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We study equilibrium measures (Kaenmaki 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 the important coordinate projection of the measure. In particular, we do this by showing that the Kaenmaki measure is equal to the sum of (the pushforwards) of two Gibbs measures on an associated subshift of finite type.
We prove that the upper box dimension of an inhomogeneous self-affine set is bounded above by the maximum of the affinity dimension and the dimension of the condensation set. In addition, we determine sufficient conditions for this upper bound to be
Given a non-empty bounded subset of hyperbolic space and a Kleinian group acting on that space, the orbital set is the orbit of the given set under the action of the group. We may view orbital sets as bounded (often fractal) subsets of Euclidean spac
In this paper we prove that if ${varphi_i(x)=lambda x+t_i}$ is an equicontractive iterated function system and $b$ is a positive integer satisfying $frac{log b}{log |lambda|} otinmathbb{Q},$ then almost every $x$ is normal in base $b$ for any non-atomic self-similar measure of ${varphi_i}$.
In this paper we compute the dimension of a class of dynamically defined non-conformal sets. Let $Xsubseteqmathbb{T}^2$ denote a Bedford-McMullen set and $T:Xto X$ the natural expanding toral endomorphism which leaves $X$ invariant. For an open set $
For self-similar sets on $mathbb{R}$ satisfying the exponential separation condition we show that the natural projections of shift invariant ergodic measures is equal to $min{1,frac{h}{-chi}}$, where $h$ and $chi$ are the entropy and Lyapunov exponen