Projections of Gibbs measures on self-conformal sets

Abstract

We show that for Gibbs measures on self-conformal sets in $\mathbb{R}^d$ $(d\ge2)$ satisfying certain minimal assumptions, without requiring any separation condition, the Hausdorff dimension of orthogonal projections to $k$-dimensional subspaces is the same and is equal to the maximum possible value in all directions. As a corollary we show that Falconer's distance set conjecture holds for this class of self-conformal sets satisfying the open set condition.