Multifractal analysis of time averages for continuous vector functions on configuration space

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