Weak limits of powers of Chacon's automorphism

TL;DR

This paper characterizes the weak limits of powers of the Koopman operator for Chacon's automorphism, revealing they are either the constant projector or specific polynomials, and shows it is not alpha-weakly mixing.

Contribution

It provides a complete description of the weak closure of the powers of the Koopman operator for Chacon's automorphism, including explicit limit forms.

Findings

Weak limits are the orthogonal projector to constants and explicit polynomials.

Chacon's automorphism is not alpha-weakly mixing.

The weak closure of powers is fully characterized.

Abstract

We completely describe the weak closure of the powers of the Koopman operator associated to Chacon's classical automorphism. We show that weak limits of these powers are the ortho-projector to constants and an explicit family of polynomials. As a consequence, we answer negatively the question of alpha-weak mixing for Chacon's automorphism.