Hausdorff dimension of planar self-affine sets and measures with overlaps

Michael Hochman; Ariel Rapaport

doi:10.4171/JEMS/1127

Hausdorff dimension of planar self-affine sets and measures with overlaps

Michael Hochman, Ariel Rapaport

The Hebrew University of Jerusalem, Technion - Israel Institute of Technology

Research output: Contribution to journal › Article › peer-review

Abstract

We prove that if μ is a self-affine measure in the plane whose defining IFS acts totally irreducibly on RP1 and satisfies an exponential separation condition, then its dimension is equal to its Lyapunov dimension.We also treat a class of reducible systems. This extends our previous work on the subject with Bárány to the overlapping case.

Original language	English
Pages (from-to)	2361-2441
Number of pages	81
Journal	Journal of the European Mathematical Society
Volume	24
Issue number	7
DOIs	https://doi.org/10.4171/JEMS/1127
State	Published - 2022

Keywords

Hausdorff dimension
Lyapunov dimension
self-affine measure
self-affine set

All Science Journal Classification (ASJC) codes

General Mathematics
Applied Mathematics

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10.4171/JEMS/1127

https://doi.org/10.4171/jems/1127

Cite this

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title = "Hausdorff dimension of planar self-affine sets and measures with overlaps",

abstract = "We prove that if μ is a self-affine measure in the plane whose defining IFS acts totally irreducibly on RP1 and satisfies an exponential separation condition, then its dimension is equal to its Lyapunov dimension.We also treat a class of reducible systems. This extends our previous work on the subject with B{\'a}r{\'a}ny to the overlapping case.",

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AB - We prove that if μ is a self-affine measure in the plane whose defining IFS acts totally irreducibly on RP1 and satisfies an exponential separation condition, then its dimension is equal to its Lyapunov dimension.We also treat a class of reducible systems. This extends our previous work on the subject with Bárány to the overlapping case.

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KW - Lyapunov dimension

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KW - self-affine set

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Hausdorff dimension of planar self-affine sets and measures with overlaps

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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