Boris Gurevich (Moscow State University and Penn State)

Multifractal analysis of ergodic averages for multidimensional time parameter.

Abstract. This is a joint paper with Arkady Tempelman. Let X be the space of all functions on $\mathbb Z^d,\,d\ge1$, with values in a finite set. Let $\tau_s,\,s\in\mathbb Z^d,$ be the translation group on X and $T_n=[-n,n]^d\cap\mathbb Z^d$. For continuous real-valued functions $f_1,\dots,f_m$$(m\ge1)$ and for any $a_1,\dots,a_m\in\mathbb R$ we evaluate the Hausdorff dimension of the set

$$\left\{x\in X:\lim_{n\to\infty}\frac{1}{(2n)^d}\sum_{s\in T_n} f_i(\tau_sx)=\alpha_i,\,i=1,\dots,m\right\}$$

in terms of a Statistical Physics model related to $f_1,\dots,f_m$. We also evaluate the Hausdorff dimension of the set of generic points for translation invariant Gibbs measures. The main challenge here is due to the possibility of phase transitions when $d\ge1$. Our proof is based on a generalization of a method invented by Cajar.