\Phi_x - \Phi_y \circ I_{y \gets x} \big\| \le K {\mathrm{d}}(x,y)^\theta \, .$$

A proof is provided in \[ss.basic\].

Next, we want to topologize the set of $\theta$-Hölder automorphisms.

Let ${\mathrm{End}}^\theta({\mathbb{E}},T)$ denote the vector space of $\theta$-Hölder endomorphisms $\Phi \colon {\mathbb{E}}\to {\mathbb{E}}$ covering $T$. Define the *$C^0$ norm*: $$\label{e.C0_norm}
\|\Phi\|_0 \coloneqq \sup_{x \in X} \|\Phi_x\| \, .$$ The stronger *$\theta$-Hölder norm* makes ${\mathrm{End}}^\theta({\mathbb{E}},T)$ a Banach space: $$\label{e.Holder_norm}
\|\Phi\|_\theta \coloneqq \max \left\{ \|\Phi\|_0 , \ \sup_{x \neq y} \frac{\|I_{Ty \gets Tx} \circ \Phi_x - \Phi_y \circ I_{y \gets x}\|}{{\mathrm{d}}(x,y)^\theta} \right\} \, .$$ The set ${\mathrm{Aut}}^\theta({\mathbb{E}},T)$ of $\theta$-Hölder automorphisms is a $C^0$-open subset of ${\mathrm{End}}^\theta({\mathbb{E}},T)$. Given $K\ge 1$, let: $$\label{e.bounded_set}
{\mathrm{Aut}}^\theta_K({\mathbb{E}},T) \coloneqq \big\{ \Phi \in {\mathrm{Aut}}^\theta({\mathbb{E}},T) {\;\mathord{;}\;}\|\Phi\|_\theta \le K, \ \|\Phi^{-1}\|_\theta \le K \big\} \, .$$

Hyperbolic homeomorphisms {#ss.hyp_homeo}
-------------------------

The concept of hyperbolicity in differentiable dynamical systems was introduced by Anosov [@Anosov] and Smale [@Smale]. Even without recourse to a differentiable structure, it is possible to define hyperbolicity (and to prove interesting theorems); this has been done in various ways: [@Bowen; @Ruelle; @AY; @Akin; @AH]. In this paper, we will use a minor variation of the definition of hyperbolic homeomorphism given by Sakai [@Sakai] (see \[r.hyperb\] below).

Recall that $X$ is 