Surface dimension, tiles, and synchronising automata

TL;DR

This paper investigates the surface regularity of compact sets, computes this regularity for self-affine attractors and tiles, and explores its applications to wavelet regularity and synchronising automata.

Contribution

It introduces a measure of surface regularity, explicitly computes it for certain fractal sets, and links it to wavelet regularity and automata synchronization.

Findings

Surface regularity characterizes set smoothness and is explicitly computable for some fractals.

A refined regularity scale for multivariate Haar wavelets is established.

Surface regularity relates to the parameter of synchronization in automata theory.

Abstract

We study the surface regularity of compact sets $G \subset R^n$ which is equal to the supremum of numbers $s\ge 0$ such that the measure of the set $G_{\varepsilon}\setminus G$ does not exceed $C\varepsilon^{s}, \varepsilon > 0$, where $G_{\varepsilon}$ denotes the $\varepsilon$-neighbourhood of~$G$. The surface dimension is by definition the difference between~$n$ and the surface regularity. Those values provide a natural characterisation of regularity for sets of positive measure. We show that for self-affine attractors and tiles those characteristics are explicitly computable and find them for some popular tiles. This, in particular, gives a refined regularity scale for the multivariate Haar wavelets. The classification of attractors of the highest possible regularity is addressed. The relation between the surface regularity and the H\"older regularity of multivariate refinable functions and wavelets is found. Finally, the surface regularity is applied to the theory of synchronising automata, where it corresponds to the concept of parameter of synchronisation.