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

Thomas Jordan,Ariel Rapaport

doi:10.1112/plms.12337

Abstract

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 exponent respectively. The proof relies on Shmerkin's recent result on the $L^{q}$ dimension of self-similar measures. We also use the same method to give results on convolutions and orthogonal projections of ergodic measures projected onto self-similar sets.

Highlights

Introduction and statement of resultsThe dimension of self-similar measures on the line has been the subject of much attention going back over 40 years, since [8]
A Borel probability measure θ on R is said to be exact dimensional if there exists a number s 0 with lim log θ(B(x, δ)) = s for θ-a.e. x ∈ R, δ↓0 log δ in which case we write dim θ = s
We first construct suitable self-similar measures, and apply Shmerkin’s results on the Lq dimension to these measures. We show that these results, together with the connection between the self-similar and ergodic measures, yield that the dimension can only drop by an amount which can be made arbitrarily small

Summary

Introduction and statement of results

The dimension of self-similar measures on the line has been the subject of much attention going back over 40 years, since [8]. In [5, Theorem 1.8; 6, Theorem 1.10] Hochman has shown that in quite general parametrized families of self-similar iterated function systems, the exponential separation condition holds outside of a set of parameters of packing and Hausdorff co-dimension at least 1. A Borel probability measure θ on R is said to be exact dimensional if there exists a number s 0 with lim log θ(B(x, δ)) = s for θ-a.e. x ∈ R, δ↓0 log δ in which case we write dim θ = s. We can use Shmerkin’s result on the Lq dimension of self-similar measures with exponential separation. Since it always holds that dim Πμ β (see [4, Theorem 2.8; 13, Section 3] for details of how to prove this), this completes the proof of Theorem 1.1

Convolutions of ergodic measures

Orthogonal projections of ergodic measures

Applications and remarks