JournalsjfgVol. 7, No. 1pp. 87–111

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

  • Jonathan M. Fraser

    University of St Andrews, UK
  • Thomas Jordan

    University of Bristol, UK
  • Natalia Jurga

    University of Surrey, Guildford, UK
Dimensions of equilibrium measures on a class of planar self-affine sets cover

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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.

Cite this article

Jonathan M. Fraser, Thomas Jordan, Natalia Jurga, Dimensions of equilibrium measures on a class of planar self-affine sets. J. Fractal Geom. 7 (2020), no. 1, pp. 87–111

DOI 10.4171/JFG/85