and Shub [@HPS]. Earlier papers [@BGV; @BV] use a different terminology (“dominated cocycles”), but subsequently the term “fiber-bunched cocycles” prevailed: [@AV_Portugalia; @AV_Inventiones; @KalSad].

If $L$ is a linear isomorphism between inner product spaces, we define its *bolicity*[^6] as $$\label{e.def_bol}
\operatorname{bol}(L) \coloneqq \|L\| \, \|L^{-1}\| \, ,$$ which measures the lack of conformality of $L$ (see \[p.Lip\_bol\]).

Let ${\mathbb{E}}$ be a $\theta$-Hölder $d$-dimensional vector bundle over $X$. Assume that $T$ is a hyperbolic homeomorphism, and that $\Phi$ is a $\theta$-Hölder automorphism of ${\mathbb{E}}$ covering $T$. We say that ${\mathbb{E}}$ is *fiber-bunched* if there exists a Riemannian norm (sometimes called an *adapted norm*) such that for all $x \in X$, $$\label{e.def_fiber_bunched}
\log \operatorname{bol}(\Phi_x)
<  \min \left\{ \theta \lambda_{\mathrm{u}}(x), \theta \lambda_{\mathrm{s}}(x)\right\} \, ,$$ where $\lambda_{\mathrm{u}}$, $\lambda_{\mathrm{s}}$ are the hyperbolicity rates of $T$. By perturbing the adapted norm if necessary, we can assume it is also $\theta$-Hölder.

Consider the subset of fiber-bunched automorphisms in the space ${\mathrm{Aut}}^\theta({\mathbb{E}},T)$ of $\theta$-Hölder automorphisms; then this set is open with respect to the $C^0$ norm , and therefore also open with respect to the stronger $\theta$-Hölder norm .

Sometimes we need stronger bunching: we say that it is *$(\eta_{\mathrm{u}},\eta_{\mathrm{s}})$-bunched* for certain constants $\eta_{\mathrm{u}}$, $\eta_s \in (0,\theta]$ if, for 