Projections of planar Mandelbrot measures

Abstract

Let $\mu$ be a planar Mandelbrot measure and $\pi_*\mu$ its orthogonal projection on one of the main axes. We study the thermodynamic and geometric properties of $\pi_*\mu$. We first show that $\pi_*\mu$ is exactly dimensional, with $\dim(\pi_*\mu)=\min(\dim(\mu),\dim(\nu))$, where~$\nu$ is the Bernoulli product measure obtained as the expectation of $\pi_*\mu$. We also prove that $\pi_*\mu$ is absolutely continuous with respect to $\nu$ if and only if $\dim(\mu)>\dim(\nu)$, 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 $\pi_*\mu$, as well as a new variational interpretation. We obtain the free energy function $\tau_{\pi_*\mu}$ of $\pi_*\mu$ 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 $\min(\tau_\nu,\tau_\mu)$, which can exhibit first order phase transitions. This is in contrast with the analyticity of $\tau_\mu$ over $[0,q_c)$. Also, we prove the validity of the multifractal formalism for $\pi_*\mu$ at each $\alpha\in (\tau_{\pi_*\mu}'(q_c-),\tau_{\pi_*\mu}'(0+)]$.