Dynamical Cantor Staircase Functions and The Small Flow Boundary Property

Abstract

The small flow boundary property (SFBP), introduced by Burguet for fixed-point free topological flows, is a non-trivial generalization of the small boundary property (SBP). We characterize when a time-discretization of such a flow satisfies the SBP and deduce that an SFBP flow admitting an aperiodic time-discretization has vanishing mean dimension. Furthermore, we introduce a new quantity, \textit{flow-generated entropy}, for a factor between a flow and a time-discretization, quantifying the dynamical complexity inherited from the flow itself. This is used in order to establish that any time-discretization of a flow with SFBP admits factors of arbitrarily small flow-generated entropy separating any fixed pair of distinct points. The argument relies on a construction of a dynamical version of the Cantor staircase function. Finally, the appendix includes proofs of fundamental properties of the marker property which have not yet appeared in the literature.