Lecture notes on ergodic transformations

TL;DR

This paper provides a comprehensive overview of ergodic transformations, covering fundamental properties, theorems, construction methods, entropy invariants, spectral analysis, and recurrence phenomena in ergodic theory.

Contribution

It synthesizes key concepts, theorems, and methods in ergodic transformations, offering a detailed lecture-based exposition and new insights into their properties and constructions.

Findings

Properties equivalent to ergodicity and weak mixing

Construction techniques for transformations

Analysis of entropy invariants and spectral properties

Abstract

Content of the lectures is the following. Properties of transformations equivalent to ergodicity. Birkhoff's Theorem. Properties equivalent to weak mixing. On typical properties of transformations. Lego to construct transformations. Typical entropy invariants. Poisson suspensions with completely positive P-entropy. Spectral theorem for unitary operators. Compact factors, Kronecker algebra. Progression recurrence for weakly mixing transformations. Double recurrence for ergodic transformations.