Micromeasure distributions and applications for conformally generated fractals

TL;DR

This paper investigates the scaling behavior of Gibbs measures on self-conformal fractals and explores applications to distance set problems and projection dimensions, covering various fractal types.

Contribution

It introduces a new analysis of Gibbs measures' scaling scenery on diverse self-conformal fractals with applications to geometric measure theory.

Findings

Characterization of Gibbs measures on self-conformal fractals

Results on Falconer's distance set problem

Dimension estimates for projections of fractals

Abstract

We study the scaling scenery of Gibbs measures for subshifts of finite type on self-conformal fractals and applications to Falconer's distance set problem and dimensions of projections. Our analysis includes hyperbolic Julia sets, limit sets of Schottky groups and graph-directed self-similar sets.