Geometry of canonical self-similar tilings

TL;DR

This paper characterizes when the parallel set of a self-similar set can be described by a canonical tiling, enabling tube formulas for volume, and generalizes tiling constructions to self-affine sets with empty interior.

Contribution

It provides geometric characterizations for the relation between parallel sets and tilings in self-similar sets and extends tiling methods to self-affine sets with empty interior.

Findings

Characterization of when $F_5epsilon$ equals $T_{-5epsilon} b7 C_5epsilon$

Derivation of tube formulas for self-similar sets

Generalization of tiling construction to self-affine sets with empty interior

Abstract

We give several different geometric characterizations of the situation in which the parallel set $F_\epsilon$ of a self-similar set $F$ can be described by the inner $\epsilon$-parallel set $T_{-\epsilon}$ of the associated canonical tiling $\mathcal T$, in the sense of \cite{SST}. For example, $F_\epsilon=T_{-\epsilon} \cup C_\epsilon$ if and only if the boundary of the convex hull $C$ of $F$ is a subset of $F$, or if the boundary of $E$, the unbounded portion of the complement of $F$, is the boundary of a convex set. In the characterized situation, the tiling allows one to obtain a tube formula for $F$, i.e., an expression for the volume of $F_\epsilon$ as a function of $\epsilon$. On the way, we clarify some geometric properties of canonical tilings. Motivated by the search for tube formulas, we give a generalization of the tiling construction which applies to all self-affine sets $F$ having empty interior and satisfying the open set condition. We also characterize the relation between the parallel sets of $F$ and these tilings.