Genericity in Topological Dynamics

For a compact metric space ∆ we consider the compact space S(∆) of shift-invariant subsystems of the full shift ∆, endowed with the Hausdorf metric.. We show that when ∆ is perfect the space S(∆) and the homeomorphism group Homeo(K) of the Cantor set (with the topology of uniform convergence) are topologically generically equivalent in the sense of Glasner and King [7], ie a dynamical property is generic/exotic in one iff it is generic/exotic in the other. We show a similar correspondence for a wide class of subspaces of S(∆), and establish a zero-one law for Baire measurable dynamical properties in these subspaces. We identify the genericity status (in terms of Baire category) of some classical dynamical properties in this setting. We show that the periodic systems are dense and the odometers are generic in the Gδ space of forward-transitive systems. Within the Polish subspace of totally transitive systems the isomorphism class of every minimal zero-dimensional system is dense; and the weak mixing, strong mixing and doubly minimal systems are generic; and so are the systems disjoint from a fixed totally transitive system. We also prove a partial connection between genericity of a dynamical property in the measure-theoretic category and genericity of systems supporting an invariant measure with the same property.