On generic subsystems of a Smale space

Abstract

We study properties of generic compact invariant subsystems of a fixed Smale space in the Hausdorff topology. For a transitive Smale space, we show that the hyperspace of all subsystems contains an open and dense family of NMC subsystems, a dynamically degenerate class with finite orbit structure. A substantially different picture emerges in the closure of the infinite transitive locus: minimality, unique ergodicity, and zero topological entropy are generic, while subsystems conjugate to regular Toeplitz shifts are dense. Despite generic zero entropy, typical subsystems exhibit large oscillations in complexity: we prove that they have infinite slow entropy at the polynomial scale, and their slow entropy at the stretched-exponential scale is equal to one. For mixing Smale spaces, we further study the closure of the infinite totally transitive subsystems. In this hyperspace, minimality, unique ergodicity, and zero entropy remain generic; the conjugacy class of any Sturmian subshift is dense; and uniquely ergodic subsystems whose invariant measure is weakly mixing form a residual set. In particular, topological weak mixing is generic in this space.