Abstract
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.
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 language | English |
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Pages (from-to) | 87-111 |
Journal | Journal of Fractal Geometry |
Volume | 7 |
Issue number | 1 |
Early online date | 13 Nov 2019 |
DOIs | |
Publication status | E-pub ahead of print - 13 Nov 2019 |