Dimension theory of iterated function systems

Abstract

Let fS i gì D1 be an iterated function system (IFS) on R d with attractor K. Let . †; / denote the one-sided full shift over the alphabet f1; : : : ;`g. We define the projection entropy function h on the space of invariant measures on † associated with the coding map W † ! K and develop some basic ergodic properties about it. This concept turns out to be crucial in the study of dimensional properties of invariant measures on K. We show that for any conformal IFS (respectively, the direct product of finitely many conformal IFSs), without any separation condition, the projection of an ergodic measure under is always exactly dimensional and its Hausdorff dimension can be represented as the ratio of its projection entropy to its Lyapunov exponent (respectively, the linear combination of projection entropies associated with several coding maps). Furthermore, for any conformal IFS and certain affine IFSs, we prove a variational principle between the Hausdorff dimension of the attractors and that of projections of ergodic measures.

Slicing theorems and rigidity phenomena for self‐affine carpets

“…A measure is called a self‐similar measure if there exists a fully supported Bernoulli measure such that (recall the map from ()). These measures are known to be exact dimensional (in much greater generality, see [11]) of dimension .…”

Dimension of ergodic measures projected onto self‐similar sets with overlaps

“…in which case we write dim θ = s. Given a Borel probability measure µ on Ω we write Πµ for the push-forward of µ by Π. Assuming µ is σ-invariant and ergodic, it follows from [FH,Theorem 2.8] that Πµ is exact dimensional. We write h µ for the entropy of µ and χ µ for its Lyapunov exponent with respect to {r λ } λ∈Λ , i.e.…”

Self-Similar Sets: Projections, Sections and Percolation

“…The first step is to show that almost surely µ is exact dimensional (that is lim r→0 log µ(B(x, r))/ − log r exists and takes a constant value µ-almost everywhere), and also that for almost all θ , proj θ µ is exact dimensional with dim H proj θ µ = min{dim H µ, 1}. This is a random extension of the deterministic result of Feng and Hu [11], and uses an ergodic-theoretic argument to show that a natural 'shift-like' operator T on the set comprising sequences and random variables…”