Genericity in Topological Dynamics

Abstract

We study genericity of dynamical properties in the space of homeomorphisms of the Cantor set and in the space of subshifts of a suitably large shift space. These rather different settings are related by a Glasner-King type correspondence: genericity in one is equivalent to genericity in the other. By applying symbolic techniques in the shift-space model we derive new results about genericity of dynamical properties for transitive and totally transitive homeomorphisms of the Cantor set. We show that the isomorphism class of the universal odometer is generic in the space of transitive systems. On the other hand, the space of totally transitive systems displays much more varied dynamics. In particular, we show that in this space the isomorphism class of every Cantor system without periodic points is dense, and the following properties are generic: minimality, zero entropy, disjointness from a fixed totally transitive system, weak mixing, strong mixing, and minimal self joinings. The last two stand in striking contrast to the situation in the measure-preserving category. We also prove a correspondence between genericity of dynamical properties in the measure-preserving category and genericity of systems supporting an invariant measure with the same property.

48 pages, to appear in Ergodic Theory Dynamical Systems. v2: revised exposition, added proof that the universal odometer is generic among transitive Cantor homeomorphisms

Cited by in corpus (8)

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• Folner tilings for actions of amenable groups
• Linear Dynamics Induced by Odometers
• Minimal ambient nuclear C*-algebras
• Generic Stationary Measures and Actions