x} = H^\star_{Ty \gets Tx} \circ \Phi_x$.

4.  \[i.holonomy\_Holder\] There exists a constant $C>0$ such that: $$\label{e.holonomy_Holder}
    y \in W^\star_{{\varepsilon}_0}(x) \quad \Rightarrow \quad
    \| H^\star_{y \gets x} - I_{y \gets x} \| \le C {\mathrm{d}}(x,y)^\theta \, .$$

5.  \[i.holonomy\_cont\] The following map is continuous: $$\label{e.joint_holonomy}
    \begin{aligned}
    \big\{ (u,y) \in {\mathbb{E}}\times X {\;\mathord{;}\;}y \in W^\star_{{\varepsilon}_0}(\pi(u)) \big\} &\to {\mathbb{E}}\\
    (u,y) &\mapsto H^\star_{y \gets \pi(u)} (u)
    \end{aligned}$$

Furthermore, the constant $C$ in works for all automorphisms in a $C^0$-neighborhood of $\Phi$ in ${\mathrm{Aut}}^\theta_K({\mathbb{E}},T)$, and the the right-hand side in depends continuously on the automorphism in that neighborhood.

The maps $H^{\mathrm{u}}$ and $H^{\mathrm{s}}$ are called *unstable* and *stable* holonomies, respectively. Properties (\[i.groupoid\_1\]) and (\[i.groupoid\_2\]) are called *groupoid properties*, and property (\[i.equivariance\]) is called *equivariance*.

The stable holonomies are actually defined by the following formula: $$H^{\mathrm{s}}_{y \gets x} \coloneqq \lim_{n \to + \infty} (\Phi_y^n)^{-1} \circ I_{T^n y \gets T^n x} \circ \Phi_x^n \, ,$$ and unstable holonomies are defined likewise, taking $n \to -\infty$ instead. The proof of \[p.holonomies\] consists essentially in proving uniform convergence in these formulas, and it turns out that fiber-bunching is the precise condition for this to work. Except for minor adjustments, the argument is the same as in [@BGV [§]{}1.4], 