The primitive spectrum of a semigroup of Markov operators

TL;DR

This paper investigates the structure of semigroups of Markov operators on continuous functions, using invariant ideals to analyze properties like ergodicity and fixed points, with applications to dynamical systems.

Contribution

It introduces a novel approach based on primitive S-ideals and topology to describe qualitative properties of Markov semigroups, inspired by C*-algebra theory.

Findings

Characterization of mean ergodicity via invariant ideals

Description of fixed space structure using primitive S-ideals

Identification of centers of attraction in dynamical systems

Abstract

For a semigroup S of Markov operators on a space of continuous functions, we use S-invariant ideals to describe qualitative properties of S such as mean ergodicity and the structure of its fixed space. For this purpose we focus on primitive S-ideals and endow the space of those ideals with an appropriate topology. This approach is inspired by the representation theory of C*-algebras and can be adapted to our dynamical setting. In the particularly important case of Koopman semigroups, we characterize the centers of attraction of the underlying dynamical system in terms of the invariant ideal structure of S.