Scaling Limits of Self-Conformal and Self-similar Measures

We prove that every self-conformal measure on ℝ and every self-similar measure on ℝ^d is uniformly scaling and generates an ergodic fractal distribution, generalizing previous results by eliminating the need for separation conditions. Additionally, we establish applications to the prevalence of normal numbers in self-conformal sets, resonance between self-conformal measures on the line, and projections of self-affine measures on carpets. The talk is based on recent research with Bárány, Pyörälä, and Wu.