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

Jonathan MacDonald Fraser, Thomas Jordan, Natalia Jurga

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

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.
Original languageEnglish
Pages (from-to)87–111
JournalJournal of Fractal Geometry
Volume7
Issue number1
Early online date13 Nov 2019
DOIs
Publication statusPublished - 2020

Keywords

  • Self-affine set
  • Käenmäki measure
  • Quasi-Bernoulli measure
  • Exact dimensional
  • Ledrappier-Young formula

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