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arXiv · 2509.22084

Distinct dimensions for attractors of bi-Lipschitz iterated function systems

Simon Baker·Amlan Banaji·De-Jun Feng·Chun-Kit Lai·Ying Xiong
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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.

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Simon Baker, Amlan Banaji, De-Jun Feng, Chun-Kit Lai, Ying Xiong. 2025-09-26. Distinct dimensions for attractors of bi-Lipschitz iterated function systems. https://arxiv.org/abs/2509.22084

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