[@KalSad [§]{}4.1], but for completeness and convenience of the reader we present the proof in \[ss.holonomies\].

Fiber-bunched automorphisms satisfy a non-commutative version of Walters’ condition [@Bousch_Walters], namely: $$\forall {\varepsilon}>0 \ \exists \delta> 0 \ \text{such that } \sup_{i \in \ldbrack 0,n \rdbrack} {\mathrm{d}}(T^i x, T^i y) < \delta  \ \Rightarrow \ \big\| \Phi_y^n \circ I_{y \gets x} - I_{T^n y \gets T^n x} \circ \Phi_x^n \big\| < {\varepsilon}\, .$$ Indeed, consider $z \coloneqq [x,y]$ and note the following identity: $$\Phi_y^n = H^{\mathrm{s}}_{T^n x \gets T^n z} \circ H^{\mathrm{u}}_{T^n z \gets T^n x} \circ \Phi^n_x \circ H^{\mathrm{u}}_{x \gets z} \circ H^{\mathrm{s}}_{z \gets y} \, .$$ Then, using the continuity of the bracket and the regularity of the holonomies, it is straightforward to obtain the non-commutative Walters’ condition.

We use the holonomies to define certain subsets of ${\mathbb{E}}$. For ${\varepsilon}>0$, $u \in {\mathbb{E}}$, and $x = \pi(u)$, let: $$\begin{aligned}
{\mathbb{W}}^{\mathrm{u}}_{\varepsilon}(u)  &\coloneqq \big\{ H^{\mathrm{u}}_{y \gets x}(u) {\;\mathord{;}\;}y \in W^{\mathrm{u}}_{\varepsilon}(x) \big\} \, ,  \\
{\mathbb{W}}^{\mathrm{u}}(u)           &\coloneqq \big\{ H^{\mathrm{u}}_{y \gets x}(u) {\;\mathord{;}\;}y \in W^{\mathrm{u}}(x)      \big\}
= \bigcup_{n\ge 0} \Phi^n({\mathbb{W}}^{\mathrm{u}}_{{\varepsilon}_0}(\Phi^{-n}(u)))\, ,\end{aligned}$$ Analogously we define ${\mathbb{W}}^{\mathrm{s}}_{\varepsilon}(u)$ and ${\mathbb{W}}^{\mathrm{s}}(u)$. The sets ${\mathbb{W}}^{\mathrm{u}}$ (resp. ${\mathbb{W}}^{\mathrm{s}}$) form a $\Phi$-invariant partition of ${\mathbb{E}}$ and project by $\pi$ onto the sets $W^{\mathrm{u}}$ 