# Decompositions of dynamical systems induced by the Koopman operator

## TL;DR

This paper characterizes how the state space of a topological dynamical system decomposes based on the fixed points of its Koopman operator, revealing conditions for topological ergodicity.

## Contribution

It introduces a hierarchy of generalized orbits and provides the finest decomposition into Lyapunov stable sets, linking fixed space dimension to ergodicity.

## Key findings

- Decomposition of state space via Koopman fixed points
- Introduction of generalized orbits hierarchy
- System is topologically ergodic iff Koopman fixed space is one-dimensional

## Abstract

For a topological dynamical system we characterize the decomposition of the state space induced by the fixed space of the corresponding Koopman operator. For this purpose, we introduce a hierarchy of generalized orbits and obtain the finest decomposition of the state space into absolutely Lyapunov stable sets. Analogously to the measure-preserving case, this yields that the system is topologically ergodic if and only if the fixed space of its Koopman operator is one-dimensional.