On the dimension and measure of inhomogeneous attractors

TL;DR

This paper advances the understanding of inhomogeneous attractors by establishing a general upper bound on their dimension using the maximum of the condensation set's dimension and a pressure-related measure, and explores their Hausdorff measure.

Contribution

It extends the dimension theory of inhomogeneous attractors to arbitrary bi-Lipschitz contractions and introduces the upper Lipschitz dimension as a key upper bound.

Findings

Maximum of condensation set dimension and upper Lipschitz dimension bounds attractor dimension

Developed methods to analyze Hausdorff measure of inhomogeneous attractors

Applicable to affine systems with low affinity dimension and conformal systems

Abstract

A central question in the field of inhomogeneous attractors has been to relate the dimension of an inhomogeneous attractor to the condensation set and associated homogeneous attractor. This has been achieved only in specific settings, with notable results by Olsen, Snigireva, Fraser and K\"aenm\"aki on inhomogeneous self-similar sets, and by Burrell and Fraser on inhomogeneous self-affine sets. This paper is devoted to filling a significant gap in the dimension theory of inhomogeneous attractors, by studying those formed from arbitrary bi-Lipschitz contractions. We show that the maximum of the dimension of the condensation set and a quantity related to pressure, which we term upper Lipschitz dimension, forms a natural and general upper bound on the dimension. Additionally, we begin a new line of enquiry; the methods developed are used to investigate the Hausdorff measure of inhomogeneous attractors. Our results have applications for affine systems with affinity dimension less than or equal to one and systems satisfying bounded distortion, such as conformal systems in dimensions greater than one.