Projections of planar Mandelbrot measures

TL;DR

This paper investigates the geometric and thermodynamic properties of projections of planar Mandelbrot measures, revealing their dimensionality, measure equivalences, phase transitions, and multifractal formalism validity.

Contribution

It provides new results on the dimensionality, measure equivalence, phase transitions, and multifractal analysis of projected Mandelbrot measures, including a new proof of the Dekking-Grimmett-Falconer formula.

Findings

Projection measure is exactly dimensional with dimension min of original and Bernoulli measure.

Absolute continuity of the projection measure occurs if and only if the original measure's dimension exceeds that of the Bernoulli measure.

The free energy function exhibits phase transitions of order larger than 1, with different behaviors for q in [0,1] and q > 1.

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 $\alpha\in (\tau_{\pi_*\mu}'(q_c-),\tau_{\pi_*\mu}'(0+)]$.