\kern-0.25ex\vert \le K_4 e^{\bar \theta \kappa_0^{( n)}} \le K_5 (r_{n+1}) ^{ \bar \alpha} \end{align} $$

We have that

$$ \begin{align*} \log(\|\Delta_n(x,v) \| ) &= \sum_{j= 0 } ^{n-1} \log \|D_{x_{j}}f_{j} v_{j} \| - \log \|D_{y_{j}}f_{j} w_{j} \|. \end{align*} $$

There exists a uniform choice of $L_2\ge 1$ with

$$ \begin{align*}\operatorname{Lip}({h}|_{W^c_0(p, 1)}):= \mathrm{H}\ddot{\rm o}\mathrm{l}^{{1}}({h}|_{W^c_0(p, 1)}) \le L_2.\end{align*} $$

Then for all $0\le j\le n$ we have the following assertions.

• • $d(x_{j}, x^{\prime }_{j})\le e^{\hat \gamma _0^{(j)}}d(x,x')\le \rho _1 e^{\hat \gamma _0^{(j)} + a_0 \kappa _0 ^{(n)}}$ , whence $d(y_j, y^{\prime }_j)\le L_2 \rho _1 e^{\hat \gamma _0^{(j)} + a_0 \kappa _0 ^{(n)}}$ .

• • By Claim 3.8,

  $$ \begin{align*} d(v_{j}, v^{\prime}_{j}) \le e^{c_0 j} d ((x,v),(x',v')) ^{\beta} \le e^{c_0 j} \rho_1^{\beta} e^{a_0 \beta \kappa_0^{(n)}}. \end{align*} $$

• • From Claims 3.9 and 3.8,

  $$ \begin{align*} d(w_{j}, w^{\prime}_{j}) \le K_3 d(\xi_{j}, \xi^{\prime}_{j})^{\bar \alpha} \le K_3 e^{\bar \alpha c_0 j} e^{a_0 \bar \alpha \beta \kappa_0^{(n)}}. \end{align*} $$

We remark that $v\mapsto \log \|D_x f(v)\|$ is $C_1 ^2$ -Lipschitz for every x. Then for some uniform choice of $K_6$ , $K_7$ , and $K_8$ we have

(24) $$ \begin{align} |& \log \| \Delta_n(x,v)\| - \log \|\Delta_n(x',v')\| | \nonumber \\[-1.5pt] &\le \sum_{j= 0 } ^{n-1} | \log\| D_{x_{j}}f_{j} v_{j}\| - \log \|D_{y_{j}}f_{j} w_{j} \| - \log \| D_{x^{\prime}_{j}}f_{j} v^{\prime}_{j} \| +\log \|D_{y^{\prime}_{j}}f_{j} w^{\prime}_{j} \| | \nonumber \\[-1.5pt] &\le \sum_{j= 0 } ^{n-1} | \log \|D_{x_{j}}f_{j} v_{j} \| - \log \| D_{x^{\prime}_{j}}f_{j} v^{\prime}_{j} \| | \nonumber + \sum_{j= 0 } ^{n-1} | \log\| D_{y_{j}}f_{j} w_{j} \| - \log\| D_{y^{\prime}_{j}}f_{j} w^{\prime}_{j} \| | \nonumber \\[-1.5pt] &\le \sum_{j= 0 } ^{n-1} C_1^2 d(v_{j}, v_{j}') + C_3 d(x_{j}, x_{j}') ^{\beta} + C_1^2 d(w_j, w^{\prime}_j) + C_3d(y_{j}, y_{j}')^{\beta} \nonumber \\[-2pt] & \le (K_6 n) \max_{j} \{( e^{\beta \hat \gamma_0^{(j)} + \beta a_0 \kappa_0 ^{(n)}} + e^{c_0 j} e^{a_0 \beta \kappa_0^{(n)}} + e^{\bar \alpha c_0 j} e^{\bar \alpha a_0 \beta \kappa_0^{(n)}} ) \} \nonumber \\[-1.5pt]& \le K_7\max_j\{ ( e^{\epsilon_1 n+ \beta \hat \gamma_0^{(j)} + \beta a_0 \kappa_0 ^{(n)}} + e^{(\epsilon_1 + c_0)n} e^{a_0 \beta \kappa_0^{(n)}} + e^{\bar \alpha ( \epsilon_1 + c_0)n} e^{\bar \alpha a_0 \beta \kappa_0^{(n)}} ) \} \\[-1.5pt] & \le K_7 (e^{ \kappa_0^{(n)} } +e^{ \kappa_0^{(n)} } +e^{\bar \alpha \kappa_0^{(n)} })\nonumber \\[-1.5pt] & \le K_8 (r_{n+1})^{ \bar \alpha a_0{}^{-1}} \nonumber \end{align} $$

where we bound the first term of (24) by

$$ \begin{align*}\epsilon_1 + \beta \hat \gamma_{\ell} + \beta a_0\kappa_{\ell} \le \kappa_{\ell}\end{align*} $$

when $ 0\le \ell <j$ and

$$ \begin{align*}\epsilon_1 +\beta a_0\kappa_{\ell} \le \beta a_0 \kappa_{\ell} + \epsilon_1+ c_0 \le \kappa_{\ell}\end{align*} $$

when $ j\le \ell <n.$ Combined with (23), it follows that

$$ \begin{align*}(x,v)\mapsto \| D_xh(v)\|\end{align*} $$

is $ ( \bar \alpha a_0{}^{-1} )$ -Hölder on $SW^c(p,1)$ .

4.1 Lyapunov charts

Fix M, $f\colon M\to M$ , and $\mu $ as above. Let $\Lambda $ denote the set of bi-regular points for $\mu $ , let $\lambda _1>\lambda _2 >\cdots >\lambda _p$ denote the Lyapunov exponents, and let $T_x M = \bigoplus _{i=1}^p E^i(x)$ denote Oseledec’s splitting for $x\in \Lambda $ . Fix a decomposition $\mathbb {R}^k = \bigoplus _{i=1}^p \mathbb {R}_i$ where $\dim \mathbb {R}_i = m_i$ is the dimension of $E_i(x)$ for $x\in \Lambda $ . Define the norm $\|\cdot \|'$ on $\mathbb {R}^k$ as follows: writing $v= \sum _{i=1}^p v_i$ where $v_i \in \mathbb {R}_i$ for every $1\le i\le p$ , set $\|v\|'= \max \{ \|v_i\|\}$ where $\|v_i\|$ restricts to the standard Euclidean norm on each $\mathbb {R}_i$ . Let $\lambda _0= \max \{|\lambda _1|, | \lambda _p|\}$ . We denote by $B(0,r)$ the ball in $\mathbb {R}^k$ of radius r centered at $0$ in the norm  $\|\cdot \|'$ .

Fix a background Riemannian metric and induced distance on M. We have the following standard construction which follows from the construction of a Lyapunov inner product and standard estimates. (See, for example, [Reference Ledrappier and YoungLY1, Appendix] or [Reference Fathi, Herman and YoccozFHY, §2].)

Given $0< \hat \epsilon <1$ and corresponding function $\hat \ell \colon \Lambda \to [1, \infty ) $ as in Proposition 4.1, define new charts $ \Phi _x \colon B(0, 1) \to M$ by rescaling:

$$ \begin{align*} \Phi_x(v) := \hat \Phi_x(\hat \ell(x)^{-1} v). \end{align*} $$

We check that with $\epsilon = 4\hat \epsilon $ and with $\ell (x) = (\hat \ell (x))^2$ , for every $x\in \Lambda $ the following hold:

1. (a) $\Phi _x\colon B(0, 1)\to M$ is a $C^{\infty }$ diffeomorphism onto a neighborhood of x with $\Phi _x(0) \,{=}\, x$ ;

2. (b) $D_0 \Phi _x\mathbb {R}_{i}= E^i(x)$ ;

3. (c) the map $\tilde f_x \colon B(0, e^{-\lambda _0 - 2 \epsilon } ) \to B(0, 1)$ given by

   $$ \begin{align*} \tilde f_x(v) := \Phi_{f(x) }^{-1} \circ f\circ \Phi_x(v) = \hat \ell(f(x)) (\hat f_x (\hat \ell(x)^{-1} v)) \end{align*} $$
   is well defined;

4. (d) $D_0\tilde f_x\mathbb {R}_i= \mathbb {R}_i$ , and for $v\in \mathbb {R}_i$ ,

   $$ \begin{align*} e^{\lambda_i- \epsilon} \|v\|' \le \|D_0\tilde f_x v\|'\le e^{\lambda_i + \epsilon}\|v\|'; \end{align*} $$

5. (e) ${\mathrm{H}\ddot{\rm o}\mathrm{l}}^{{\beta }} (D\tilde f_x)\le \epsilon $ , whence $\operatorname {Lip}(\tilde f_x - D_0\tilde f_x) \le \epsilon $ ;

6. (f) similar statements to (c), (d), and (e) hold for $f{}^{-1}$ ;

7. (g) there is a uniform $k_0$ so that $ \operatorname {Lip}( \Phi _x)\le k_0$ and $\operatorname {Lip}( \Phi _x^{-1} ) \le \ell (x)$ ;

8. (h) $e^{- \epsilon } \le \dfrac {\ell (f(x))}{\ell (x)}\le e^{ \epsilon }$ .

Indeed, (a), (b), (c), (g), and (h) follow immediately from construction. For (d) and (e), note that for $u\in B(0,1)$ and $\xi \in \mathbb {R}^k$ with $\|\xi \|'= 1$ ,

$$ \begin{align*} D_u\tilde f_x (\xi) = \hat \ell(f(x)) D_{\hat \ell(x)^{-1} u }\hat f_x (\hat \ell (x)^{-1}\xi )= \dfrac{\hat \ell(f(x))}{\hat \ell(x)}D_{\hat \ell(x)^{-1} u }\hat f_x(\xi), \end{align*} $$

hence

$$ \begin{align*} D_0\tilde f_x (\xi) = \dfrac{\hat \ell(f(x))}{\hat\ell(x)} D_0 \hat f_x(\xi) \end{align*} $$

and

$$ \begin{align*} \|D_u\tilde f_x(\xi) - D_v\tilde f_x(\xi)\|' & = \dfrac{\hat \ell(f(x))}{\hat \ell(x)}\|D_{\hat \ell(x)^{-1} u }\hat f_x (\xi )- D_{\hat \ell(x)^{-1} v }\hat f_x (\xi )\|'\\ & \le \dfrac{\hat \ell(f(x))}{\hat \ell(x)}\mathrm{H}\ddot{\rm o}\mathrm{l}^{{\beta}}(D\hat f_x ) {\|\hat \ell(x)^{-1} u - \hat \ell(x)^{-1} v\|'}^{\beta} \\ & \le \dfrac{\hat \ell(f(x))}{\hat \ell(x)}\hat \epsilon \hat \ell(x) ^{\beta} (\hat \ell(x)^{-1} )^{\beta} {\| u - v\|'}^{\beta} \\ & = \dfrac{\hat \ell(f(x))}{\hat \ell(x)}\hat \epsilon {\| u - v\|'}^{\beta}. \end{align*} $$

The family of maps $\{\Phi _x, x\in \Lambda \}$ is called a family of $(\epsilon ,\ell )$ -charts. Fix a suitable bump function $\Theta \colon \mathbb {R}^k\to [0,1]$ with $\Theta (v) = 0 $ if $\|v\|'\ge e^{-\lambda _0 - 2\epsilon } $ and $\Theta (v) = 1 $ if $\|v\|'\le \tfrac 1 2 e^{-\lambda _0 - 2\epsilon }. $ Let

$$ \begin{align*} F_x = D_0\tilde f_x + \Theta \cdot (\tilde f_x - D_0\tilde f_x) \end{align*} $$

and write $\phi _x = \Theta \cdot (\tilde f_x - D_0\tilde f_x)$ . We have $\operatorname {Lip}(\phi _x)\le \operatorname {Lip}(\tilde f_x - D_0\tilde f_x)$ , whence $\|\phi _x\|_{C^1}\le \epsilon .$ Taking $\epsilon>0$ sufficiently small, $F_x$ is a diffeomorphism and

$$ \begin{align*} F_x^{-1} - (D_0 \tilde f_x)^{-1} = - (D_0 \tilde f_x)^{-1} \circ \phi_x \circ (F_x^{-1}). \end{align*} $$

In particular, given $\epsilon '>0$ (sufficiently small to apply Proposition 2.1), we may take $\epsilon>0$ sufficiently small so that for every $x\in \Lambda $ :

1. (1) $F_x (u)= \tilde f_x (u)$ and $F_x^{-1} (u)= (\tilde f_x){}^{-1} (u)$ for all u with $\|u\|\le \tfrac 1 2e^{-2\lambda _0 - 4\epsilon }$ ;

2. (2) $\|F_x- D_0\tilde f_x\|_{C^1}<\epsilon '$ ;

3. (3) $\|F_x{}^{-1} - (D_0\tilde f_x){}^{-1}\|_{C^1}<\epsilon '$ .

Additionally, there is $C_0>0$ so that:

1. (4) ${\mathrm{H}\ddot{\rm o}\mathrm{l}^{{\beta }}}_{\|\cdot \|'} (DF_x ) \le C_0$ ;

2. (5) ${\mathrm{H}\ddot{\rm o}\mathrm{l}}^{{\beta }} _{\|\cdot \|'}(DF_x^{-1} ) \le C_0$ .

Furthermore, taking $\epsilon>0$ sufficiently small, we can ensure all relevant estimates remain true in the Euclidean norm $\|\cdot \|$ .

Given sufficiently small $\epsilon>0$ , fix a family of $(\epsilon ,\ell )$ -charts $\{ \Phi _x : x\in \Lambda \}$ as above. Let $0<\delta <1$ be a reduction factor. (Say $\delta <\tfrac 1 4 e^{-3\lambda _0 - 6\epsilon }$ to adapt the arguments in [Reference Ledrappier and YoungLY1, equations (2.2) and (2.3)].) For $x\in \Lambda $ , let

$$ \begin{align*} S^{cu}_{\delta,x}:= \{ y\in \mathbb {R}^k: \| \Phi_{f^{-n}(x)}^{-1} \circ f^{-n} \circ \Phi_x (y)\|<\delta \text{ for all } n\ge 0\}. \end{align*} $$

For $x\in \Lambda $ , $\star \in \{s,u,c,su,cu\}$ , and $v\in \mathbb {R}^k$ , let $\tilde W^{\star }_x(v)$ be the corresponding ‘fake’ manifold through the point v constructed in Proposition 2.1 using the sequence of globalizations $\{F_{f^j(x)}\}$ along the orbit $f^j(x)$ .

From the uniformly partially hyperbolic dynamics inside charts we obtain the following claim.

From Corollary 2.3, it follows (for sufficiently small $\delta>0$ ) that the Lipschitz property of holonomies along unstable manifolds inside the center-unstable sets $S^{cu}_{\delta ,x}$ derived in [Reference Ledrappier and YoungLY1, equation (4.2)] for $C^2$ maps holds for $C^{1+\beta }$ maps. We similarly obtain that the holonomies along ‘fake’ $W^i$ manifolds is Lipschitz inside $W^{i+1}$ manifolds for $\lambda _i>\lambda _{i+1}>0$ ; this replaces the Lipschitz estimates [Reference Ledrappier and YoungLY2, Lemma 8.2.5 and equation (8.4)]. Note also that since the coordinate changes intertwining the charts $\Phi _x$ and $\hat \Phi _x$ are linear we also obtain an analogous Lipschitzness of holonomies relative to the original charts $\hat \Phi _x$ . In particular, [Reference Ledrappier and YoungLY1, Proposition 5.1] and [Reference Ledrappier and YoungLY2, Proposition 11.2] remain valid for $C^{1+\beta }$ diffeomorphisms. It follows that the results of [Reference Ledrappier and YoungLY1, Reference Ledrappier and YoungLY2] hold for $C^{1+\beta }$ diffeomorphisms.