Zero-dimensional and symbolic extensions of topological flows

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 $0^\infty$.