theta-Hölder bundle automorphisms {#ss.auto}
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Assume that ${\mathbb{E}}$ is a $\theta$-Hölder vector bundle over the compact metric space $X$. Fix a $\theta$-Hölder Riemannian norm on ${\mathbb{E}}$.

A map $\Phi \colon {\mathbb{E}}\to {\mathbb{E}}$ is called an *endomorphism* of ${\mathbb{E}}$ if there exists a continuous map $T \colon X \to X$ such that $\pi \circ \Phi = T \circ \pi$ (we say that $\Phi$ *covers* $T$) and for each $x \in X$, the restriction of $\Phi$ to the fiber ${\mathbb{E}}_x$ is a linear map $\Phi_x$ to the fiber ${\mathbb{E}}_{Tx}$. If $T$ is a homeomorphism and each $\Phi_x$ is a isomorphism then we say that $\Phi$ is an *automorphism*.

We say that the endomorphism $\Phi$ covering $T$ is *$\theta$-Hölder* if $T$ is Lipschitz and the maps $$x \in U_i \cap T^{-1}(U_j) \mapsto [h_j(Tx)]^{-1} \circ \Phi_x \circ h_i(x) \in {\mathit{GL}}(d,{\mathbb{R}})$$ are $\theta$-Hölder.[^5] As an immediate consequence, the function $x \in X \mapsto \|\Phi_x\|$ is $\theta$-Hölder.

We can characterize $\theta$-Hölder automorphisms in terms of the transport maps from \[p.transport\]:

\[p.endo\_Holder\] An endomorphism $\Phi \colon {\mathbb{E}}\to {\mathbb{E}}$ covering a Lipschitz map $T$ is $\theta$-Hölder if and only if there exists $K>0$ such that for all $x$, $y \in X$, we have $$\big\| I_{Ty \gets Tx} \circ 