## 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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Journal | Journal of Fractal Geometry |

Publication status | Accepted/In press - 18 Jun 2018 |