some adapted norm, and all $x\in X$, $$\label{e.def_strongly_bunched}
\log \operatorname{bol}(\Phi_x)
<  \min \left\{ \eta_{\mathrm{u}}\lambda_{\mathrm{u}}(x), \eta_{\mathrm{s}}\lambda_{\mathrm{s}}(x)\right\} \, .$$

\[r.pointwise\_vs\_absolute\] We have used the *pointwise* definition of fiber-bunching; the more stringent notion of *absolute* fiber-bunching requires the same condition with *constant* hyperbolicity exponents $\lambda_{\mathrm{u}}$, $\lambda_{\mathrm{s}}$. Furthermore, our definition of fiber-bunching is *immediate* in the sense that it manifests itself in a single iterate; one can also define a notion of *eventual* fiber-bunching.

The most basic and fruitful consequence of fiber-bunching is the existence of certain unstable and stable holonomy maps. Like the transport maps from \[p.transport\], unstable and stable holonomy maps provide a way of linearly transporting vectors from a fiber ${\mathbb{E}}_x$ to another fiber ${\mathbb{E}}_y$ (as long as the points $x$, $y$ belong to the same unstable or stable set), but with several extra properties:

\[p.holonomies\] Let $\Phi \in {\mathrm{Aut}}^\theta_K({\mathbb{E}},T)$ be a fiber-bunched automorphism. For each $\star \in \{{\mathrm{u}},{\mathrm{s}}\}$, there exist a unique family of linear maps $H^\star_{y \gets x} \colon {\mathbb{E}}_x \to {\mathbb{E}}_y$, defined whenever $y \in W^\star(x)$, such that the following properties hold:

1.  \[i.groupoid\_1\] $H^\star_{x \gets x} = {\mathrm{id}}$.

2.  \[i.groupoid\_2\] $H^\star_{z \gets y} \circ H^\star_{y \gets x} = H^\star_{z \gets x}$.

3.  \[i.equivariance\] $\Phi_y \circ H^\star_{y \gets 