Zero-dimensional and symbolic extensions of topological flows

TL;DR

This paper investigates the existence and properties of zero-dimensional and symbolic extensions of topological flows, establishing conditions under which such extensions exist and how they relate to orbit equivalence and the nature of the roof function.

Contribution

It demonstrates that every topological flow has a principal zero-dimensional extension and characterizes when symbolic extensions exist, especially for singular suspension flows.

Findings

Any topological flow admits a principal zero-dimensional extension.

Existence of symbolic extensions depends on the time-$t$ map and orbit equivalence.

For singular flows, the existence of symbolic extensions relates to the smoothness of the roof function.

Abstract

A zero-dimensional (resp. symbolic) flow is a suspension flow over a zero-dimensional system (resp. a subshift). We show that any topological flow admits a principal extension by a zero-dimensional flow. Following [Bur19] we deduce that any topological flow admits an extension by a symbolic flow if and only if its time-$t$ map admits an extension by a subshift for any $t\neq 0$. Moreover the existence of such an extension is preserved under orbit equivalence for regular topological flows, but this property does not hold more true for singular flows. Finally we investigate symbolic extensions for singular suspension flows. In particular, the suspension flow over the full shift on $\{0,1\}^{\mathbb Z}$ with a roof function $f$ vanishing at the zero sequence $0^\infty$ admits a principal symbolic extension or not depending on the smoothness of $f$ at $0^\infty$.