Projections of planar Mandelbrot measures

Abstract

Let $\mu$ be a planar Mandelbrot measure and ${\pi }_{\ast }\mu$ its orthogonal projection on one of the main axes. We study the thermodynamic and geometric properties of ${\pi }_{\ast }\mu$. We first show that ${\pi }_{\ast }\mu$ is exactly dimensional, with $\dim ({\pi }_{\ast }\mu )=\min (\dim (\mu ),\dim (\nu ))$, where~$\nu$ is the Bernoulli product measure obtained as the expectation of ${\pi }_{\ast }\mu$. We also prove that ${\pi }_{\ast }\mu$ 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 ${\pi }_{\ast }\mu$, as well as a new variational interpretation. We obtain the free energy function ${\tau }_{{\pi }_{\ast }\mu }$ of ${\pi }_{\ast }\mu$ 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 ${\pi }_{\ast }\mu$ at each $\alpha \in ({\tau }_{{\pi }_{\ast }\mu }^{\prime }(q_c-),{\tau }_{{\pi }_{\ast }\mu }^{\prime }(0+)]$.

83 pages, 7 figures

References in corpus (1)

• The dimension of projections of fractal percolations