Equi-homogeneity, Assouad Dimension and Non-autonomous Dynamics

TL;DR

This paper investigates the regularity and dimensional properties of fractal sets generated by both autonomous and non-autonomous iterated function systems, introducing the concept of equi-homogeneity and analyzing its implications for various fractal dimensions.

Contribution

It introduces the notion of equi-homogeneity for fractal sets, explores its relationship with other regularity and dimensional properties, and provides conditions for non-autonomous IFS attractors to be equi-homogeneous.

Findings

Self-similar sets satisfying Moran open-set condition are equi-homogeneous.

Equi-homogeneity is weaker than Ahlfors-David regularity and distinct from previous notions of dimensional equivalence.

Under certain conditions, multiple fractal dimensions coincide for equi-homogeneous sets.

Abstract

We show that self-similar sets arising from iterated function systems that satisfy the Moran open-set condition, a canonical class of fractal sets, are `equi-homogeneous'. This is a regularity property that, roughly speaking, means that at each fixed length-scale any two neighbourhoods of the set have covers of approximately equal cardinality. Self-similar sets are notable in that they are Ahlfors-David regular, which implies that their Assouad and box-counting dimensions coincide. More generally, attractors of non-autonomous iterated functions systems (where maps are allowed to vary between iterations) can have distinct Assouad and box-counting dimensions. Consequently the familiar notion of Ahlfors-David regularity is too strong to be useful in the analysis of this important class of sets, which include generalised Cantor sets and possess different dimensional behaviour at different length-scales. We further develop the theory of equi-homogeneity showing that it is a weaker property than Ahlfors-David regularity and distinct from any previously defined notion of dimensional equivalence. However, we show that if the upper and lower box-counting dimensions of an equi-homogeneous set are equal and `attained' in a sense we make precise then the lower Assouad, Hausdorff, packing, lower box-counting, upper box-counting and Assouad dimensions coincide. Our main results provide conditions under which the attractor of a non-autonomous iterated function system is equi-homogeneous and we use this to compute the Assouad dimension of a certain class of these highly non-trivial sets.