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Hausdorff dimension of multidimensional multiplicative subshifts

Part of: Smooth dynamical systems: general theory Classical measure theory

Published online by Cambridge University Press:  19 July 2023

JUNG-CHAO BAN ,
WEN-GUEI HU  and
GUAN-YU LAI Open the ORCID record for GUAN-YU LAI [Opens in a new window]
JUNG-CHAO BAN
Affiliation:
Department of Mathematical Sciences, National Chengchi University, Taipei 11605, Taiwan, ROC Math. Division, National Center for Theoretical Science, National Taiwan University, Taipei 10617, Taiwan, ROC (e-mail: jcban@nccu.edu.tw)
WEN-GUEI HU
Affiliation:
College of Mathematics, Sichuan University, Chengdu 610064, China (e-mail: wghu@scu.edu.cn)
GUAN-YU LAI*
Affiliation:
Department of Mathematical Sciences, National Chengchi University, Taipei 11605, Taiwan, ROC
*
e-mail: gylai@nccu.edu.tw
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Abstract

The purpose of this study is two-fold. First, the Hausdorff dimension formula of the multidimensional multiplicative subshift (MMS) in $\mathbb {N}^d$ is presented. This extends the earlier work of Kenyon et al [Hausdorff dimension for fractals invariant under multiplicative integers. Ergod. Th. & Dynam. Sys. 32(5) (2012), 1567–1584] from $\mathbb {N}$ to $\mathbb {N}^d$. In addition, the preceding work of the Minkowski dimension of the MMS in $\mathbb {N}^d$ is applied to show that their Hausdorff dimension is strictly less than the Minkowski dimension. Second, the same technique allows us to investigate the multifractal analysis of multiple ergodic average in $\mathbb {N}^d$. Precisely, we extend the result of Fan et al, [Multifractal analysis of some multiple ergodic averages. Adv. Math. 295 (2016), 271–333] of the multifractal analysis of multiple ergodic average from $\mathbb {N}$ to $\mathbb {N}^d$.

Keywords

multiplicative subshiftsmultiple ergodic averagesHausdorff dimensionmultifractal analysis

MSC classification

Primary: 28A80: Fractals 37C45: Dimension theory of dynamical systems 28A78: Hausdorff and packing measures
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 5 , May 2024 , pp. 1239 - 1268
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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