Distributional Chaos on Compact Metric Spaces via Specification Properties

AbstractIn this paper we show that a continuous function on a compact metric space exhibits distributional chaos as introduced in [B. Schweizer and J. Smítal, Trans. Amer. Math. Soc.344 (1994), 737–754] and elucidated in [B. Schweizer, A. Sklar, and J. Smital, to appear] if the function has either a weaker form of the specification property (see [M. Denker, C. Grillenberger, and K. Sigmund, Springer Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, New York/Heidelberg/Berlin, 1976]) or the generalized specification property introduced in [F. Balibrea, B. Schweizer, A. Sklar, and J. Smítal, to appear]. In particular, any Anosov diffeomorphism is distributionally chaotic, regardless of the fact that in this case the trajectories of a.e. pair of points exhibit regular, non-chaotic behavior