the identity.

2.  For every pair of indices $i$, $j$, the matrix-valued map $$(x,y)  \in U_i \times U_j \mapsto [h_j(y)]^{-1} \circ I_{y \gets x} \circ h_i(x)$$ is $\theta$-Hölder.

See \[ss.basic\] for the proof of \[p.transport\]. The next propositions, also proved in \[ss.basic\], give additional quantitative properties of the transport maps that will be useful in subsequent calculations. Recall that we agree to denote a norm and its induced operator norm by the same symbol, as in .

\[p.transport\_groupoid\] Let ${\mathbb{E}}$ be a $\theta$-Hölder vector bundle, endowed with a Finsler norm. Let $\{I_{y\gets x}\}$ be the family of transport maps provided by \[p.transport\]. Then there is $C>0$ such that for all points $x$, $y$, $z \in X$, $$\| I_{y \gets z} \circ I_{z \gets x} - I_{y \gets x} \| \le C \max\{ {\mathrm{d}}(x,z)^\theta, {\mathrm{d}}(y,z)^\theta \} \, ,$$

\[p.norm\_Holder\] Let ${\mathbb{E}}$ be a $\theta$-Hölder vector bundle, endowed with a Finsler norm $\| \mathord{\cdot}\|$. Let $\{I_{y\gets x}\}$ be the family of transport maps provided by \[p.transport\]. Then the Finsler norm $\|\mathord{\cdot}\|$ is $\theta$-Hölder if and only if there exists $C>0$ such that for all points $x$, $y \in X$, $$\big| \|I_{y \gets x}\| - 1 \big| \le C {\mathrm{d}}(x,y) ^\theta \, .$$

