Hypercyclic operators, Gauss measures and Polish dynamical systems

TL;DR

This paper explores hypercyclic operators within Polish dynamical systems, analyzing Gaussian measure preservation, spectral types, and mixing properties, and classifies hypercyclic operators as various types of dynamical systems.

Contribution

It provides a detailed spectral analysis of hypercyclic operators with Gaussian measures and demonstrates the existence of mildly but not strongly mixing hypercyclic operators.

Findings

Constructed hypercyclic operators preserving Gaussian measures.

Described the spectral type of associated Koopman operators.

Proved existence of mildly but not strongly mixing hypercyclic operators.

Abstract

In this work we consider hypercyclic operators as a special case of Polish dynamical systems. In the first section we analyze the construction of Bayart and Grivaux of a hypercyclic operator which preserves a Gaussian measure, and derive a description of the maximal spectral type of the Koopman operator associated to the corresponding measure preserving dynamical system. We then use this information to show the existence of a mildly but not strongly mixing hypercyclic operator on Hilbert space. In the last two sections we study hypercyclic and frequently hypecyclic operators which, as Polish dynamical systems are, M-systems, E-systems, and syndetically transitive systems.