Multifractal spectrum for barycentric averages

VERICAT, FERNANDO; MESON, ALEJANDRO M.

JOURNAL OF DYNAMICAL AND CONTROL SYSTEMS

SPRINGER/PLENUM PUBLISHERS

Lugar: New York; Año: 2016 vol. 22 p. 623 - 635

1079-2724

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 {E_{N,ϕ}(x)} of empirical measures on Y can be introduced. The sequence {bar(E_{N,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 [T. Austin, 2011]. In this work we shall consider the following multifractal decomposition in X: K_{y,ϕ}={x:lim_{N→∞}bar(E_{N,ϕ}(x))=y}, and we will obtain a variational formula for this multifractal spectrum.