Multifractal Spectrum for Barycentric Averages

Abstract

Let (X, ν) and Y be a measured space and a CAT (0) space, respectively. If M2(Y ) is the set of measures on Y with finite second moment then a map bar : M2(Y ) → Y can be defined. Also, for any x ∈ X and for a map ϕ : X → Y , a sequence EN ,ϕ(x) of empirical measures on Y can be introduced. The sequence bar EN ,ϕ(x) replaces in CAT (0) spaces the usual ergodic averages for real valuated maps. It converges in Y (to a map ϕ (x)) almost surely for any x ∈ X (Austin J Topol Anal. 2011;3: 145–152). In this work, we shall consider the following multifractal decomposition in X : Ky,ϕ = x : lim N→∞ bar EN ,ϕ(x) = y , and we will obtain a variational formula for this multifractal spectrum.