a compact metric space. Let $T \colon X \to X$ be a homeomorphism. Given $x \in X$ and ${\varepsilon}>0$, we define the following sets:

-   *local unstable set* $W^{\mathrm{u}}_{\varepsilon}(x) \coloneqq \big\{ y \in X {\;\mathord{;}\;}{\mathrm{d}}(T^{-n} y, T^{-n} x) \le {\varepsilon}\text{ for all } n\ge 0\big\}$;

-   *local stable set* $W^{\mathrm{s}}_{\varepsilon}(x) \coloneqq \big\{ y \in X {\;\mathord{;}\;}{\mathrm{d}}(T^n y, T^n x) \le {\varepsilon}\text{ for all } n\ge 0\big\}$.

We say that $T$ is a *hyperbolic homeomorphism* if the following axioms hold:

1.  \[i.biLip\] $T$ is bi-Lipschitz, i.e., both $T$ and $T^{-1}$ are Lipschitz;

2.  \[i.lambdas\] there exist a constant ${\varepsilon}_0 > 0$ and a pair of continuous positive functions $\lambda_{\mathrm{u}}$, $\lambda_{\mathrm{s}}$ (called the *hyperbolicity exponents*) such that: $$\begin{aligned}
    {4}
    \label{e.lambda_u}
    x &\in X, &\  x', x'' &\in W^{\mathrm{u}}_{{\varepsilon}_0}(x) &\quad &\Rightarrow &\quad
    {\mathrm{d}}(T^{-1} x', T^{-1} x'') &\le e^{-\lambda_{\mathrm{u}}(x)} \, {\mathrm{d}}(x',x'') \, , \\
    \label{e.lambda_s}
    y &\in X, &\  y', y'' &\in W^{\mathrm{s}}_{{\varepsilon}_0}(y) &\quad &\Rightarrow &\quad
    {\mathrm{d}}(T y', T y'') &\le e^{-\lambda_{\mathrm{s}}(y)} \, {\mathrm{d}}(y', y'') \, ;\end{aligned}$$

3.  \[i.bracket\] there exists a constant ${\varepsilon}_1 \in (0,{\varepsilon}_0)$ such that for any pair of points $x$, $y \in X$ with ${\mathrm{d}}(x,y) \le 2{\varepsilon}_1$, the intersection $W^{\mathrm{u}}_{{\varepsilon}_0}(x) \cap W^{\mathrm{s}}_{{\varepsilon}_0}(y)$ contains exactly one point, denoted by $[x,y]$ and 