Nonstandard analysis of the behavior of ergodic means of dynamical systems on very big finite probability spaces

Abstract

The trivial proof of the ergodic theorem for a finite set $Y$ and a permutation $T:Y\to Y$ shows that for an arbitrary function $f:Y\to{\mathbb R}$ the sequence of ergodic means $A_n(f,T)$ stabilizes for $n \gg |T|$. We show that if $|Y|$ is very large and $|f(y)| \ll |Y|$ for almost all $y$, then $A_n(f,T)$ stabilizes for significantly long segments of very large numbers $n$ that are, however, $\ll |T|$. This statement has a natural rigorous formulation in the setting of nonstandard analysis, which is, in fact, equivalent to the ergodic theorem for infinite probability spaces. Its standard formulation in terms of sequences of finite probability spaces is complicated. We also discuss some other properties of the sequence $A_n(f,T)$ for very large finite $|Y|$ and $n$. A special consideration is given to the case, when a very big finite space $Y$ and its permutation $T$ approximate a dynamical system $(X,\nu, \tau)$, where $X$ is compact metric space, $\nu$ is a Borel measure on $X$ and $\tau:X\to X$ is a measure preserving transformation. The definition of approximation introduced here is new to our knowledge.