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

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.