The specification property and chaos in multidimensional shift spaces and general compact metric spaces.


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Rufus Bowen introduced the specification property for maps on a compact metric space. In this dissertation, we consider some implications of the specification property for Zᵈ-actions on subshifts of Ʃ^Zᵈ as well as on a general compact metric space. In particular, we show that if σ :X to X is a continuous Zᵈ-action with a weak form of the specification property on a d-dimensional subshift of Ʃ^Zᵈ, then σ exhibits both ω-chaos, introduced by Li, and uniform distributional chaos, introduced by Schweizer and Smítal. The ω-chaos result is further generalized for some broader, directional notions of limit sets and general compact metric spaces with uniform expansion at a fixed point.



Specification property. Shift space. Symbolic dynamics. Group action. Zd-action. Omega-chaos. Distributional chaos.