The dimension of projections of self-affine sets and measures

Kenneth John Falconer; Thomas Michael William Kempton

doi:10.5186/aasfm.2017.4232

The dimension of projections of self-affine sets and measures

Kenneth John Falconer, Thomas Michael William Kempton

Pure Mathematics

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

Abstract

Let E be a plane self-affine set defined by affine transformations with linear parts given by matrices with positive entries. We show that if μ is a Bernoulli measure on E with dim_Hμ = dim_Lμ, where dim_H and dim_L denote Hausdorff and Lyapunov dimensions, then the projection of μ in all but at most one direction has Hausdorff dimension min{dim_Hμ, 1}. We transfer this result to sets and show that many self-affine sets have projections of dimension min{dim_HE, 1} in all but at most one direction

Original language	English
Pages (from-to)	473-486
Number of pages	14
Journal	Annales Academiae Scientiarum Fennicae-Mathematica
Volume	42
DOIs	https://doi.org/10.5186/aasfm.2017.4232
Publication status	Published - 6 Feb 2017

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10.5186/aasfm.2017.4232

http://arxiv.org/abs/1511.03556

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title = "The dimension of projections of self-affine sets and measures",

abstract = "Let E be a plane self-affine set defined by affine transformations with linear parts given by matrices with positive entries. We show that if μ is a Bernoulli measure on E with dimHμ = dimLμ, where dimH and dimL denote Hausdorff and Lyapunov dimensions, then the projection of μ in all but at most one direction has Hausdorff dimension min{dimHμ, 1}. We transfer this result to sets and show that many self-affine sets have projections of dimension min{dimHE, 1} in all but at most one direction",

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AU - Kempton, Thomas Michael William

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N2 - Let E be a plane self-affine set defined by affine transformations with linear parts given by matrices with positive entries. We show that if μ is a Bernoulli measure on E with dimHμ = dimLμ, where dimH and dimL denote Hausdorff and Lyapunov dimensions, then the projection of μ in all but at most one direction has Hausdorff dimension min{dimHμ, 1}. We transfer this result to sets and show that many self-affine sets have projections of dimension min{dimHE, 1} in all but at most one direction

AB - Let E be a plane self-affine set defined by affine transformations with linear parts given by matrices with positive entries. We show that if μ is a Bernoulli measure on E with dimHμ = dimLμ, where dimH and dimL denote Hausdorff and Lyapunov dimensions, then the projection of μ in all but at most one direction has Hausdorff dimension min{dimHμ, 1}. We transfer this result to sets and show that many self-affine sets have projections of dimension min{dimHE, 1} in all but at most one direction

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JO - Annales Academiae Scientiarum Fennicae-Mathematica

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ER -

The dimension of projections of self-affine sets and measures

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Non-conformal repellers: Fractal and multifractal structure of non-conformal repellers

Kenneth John Falconer

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The dimension of projections of self-affine sets and measures

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Non-conformal repellers: Fractal and multifractal structure of non-conformal repellers

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Kenneth John Falconer

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