On The Lq dimensions of measures on Heuter-Lalley type self-affine sets

Jonathan M. Fraser; Tom Kempton

doi:10.1090/proc/13672

On The L^q dimensions of measures on Heuter-Lalley type self-affine sets

Jonathan M. Fraser, Tom Kempton

Pure Mathematics

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

Abstract

We study the L^q-dimensions of self-affine measures and the Käenmäki measure on a class of self-affine sets similar to the class considered by Hueter and Lalley. We give simple, checkable conditions under which the L^q-dimensions are equal to the value predicted by Falconer for a range of q. As a corollary this gives a wider class of self-affine sets for which the Hausdorff dimension can be explicitly calculated. Our proof combines the potential theoretic approach developed by Hunt and Kaloshin with recent advances in the dynamics of self-affine sets.

Original language	English
Pages (from-to)	161-173
Journal	Proceedings of the American Mathematical Society
Volume	146
Issue number	1
Early online date	1 Aug 2017
DOIs	https://doi.org/10.1090/proc/13672
Publication status	Published - Jan 2018

Keywords

Self-affine set
Self-affine measure
Käenmäki measure
Lq-dimensions
Affinity dimension
Potential theoretic method

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10.1090/proc/13672

Fraser_2017_Lq_energy_PAMS_AAM
© 2017, American Mathematical Society. This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1090/proc/13672
Accepted author manuscript, 296 KB

Fractal Geometry and Dimension: Fractal Geometry and dimension theory
Fraser, J. (PI)
The Leverhulme Trust
1/09/16 → 30/06/18
Project: Fellowship
Non-conformal repellers: Fractal and multifractal structure of non-conformal repellers
Falconer, K. J. (PI)
EPSRC
13/01/14 → 12/01/17
Project: Standard

Cite this

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title = "On The Lq dimensions of measures on Heuter-Lalley type self-affine sets",

abstract = "We study the Lq-dimensions of self-affine measures and the K{\"a}enm{\"a}ki measure on a class of self-affine sets similar to the class considered by Hueter and Lalley. We give simple, checkable conditions under which the Lq-dimensions are equal to the value predicted by Falconer for a range of q. As a corollary this gives a wider class of self-affine sets for which the Hausdorff dimension can be explicitly calculated. Our proof combines the potential theoretic approach developed by Hunt and Kaloshin with recent advances in the dynamics of self-affine sets.",

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author = "Fraser, {Jonathan M.} and Tom Kempton",

note = "The authors were financially supported by an LMS Scheme 4 Research in Pairs grant. The second author also acknowledges financial support from the EPSRC grant EP/K029061/1, and the first author acknowledges financial support from a Leverhulme Trust Research Fellowship (RF-2016-500).",

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N2 - We study the Lq-dimensions of self-affine measures and the Käenmäki measure on a class of self-affine sets similar to the class considered by Hueter and Lalley. We give simple, checkable conditions under which the Lq-dimensions are equal to the value predicted by Falconer for a range of q. As a corollary this gives a wider class of self-affine sets for which the Hausdorff dimension can be explicitly calculated. Our proof combines the potential theoretic approach developed by Hunt and Kaloshin with recent advances in the dynamics of self-affine sets.

AB - We study the Lq-dimensions of self-affine measures and the Käenmäki measure on a class of self-affine sets similar to the class considered by Hueter and Lalley. We give simple, checkable conditions under which the Lq-dimensions are equal to the value predicted by Falconer for a range of q. As a corollary this gives a wider class of self-affine sets for which the Hausdorff dimension can be explicitly calculated. Our proof combines the potential theoretic approach developed by Hunt and Kaloshin with recent advances in the dynamics of self-affine sets.

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KW - Käenmäki measure

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KW - Potential theoretic method

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JO - Proceedings of the American Mathematical Society

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ER -

On The Lq dimensions of measures on Heuter-Lalley type self-affine sets

Abstract

Keywords

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Projects

Fractal Geometry and Dimension: Fractal Geometry and dimension theory

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

Cite this

On The L^q dimensions of measures on Heuter-Lalley type self-affine sets