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

Jonathan Fraser, Thomas Jordan, Natalia Jurga

Research output: Contribution to journalArticle (Academic Journal)peer-review

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We study equilibrium measures (K¨aenm¨aki 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 K¨aenm¨aki measure is equal to the sum of (the pushforwards) of two Gibbs measures on an associated subshift of finite type.
Mathematics Subject Classification 2010: primary: 37C45; secondary: 28A80.
Key words and phrases: self-affine set, K¨aenm¨aki measure, quasi-Bernoulli measure, exact dimensional, Ledrappier-Young formula.
Original languageEnglish
Pages (from-to)87-111
JournalJournal of Fractal Geometry
Issue number1
Early online date13 Nov 2019
Publication statusE-pub ahead of print - 13 Nov 2019


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