Distinct dimensions for attractors of bi-Lipschitz iterated function systems
Dynamical Systems
2025-09-29 v1 Classical Analysis and ODEs
Metric Geometry
Abstract
In this paper, we construct an iterated function system on the line consisting of two bi-Lipschitz contractions whose attractor has distinct lower, Hausdorff, lower box, upper box, and Assouad dimensions, thereby providing negative answers to certain folklore questions. Furthermore, as a by-product of our study of bi-Lipschitz IFSs, we construct IFSs within this family that exhibit interesting fractal behaviour. In particular, we prove the following two statements: (i) There exists a bi-Lipschitz IFS for which the pushforward of any ergodic measure with positive entropy is not exact dimensional; (ii) There exists a bi-Lipschitz IFS whose attractor has empty interior yet positive Lebesgue measure.
Cite
@article{arxiv.2509.22084,
title = {Distinct dimensions for attractors of bi-Lipschitz iterated function systems},
author = {Simon Baker and Amlan Banaji and De-Jun Feng and Chun-Kit Lai and Ying Xiong},
journal= {arXiv preprint arXiv:2509.22084},
year = {2025}
}