Hausdorff dimension of planar self-affine sets and measures

Abstract

We calculate the Assouad dimension of a planar self-affine set X satisfying the strong separation condition and the projection condition and show that X is minimal for the conformal Assouad dimension. Furthermore, we see that such a self-affine set X adheres to very strong tangential regularity by showing that any two points of X, which are generic with respect to a self-affine measure having simple Lyapunov spectrum, share the same collection of tangent sets.

Dimension of self-affine sets for fixed translation vectors

“…Very recently, after this article was finished, Bárány, Hochman, and Rapaport showed that the dimension of planar self‐affine measures is equal to the Lyapunov dimension if the strong open set condition holds and the matrix tuple is strongly irreducible. Furthermore, we point out that, under the assumption that is linearly independent (which, in particular, forces ), one can prove the result of Theorem for every ergodic measure.…”

Analyticity of the affinity dimension for planar iterated function systems with matrices which preserve a cone

“…Therefore without loss of generality we can assume it is positive. Therefore ψ A (1) so it is sufficient to show that log(1 + t 3 (t−t 1 ) t 1 t 3 −t 2 t 4 ) can be written as a convergent power series in (t − t 1 ). Indeed…”

“…A large part of the dimension theory of self-similar sets is well understood. For example, if we denote the contraction ratios of the similarities S (i) by r := {r i } i∈I then under suitable separation assumptions on the pieces {S (i) (F )} i∈I it is well known that all notions of dimension of F coincide and the common value is given by the solution s to the pressure-type formula P r (s) = i∈I r s i = 1 (1) Date: 17th April 2019. 1 which we call the similarity dimension.…”

“…For example, if we denote the contraction ratios of the similarities S (i) by r := {r i } i∈I then under suitable separation assumptions on the pieces {S (i) (F )} i∈I it is well known that all notions of dimension of F coincide and the common value is given by the solution s to the pressure-type formula P r (s) = i∈I r s i = 1 (1) Date: 17th April 2019. 1 which we call the similarity dimension. However, when we pass to the more general selfaffine setting where the matrices A (i) are allowed to exhibit different rates of contraction in different directions, the problem of calculating the dimension becomes drastically more complex.…”

“…We let dim H and dim B respectively denote the Hausdorff dimension and box dimension of a subset of R d . Theorem 1.1 (Theorem 1.1 [1]). Let Φ = {S (i) : R 2 → R 2 : i ∈ I} be an affine iterated function system and F = i∈I S (i) (F ) be the associated self-affine set.…”