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Dimensions of equilibrium measures on a class of planar self-affine sets

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Date
2020
Author
Fraser, Jonathan MacDonald
Jordan, Thomas
Jurga, Natalia
Keywords
Self-affine set
Käenmäki measure
Quasi-Bernoulli measure
Exact dimensional
Ledrappier-Young formula
QA Mathematics
T-NDAS
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Abstract
We study equilibrium measures (Käenmäki measures) supported on self-affine sets generated by a finite collection of diagonal and anti-diagonal matrices acting on the plane and satisfying the strong separation property. Our main result is that such measures are exact dimensional and the dimension satisfies the Ledrappier–Young formula, which gives an explicit expression for the dimension in terms of the entropy and Lyapunov exponents as well as the dimension of a coordinate projection of the measure. In particular, we do this by showing that the Käenmäki measure is equal to the sum of (the pushforwards) of two Gibbs measures on an associated subshift of finite type.
Citation
Fraser , J M , Jordan , T & Jurga , N 2020 , ' Dimensions of equilibrium measures on a class of planar self-affine sets ' , Journal of Fractal Geometry , vol. 7 , no. 1 , pp. 87–111 . https://doi.org/10.4171/JFG/85
Publication
Journal of Fractal Geometry
Status
Peer reviewed
DOI
https://doi.org/10.4171/JFG/85
ISSN
2308-1309
Type
Journal article
Rights
© 2018, European 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/10.4171/JFG/85
Description
Funding: JMF was financially supported by a Leverhulme Trust Research Fellowship (RF-2016-500).
Collections
  • University of St Andrews Research
URL
https://arxiv.org/abs/1706.06833v2
URI
http://hdl.handle.net/10023/16361

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