Related papers: Ledrappier-Young formula and exact dimensionality …
Assuming strong irreducibility and proximality, we prove that the Furstenberg measure, corresponding to a finitely supported measure on the general linear group of a finite dimensional real vector space, is exact dimensional. We also…
Ledrappier and Young introduced a relation between entropy, Lyapunov exponents and dimension for invariant measures of diffeomorphisms on compact manifolds. In this paper, we show that a self-affine measure on the plane satisfies the…
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…
In this paper we study the dimension theory of planar self-affine sets satisfying dominated splitting in the linear parts and strong separation condition. The main results of this paper is the existence of dimension maximizing Gibbs…
An affine iterated function system is a finite collection of affine invertible contractions and the invariant set associated to the mappings is called self-affine. In 1988, Falconer proved that, for given matrices, the Hausdorff dimension…
We show that, in a generic setting, self-affine and almost self-affine measures are exact dimensional, with local dimension equal almost everywhere to the information dimension and given by the zero of a superadditive pressure functional.
We prove that if $\mu$ is a self-affine measure in the plane whose defining IFS acts totally irreducibly on $\mathbb{RP}^1$ and satisfies an exponential separation condition, then its dimension is equal to its Lyapunov dimension. We also…
Let $\{S_i\}_{i\in \Lambda}$ be a finite contracting affine iterated function system (IFS) on ${\Bbb R}^d$. Let $(\Sigma,\sigma)$ denote the two-sided full shift over the alphabet $\Lambda$, and $\pi:\Sigma\to {\Bbb R}^d$ be the coding map…
We study the pointwise dimension for a new class of projection measures on arbitrary fractal limit sets without separation conditions. We prove that the pointwise dimension exists a.e. for this class of measures associated to equilibrium…
We study a natural class of invariant measures supported on the attractors of a family of nonlinear, non-conformal iterated function systems introduced by Falconer, Fraser and Lee. These are pushforward quasi-Bernoulli measures, a class…
For a self-affine measure on a Bedford-McMullen carpet we prove that its quantization dimension exists and determine its exact value. Further, we give various sufficient conditions for the corresponding upper and lower quantization…
Under mild conditions we show that the affinity dimension of a planar self-affine set is equal to the supremum of the Lyapunov dimensions of self-affine measures supported on self-affine proper subsets of the original set. These self-affine…
We show the existence of the local dimension of an invariant probability measure on an infinitely generated self-affine set, for almost all translations. This implies that an ergodic probability measure is exactly dimensional. Furthermore…
Let $X=\bigcup\varphi_{i}X$ be a strongly separated self-affine set in $\mathbb{R}^2$ (or one satisfying the strong open set condition). Under mild non-compactness and irreducibility assumptions on the matrix parts of the $\varphi_{i}$, we…
Let $\mu$ be a self-affine measure on $\mathbb{R}^{d}$ associated to an affine IFS $\Phi$ and a positive probability vector $p$. Suppose that the maps in $\Phi$ do not have a common fixed point, and that standard irreducibility and…
We study the Assouad and quasi-Assoaud dimensions of dominated rectangular self-affine sets in the plane. In contrast to previous work on the dimension theory of self-affine sets, we assume that the sets satisfy certain separation…
We study the multifractal analysis of self-similar measures arising from random homogeneous iterated function systems. Under the assumption of the uniform strong separation condition, we see that this analysis parallels that of the…
We prove that self-similar measures on the real line are absolutely continuous for almost all parameters in the super-critical region, in particular confirming a conjecture of S-M. Ngai and Y. Wang. While recently there has been much…
We study measures $\mu$ on the plane with two independent Alberti representations. It is known, due to Alberti, Cs\"ornyei, and Preiss, that such measures are absolutely continuous with respect to Lebesgue measure. The purpose of this paper…
A breakthrough result of B\'ar\'any, Hochman and Rapaport published in 2019 established that every self-affine measure on $\mathbb{R}^2$ satisfying certain mild non-degeneracy conditions has Hausdorff dimension equal to its Lyapunov…