hyperbolicity rely on it: see [@KH Chapter 19]. Nevertheless, we could not find a reference for the following property for hyperbolic homeomorphisms:

\[p.regularity\_base\] Let $T$ be a hyperbolic homeomorphism. There exist constants $0 < \kappa_{\mathrm{s}}\le 1$ and $C>0$ such that if $x$, $x'$, $y$, $y' \in X$ satisfy (see \[f.rectangle\]): $$\label{e.rectangle}
x' \in W^{\mathrm{u}}_{{\varepsilon}_0}(x), \
y' \in W^{\mathrm{u}}_{{\varepsilon}_0}(y), \
y  \in W^{\mathrm{s}}_{{\varepsilon}_0}(x), \
y' \in W^{\mathrm{u}}_{{\varepsilon}_0}(x'),$$ then: $${\mathrm{d}}(y,y') \le C \, {\mathrm{d}}(x, x')^{\kappa_{\mathrm{s}}} \, .$$

(-1,0)–(2,0) node\[right\][$W^{\mathrm{u}}$]{}; (-1,5)–(2,5) node\[right\][$W^{\mathrm{u}}$]{}; (0,-1)–(0,6.5) node\[midway,left\][$W^{\mathrm{s}}$]{}; (1,-1)–(1,6.5) node\[midway,right\][$W^{\mathrm{s}}$]{}; (0,0) circle\[radius=2pt\] node\[below left\][$x$]{}; (1,0) circle\[radius=2pt\] node\[below right\][$x'$]{}; (0,5) circle\[radius=2pt\] node\[above left\][$y$]{}; (1,5) circle\[radius=2pt\] node\[above right\][$y'$]{};

The proof, which includes an estimate for the constant $\kappa_{\mathrm{s}}$, is given in \[s.technical\]. If $T$ is the restriction of a $C^2$ diffeomorphism to a basic hyperbolic basic set then a better estimate for $\kappa_{\mathrm{s}}$ is given in [@SS]. Even better regularity estimates can be obtained under various types of extra assumptions: see [@PintoRand] and references therein.

Fiber-bunched automorphisms and their holonomies
------------------------------------------------

We now discuss the notion of *fiber-bunching*. It was introduced in a setting very similar to ours by Bonatti, Gómez-Mont, and Viana [@BGV], though related concepts can be traced back to Brin and Pesin [@BrinP] and Hirsch, Pugh, 