Ergodic theorem for a Loeb space and hyperfinite approximations of dynamical systems

TL;DR

This paper explores the ergodic theorem within hyperfinite Loeb spaces, introduces hyperfinite approximations of dynamical systems, and discusses why standard nonstandard analysis techniques do not directly extend to prove the theorem.

Contribution

It introduces the concept of hyperfinite approximations for dynamical systems and proves their existence, providing a new perspective on ergodic theorems in nonstandard analysis.

Findings

Hyperfinite approximations of dynamical systems exist.

Standard transfer techniques do not directly prove the ergodic theorem in hyperfinite spaces.

Results are formulated in terms of sequences of finite dynamical systems.

Abstract

Although the G.Birkhoff Ergodic Theorem (BET) is trivial for finite spaces, this does not help in proving it for hyperfinite Loeb spaces. The proof of the BET for this case, suggested by T. Kamae, works, actually, for arbitrary probability spaces, as it was shown by Y. Katznelson and B. Weiss. In this paper we discuss the reason why the usual approach, based on transfer of some simple facts about arbitrary large finite spaces on infinite spaces using nonstandard analysis technique, does not work for the BET. We show that the the BET for hyperfinite spaces may be interpreted as some qualitative result for very big finite spaces. We introduce the notion of a hyperfinite approximation of a dynamical system and prove the existence of such an approximation. The standard versions of the results obtained in terms of sequences of finite dynamical systems are formulated.