The Hausdorff dimension of the projections of self-affine carpets

AJ Ferguson; TM Jordan; P Shmerkin

doi:10.4064/fm209-3-1

The Hausdorff dimension of the projections of self-affine carpets

AJ Ferguson, TM Jordan, P Shmerkin

Research output: Contribution to journal › Article (Academic Journal) › peer-review

15 Citations (Scopus)

Abstract

We study the orthogonal projections of a large class of self-affine carpets, which contains the carpets of Bedford and McMullen as special cases. Our main result is that if Λ is such a carpet, and certain natural irrationality conditions hold, then every orthogonal projection of Λ in a non-principal direction has Hausdorff dimension min(γ,1), where γ is the Hausdorff dimension of Λ. This generalizes a recent result of Peres and Shmerkin on sums of Cantor sets.

Translated title of the contribution	The Hausdorff dimension of the projections of self-affine carpets
Original language	English
Pages (from-to)	193 - 213
Number of pages	21
Journal	Fundamenta Mathematicae
Volume	209, number 3
DOIs	https://doi.org/10.4064/fm209-3-1
Publication status	Published - Sep 2010

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title = "The Hausdorff dimension of the projections of self-affine carpets",

abstract = "We study the orthogonal projections of a large class of self-affine carpets, which contains the carpets of Bedford and McMullen as special cases. Our main result is that if Λ is such a carpet, and certain natural irrationality conditions hold, then every orthogonal projection of Λ in a non-principal direction has Hausdorff dimension min(γ,1), where γ is the Hausdorff dimension of Λ. This generalizes a recent result of Peres and Shmerkin on sums of Cantor sets.",

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AU - Ferguson, AJ

AU - Jordan, TM

AU - Shmerkin, P

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N2 - We study the orthogonal projections of a large class of self-affine carpets, which contains the carpets of Bedford and McMullen as special cases. Our main result is that if Λ is such a carpet, and certain natural irrationality conditions hold, then every orthogonal projection of Λ in a non-principal direction has Hausdorff dimension min(γ,1), where γ is the Hausdorff dimension of Λ. This generalizes a recent result of Peres and Shmerkin on sums of Cantor sets.

AB - We study the orthogonal projections of a large class of self-affine carpets, which contains the carpets of Bedford and McMullen as special cases. Our main result is that if Λ is such a carpet, and certain natural irrationality conditions hold, then every orthogonal projection of Λ in a non-principal direction has Hausdorff dimension min(γ,1), where γ is the Hausdorff dimension of Λ. This generalizes a recent result of Peres and Shmerkin on sums of Cantor sets.

U2 - 10.4064/fm209-3-1

DO - 10.4064/fm209-3-1

M3 - Article (Academic Journal)

VL - 209, number 3

SP - 193

EP - 213

JO - Fundamenta Mathematicae

JF - Fundamenta Mathematicae

SN - 0016-2736

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