Topologically mixing suspension flows over shift spaces

TL;DR

This work advances the understanding of topological mixing for suspension flows over symbolic bases by establishing a sharp dichotomy: a Walters roof over a transitive shift with a synchronizing word yields a non-mixing suspension precisely when the periodic-orbit time sums lie in a common lattice $\delta\mathbb{Z}$. It extends the dichotomy to subshifts of finite type and β-shifts, showing that non-mixing implies cohomology to a locally constant roof with values in $\delta\mathbb{Z}$, and that for Such shifts the mixing roof functions form a $G_{\delta}$-dense set. Beyond these dichotomies, the paper proves that non-mixing roof functions are dense among positive continuous roofs on any finite-alphabet shift, and provides concrete examples illustrating the limitations of cohomology-based criteria in non-manifold symbolic settings. The results illuminate how, in symbolic dynamics, topological mixing can fail in broadly prevalent ways, contrasting with the smoother, measure-theoretic regime and offering a nuanced view of how roof-function structure governs flow mixing and its prevalence. These insights have potential implications for understanding mixing phenomena in symbolic and cross-disciplinary dynamical systems.

Abstract

We establish necessary and sufficient conditions for suspension flows over certain families of shift spaces to be topologically mixing. We also show the similarities and differences between this case and the smooth measure theoretic setting on a manifold. Additionally, we show that the set of roof functions defined on a shift space that produce suspension flows that are not topologically mixing is dense in the set of all continuous roof functions.

Figures (7)

• Figure 1: Partition of the phase space of the flow $\varphi^t$.
• Figure 2: Base dynamics of conjugate flow $\psi^t$
• Figure 3: $\psi^t$ is a suspension flow with roof function $1$.
• Figure 4: Breakdown of how the orbit of $x_{n,m}$ is shadowed by the various points accompanied by the equation that each orbit segment corresponds to. The upper line represents the orbit of $x_{n,m}$, and the lower line represents the orbit segments of the shadowing points.
• Figure 5: Visualizing the transfer function in the phase space of the flow. The thick curves of $\overline{C}$ give the value of $g$.

Theorems & Definitions (59)

• Theorem 1.1: fish-hass
• Definition 1.2
• Remark 1.3
• Definition 1.4
• Theorem 1.5
• Theorem 1.6
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4