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On the coincidence of the Hausdorff and box dimensions for some affine-invariant sets

Part of: Limit theorems Classical measure theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  08 September 2025

ZHOU FENG Open the ORCID record for ZHOU FENG [Opens in a new window]
ZHOU FENG*
Affiliation:
Department of Mathematics, https://ror.org/00t33hh48 The Chinese University of Hong Kong , Shatin, Hong Kong
*
e-mail: zfeng@math.cuhk.edu.hk
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Abstract

Let $ K $ be a compact subset of the d-torus invariant under an expanding diagonal endomorphism with s distinct eigenvalues. Suppose the symbolic coding of K satisfies weak specification. When $ s \leq 2 $, we prove that the following three statements are equivalent: (A) the Hausdorff and box dimensions of $ K $ coincide; (B) with respect to some gauge function, the Hausdorff measure of $ K $ is positive and finite; (C) the Hausdorff dimension of the measure of maximal entropy on $ K $ attains the Hausdorff dimension of $ K $. When $ s \geq 3 $, we find some examples in which statement (A) does not hold but statement (C) holds, which is a new phenomenon not appearing in the planar cases. Through a different probabilistic approach, we establish the equivalence of statements (A) and (B) for Bedford–McMullen sponges.

Keywords

Hausdorff dimensionbox dimensionHausdorff measurenon-conformal invariant set

MSC classification

Primary: 28A80: Fractals
Secondary: 37D35: Thermodynamic formalism, variational principles, equilibrium states 60F15: Strong theorems

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Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 33
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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