Multifractal Analysis of Time Averages for Continuous Vector Functions on Configuration Space

Abstract

We consider a natural action τ of the group ${\bf Z}^d$ on the space X consisting of the functions $x: {\bf Z}^d\to S$ (S‐valued configurations on ${\bf Z}^d$), where S is a finite set. For an arbitrary continuous function $f: X\to{\bf R}^m$, we study the multifractal spectrum of its time means corresponding to the dynamical system τ and a proper “averaging” sequence of finite subsets of the lattice ${\bf Z}^d$. The main tool of the research is thermodynamic formalism.