(resp. $W^{\mathrm{s}}$).

Part (\[i.holonomy\_Holder\]) of \[p.holonomies\] basically says that the “leaves” ${\mathbb{W}}^{\mathrm{u}}$, ${\mathbb{W}}^{\mathrm{s}}$ are Hölder-continuous. We will need the transverse regularity of the holonomies:

\[p.regularity\_above\] Let $\Phi \in {\mathrm{Aut}}^\theta_K({\mathbb{E}},T)$ be a fiber-bunched automorphism. There exist $\theta_{\mathrm{s}}\in (0,\theta \kappa_{\mathrm{s}}]$ and $C>0$ such that if $x$, $x'$, $y$, $y' \in X$ satisfy conditions as in \[f.rectangle\] then: $$\left\| H^{\mathrm{u}}_{y' \gets y} \circ H^{\mathrm{s}}_{y \gets x} - H^{\mathrm{s}}_{y' \gets x'} \circ H^{\mathrm{u}}_{x'\gets x} \right\|
\le C {\mathrm{d}}(x,x')^{\theta_{\mathrm{s}}} \, .$$ Furthermore, the same constants $\theta_{\mathrm{s}}$ and $C$ work for every automorphism in a $C^0$-neighborhood of $\Phi$ in ${\mathrm{Aut}}^\theta_K({\mathbb{E}},T)$.

We were not able to find such a statement in the literature, so we provide a proof in \[ss.regularity\].

Invariant subbundles {#s.subbundles}
====================

Subbundles and rigidity {#ss.rigidity}
-----------------------

Let $E$ be an inner product space of dimension $d$, and let $p \in \ldbrack 1, d \rdbrack$. We denote by ${\mathcal{G}}_p(E)$ the *$p$-th Grassmannian* of $E$, i.e., the set of all $p$-dimensional subspaces of $E$. There are many metrics on this set that are “natural” in the sense that they are preserved by the action of orthogonal linear maps: see [@QZL]. As shown in \[ss.Grass\], we can find one such metric ${\mathrm{d}}$ with the useful properties stated 