  Within the mathematical analysis of deep convolutional neural networks, the
wavelet scattering transform introduced by St\'ephane Mallat is a unique
example of how the ideas of multiscale analysis can be combined with a cascade
of modulus nonlinearities to build a nonexpansive, translation invariant signal
representation with provable geometric stability properties, namely Lipschitz
continuity to the action of small $C^2$ diffeomorphisms - a remarkable result
for both theoretical and practical purposes, inherently depending on the choice
of the filters and their arrangement into a hierarchical architecture. In this
note, we further investigate the intimate relationship between the scattering
structure and the regularity of the deformation in the H\"older regularity
scale $C^\alpha$, $\alpha >0$. We are able to precisely identify the stability
threshold, proving that stability is still achievable for deformations of class
$C^{\alpha}$, $\alpha>1$, whereas instability phenomena can occur at lower
regularity levels modelled by $C^\alpha$, $0\le \alpha <1$. While the behaviour
at the threshold given by Lipschitz (or even $C^1$) regularity remains beyond
reach, we are able to prove a stability bound in that case, up to $\varepsilon$
losses.
