Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts

Abstract

Positive topological entropy and distributional chaos are characterized for hereditary shifts.A hereditary shift has positive topological entropy if and only if it is DC$2$-chaotic(or equivalently, DC$3$-chaotic) if and only if it is not uniquely ergodic.A hereditary shift is DC$1$-chaotic if and only if it is not proximal (has more than one minimal set).As every spacing shift and every beta shift is hereditary the results apply to those classes of shifts.Two open problems on topological entropy and distributional chaosof spacing shifts from an article of Banks et al. are solved thanks to this characterization.Moreover, it is shown that a spacing shift $\Omega_P$ has positive topological entropy if and only if$\mathbb{N}\setminus P$ is a set ofPoincaré recurrence. Using a result of Kříž an example of a proximal spacing shiftwith positive entropy is constructed.Connections between spacing shifts and difference sets are revealed and the methods of this paper are used to obtain new proofs of some results ondifference sets.