Projections of planar Mandelbrot random measures

Let μ be a planar Mandelbrot measure and π⁎μ its orthogonal projection on one of the principal axes. We study the thermodynamic and geometric properties of π⁎μ. We first show that π⁎μ is exact 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). 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 τπ⁎μ of π⁎μ on a wide subinterval [0,qc) of R+. For q∈[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⁡(τν,τμ), which can exhibit first order phase transitions. This is in contrast with the analyticity of τμ over [0,qc). Also, we prove the validity of the multifractal formalism for π⁎μ at each α∈(τπ⁎μ′(qc−),τπ⁎μ′(0+)].