Random sparse sampling in a Gibbs weighted tree

Original · EN

Let μ be the geometric realization on [0,1] of a Gibbs measure on Σ={0,1}ⁿ associated with a Hölder potential. The thermodynamic and multifractal properties of μ are well known to be linked via the multifractal formalism. In this article, the impact of a random sampling procedure on this structure is studied. More precisely, let {Iw}w∈ Σ* stand for the collection of dyadic subintervals of [0,1] naturally indexed by the set of finite dyadic words Σ*. Fix η∈(0,1), and a sequence (pw)w∈ Σ* of independent Bernoulli variables of parameters 2⁻|ʷ|⁽¹⁻η⁾ (|w| is the length of w). We consider the (very sparse) remaining values μ={μ(Iw): w∈ Σ*, pw=1}. We prove that when η<1/2, it is possible to entirely reconstruct μ from the sole knowledge of μ, while it is not possible when η>1/2, hence a first phase transition phenomenon. We show that, for all η∈ (0,1), it is possible to reconstruct a large part of the initial multifractal structure of μ, via the fine study of μ. After reorganization, these coefficients give rise to a random capacity with new remarkable scaling and multifractal properties: its Lq-spectrum exhibits two phase transitions, and has a rich thermodynamic and geometric structure.