Scaling limits of self-conformal measures

TL;DR

This paper proves that all self-conformal measures on the real line are uniformly scaling and generate ergodic fractal distributions, extending previous results without separation conditions and exploring implications for normal numbers and measure projections.

Contribution

It establishes the uniform scaling property for all self-conformal measures on the real line without separation assumptions, broadening the understanding of their fractal structure.

Findings

Self-conformal measures are uniformly scaling and generate ergodic fractal distributions.

Applications include prevalence of normal numbers in self-conformal sets.

Results impact projections of self-affine measures on carpets.

Abstract

We show that any self-conformal measure $\mu$ on $\mathbb{R}$ is uniformly scaling and generates an ergodic fractal distribution. This generalizes existing results by removing the need for any separation condition. We also obtain applications to the prevalence of normal numbers in self-conformal sets, the resonance between self-conformal measures on the line, and projections of self-affine measures on carpets.