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

• Main Authors: Brix, Kevin Aguyar, Hume, Jeremy B, Li, Xin
• Format: Journal Article
• Language: English
• Published: 21-08-2024
• Subjects: Mathematics - Dynamical Systems; Mathematics - Operator Algebras
• DOI: 10.48550/arxiv.2408.11917

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.