Hausdorff dimension of planar self-affine sets and measures

Balázs Bárány; Michael Hochman; Ariel Rapaport

doi:10.1007/s00222-018-00849-y

Hausdorff dimension of planar self-affine sets and measures

Balázs Bárány, Michael Hochman, Ariel Rapaport

Einstein Institute of Mathematics

The Hebrew University of Jerusalem

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

Abstract

Let X =^⋃ϕ_i X be a strongly separated self-affine set in R² (or one satisfying the strong open set condition). Under mild non-conformality and irreducibility assumptions on the matrix parts of the ϕ_i, we prove that dim X is equal to the affinity dimension, and similarly for self-affine measures and the Lyapunov dimension. The proof is via analysis of the dimension of the orthogonal projections of the measures, and relies on additive combinatorics methods.

Original language	English
Pages (from-to)	601-659
Number of pages	59
Journal	Inventiones Mathematicae
Volume	216
Issue number	3
DOIs	https://doi.org/10.1007/s00222-018-00849-y
State	Published - 2019

All Science Journal Classification (ASJC) codes

General Mathematics

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

abstract = "Let X =⋃ϕi X be a strongly separated self-affine set in R2 (or one satisfying the strong open set condition). Under mild non-conformality and irreducibility assumptions on the matrix parts of the ϕi, we prove that dim X is equal to the affinity dimension, and similarly for self-affine measures and the Lyapunov dimension. The proof is via analysis of the dimension of the orthogonal projections of the measures, and relies on additive combinatorics methods.",

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AU - Hochman, Michael

AU - Rapaport, Ariel

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AB - Let X =⋃ϕi X be a strongly separated self-affine set in R2 (or one satisfying the strong open set condition). Under mild non-conformality and irreducibility assumptions on the matrix parts of the ϕi, we prove that dim X is equal to the affinity dimension, and similarly for self-affine measures and the Lyapunov dimension. The proof is via analysis of the dimension of the orthogonal projections of the measures, and relies on additive combinatorics methods.

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JO - Inventiones Mathematicae

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ER -

Hausdorff dimension of planar self-affine sets and measures

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