called *the bracket of $x$ and $y$*, which depends continuously on $x$ and $y$;

4.  \[i.bounded\_angles\] there exists a constant $C>0$ such that: $$\label{e.bounded_angles} 
    x, y \in X, \ {\mathrm{d}}(x,y) \le 2{\varepsilon}_1 \ \Rightarrow \ 
    \max \big\{ {\mathrm{d}}([x,y],x) , {\mathrm{d}}([x,y],y) \big\} \le C {\mathrm{d}}(x,y) \, .$$

\[r.hyperb\] Sakai [@Sakai] uses the terminology *$\mathcal{L}$-hyperbolic homeomorphism*, while Ruelle [@Ruelle] uses *Smale spaces*. Modulo a change of metric, both definitions are equivalent to ours, and also to expansivity plus the shadowing property: see [@Sakai] and references cited there.

Let us also define other sets associated with $T$:

-   *unstable set* $W^{\mathrm{u}}(x) \coloneqq \left\{ y \in X {\;\mathord{;}\;}{\mathrm{d}}(T^{-n} y, T^{-n} x) \to 0 \text{ as } n \to +\infty \right\}$;

-   *stable set* $W^{\mathrm{s}}(x) \coloneqq \left\{ y \in X {\;\mathord{;}\;}{\mathrm{d}}(T^{n} y, T^{n} x) \to 0 \text{ as } n \to +\infty \right\}$;

If $T$ is a hyperbolic homeomorphism then, as an immediate consequence of part (\[i.lambdas\]) of the definition, for every ${\varepsilon}\in (0,{\varepsilon}_0]$ we have the following set relations: $$\label{e.longW}
W^{\mathrm{u}}(x) = \bigcup_{n\ge 0} T^n(W^{\mathrm{u}}_{\varepsilon}(T^{-n} x)) \, , \qquad
W^{\mathrm{s}}(x) = \bigcup_{n\ge 0} T^{-n}(W^{\mathrm{s}}_{\varepsilon}(T^n x)) \, .$$

The transverse regularity of the unstable and stable sets is a classical subject, and fine results about 