Regularity of non-autonomous self-similar sets
Antti Käenmäki, Alex Rutar
TL;DR
This work develops a comprehensive framework for the Assouad dimension of non-autonomous self-similar sets, showing that under the bounded neighbourhood condition the local overlap counts control global scaling so that $ ext{dim}_A K$ equals the symbolic limit $ obreak ext{lim}_{m\to\fty} ext{sup}_{n}\theta(n,m)$. The authors connect symbolic pressures, disc-packing, and coarse bi-Lipschitz properties to reduce geometric questions to combinatorial data, while clarifying the roles of OSC, cone conditions, and branching. They prove submaximality of the key pressure-derived quantities and supply sharp examples highlighting the necessity of BN and the subtle interplay with other separation assumptions. The results unify and extend prior formulas for autonomous or constrained non-autonomous IFSs and provide a flexible toolkit for analyzing fine local scaling in these inhomogeneous fractal constructions. Overall, the paper advances a precise, broadly applicable dimension theory for non-autonomous self-similar sets with practical implications for their local geometry and dimension spectra.
Abstract
Non-autonomous self-similar sets are a family of compact sets which are, in some sense, highly homogeneous in space but highly inhomogeneous in scale. The main purpose of this note is to clarify various regularity properties and separation conditions relevant for the fine local scaling properties of these sets. A simple application of our results is a precise formula for the Assouad dimension of non-autonomous self-similar sets in $\mathbb{R}^d$ satisfying a certain ``bounded neighbourhood'' condition, which generalizes earlier work of Li--Li--Miao--Xi and Olson--Robinson--Sharples. We also see that the bounded neighbourhood assumption is, in few different senses, as general as possible.