Chaos on non-compact metric spaces.

Distributional chaos, ω-chaos, and the specification property have been the subject of much inquiry in dynamical systems in the last 50 years. Several results link the specification property to both distributional chaos and ω-chaos in compact dynamical systems.

We focus our study of distributional chaos on a shift space over a countable alphabet with the product topology and the usual shift map, which is known as the Baire Space. We prove that on the Baire Space subshifts of finite type exhibit dense distributional chaos and subshifts of bounded type that are perfect and have a dense set of periodic points also have distributional chaos. We also show that in this context a subshift with a generalized form of the specification property must have distributional chaos.

For ω-chaos, we focus on Lindelöf dynamical systems. In particular, if a system exhibits a generalized version of the specification property and has at least three points with mutually separated orbit closures, then the system exhibits dense ω-chaos.