Pointwise Ergodic Theorems for Higher Levels of Mixing

TL;DR

This paper extends pointwise ergodic theorems to higher levels of mixing, providing stronger results for weakly and strongly mixing systems and characterizations of these systems.

Contribution

It introduces new pointwise ergodic theorems for weakly and strongly mixing systems that are stronger than existing theorems and characterizes these systems.

Findings

Stronger pointwise ergodic theorems for weakly and strongly mixing systems.

Characterization of weakly mixing and strongly mixing systems.

Potential extensions to other ergodic hierarchy levels.

Abstract

We prove strengthenings of the Birkhoff Ergodic Theorem for weakly mixing and strongly mixing measure preserving systems. We show that our pointwise theorem for weakly mixing systems is strictly stronger than the Wiener-Wintner Theorem. We also show that our pointwise Theorems for weakly mixing and strongly mixing systems characterize weakly mixing systems and strongly mixing systems respectively. The methods of this paper also allow one to prove an enhanced pointwise ergodic theorem for other levels of the ergodic hierarchy such as ergodicity and mild mixing but not K-mixing. The author plans to include these additional pointwise ergodic theorems in his thesis.