Projections of planar Mandelbrot measures

Abstract

Let $μ$ be a planar Mandelbrot measure and $π_*μ$ its orthogonal projection on one of the main axes. We study the thermodynamic and geometric properties of $π_*μ$. We first show that $π_*μ$ is exactly dimensional, with $\dim(π_*μ)=\min(\dim(μ),\dim(ν))$, where~$ν$ is the Bernoulli product measure obtained as the expectation of $π_*μ$. We also prove that $π_*μ$ is absolutely continuous with respect to $ν$ if and only if $\dim(μ)>\dim(ν)$, and find sufficient conditions for the equivalence of these measures. Our results provides a new proof of Dekking-Grimmett-Falconer formula for the Hausdorff and box dimension of the topological support of $π_*μ$, as well as a new variational interpretation. We obtain the free energy function $τ_{π_*μ}$ of $π_*μ$ on a wide subinterval $[0,q_c)$ of $\mathbb{R}_+$. For $q\in[0,1]$, it is given by a variational formula which sometimes yields phase transitions of order larger than~1. For $q>1$, it is given by $\min(τ_ν,τ_μ)$, which can exhibit first order phase transitions. This is in contrast with the analyticity of $τ_μ$ over $[0,q_c)$. Also, we prove the validity of the multifractal formalism for $π_*μ$ at each $α\in (τ_{π_*μ}'(q_c-),τ_{π_*μ}'(0+)]$.

83 pages, 7 figures