# Lino Haupt (Jena)

## Introduction to the Hierarchy of Topological Dynamical Systems with Discrete Spectrum

This talk will give an introduction into the method of studying a topological dynamical system via its so called maximal equicontinuous factor. It is based on my master thesis which provides collective overview of the current state of knowledge.

A topological dynamical system with continuous dependence of all states on the initial conditions is called equicontinuous. Equicontinuous dynamical systems are in many ways non-chaotic. For example they have zero entropy. Further minimal equicontinuous they can be classified as rotations of compact groups.

It is well-known that any (compact) topological dynamical system has an equicontinuous factor maximal among all equicontinuous factors. The invertibility properties as well as other regularity properties of the factor map onto this maximal equicontinuous factor (MEF) can tell us how far a system is away from the simple equicontinuous case. In fact, the literature offers a variety of equivalences between dynamical properties on the one hand and the invertibility properties of the factor map to the MEF on the other hand. Those invertibility properties can be put into a canonical hierarchy starting from homeomorphy. Via those aforementioned equivalences, a parallel hierarchy of dynamical properties starting with equicontinuity emerges.

In the realm of measure preserving dynamics those with discrete spectrum are considered most simple. The well-known Halmos-von Neumann theorem states that for ergodic systems with discrete spectra the following holds:

1. Two ergodic systems with discrete spectrum are isomorphic if and only if their spectra coincide.

2. Any subgroup of the circle can be realized as a spectrum of an ergodic system with discrete spectrum.

3. Any ergodic discrete spectrum is isomorphic to a rotation on a compact group.

Consider a topological dynamical system which has discrete spectrum if viewed as a measure preserving one. On the one hand the MEF of that system is topologically conjugate to a group rotation. On the other hand the system is measure theoretically isomorphic to a group rotation. Here we study the case in which both group rotations coincide.

One can understand the aforementioned hierarchy as a description of the landscape of topological models of an measure preserving system with discrete spectrum._{Contact: Tobias Jäger}