Minimal covers with continuity-preserving transfer operators for topological dynamical systems

TL;DR

This paper introduces minimal and essential covers with enhanced continuity properties for non-invertible dynamical systems, enabling the application of thermodynamic formalism and connecting to groupoid models in operator algebras.

Contribution

It constructs new types of covers that preserve continuity and functoriality, generalizing classical covers like Krieger and Fischer, and applies these to thermodynamic formalism and groupoid models.

Findings

Established thermodynamic formalism for a broad class of non-invertible systems.

Constructed functorial minimal and essential covers with strong continuity properties.

Linked dynamical covers to groupoid models for C*-algebras.

Abstract

Given a non-invertible dynamical system with a transfer operator, we show there is a minimal cover with a transfer operator that preserves continuous functions. We also introduce an essential cover with even stronger continuity properties. For one-sided sofic subshifts, this generalizes the Krieger and Fischer covers, respectively. Our construction is functorial in the sense that certain equivariant maps between dynamical systems lift to equivariant maps between their covers, and these maps also satisfy better regularity properties. As applications, we identify finiteness conditions which ensure that the thermodynamic formalism is valid for the covers. This establishes the thermodynamic formalism for a large class of non-invertible dynamical systems, e.g. certain piecewise invertible maps. When applied to semi-\'etale groupoids, our minimal covers produce \'etale groupoids which are models for $C^*$-algebras constructed by Thomsen. The dynamical covers and groupoid covers are unified under the common framework of topological graphs.