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Positive topological entropy and distributional chaos are characterized forhereditary shifts. A hereditary shift has positive topological entropy if andonly if it is DC2-chaotic (or equivalently, DC3-chaotic) if and only if it isnot uniquely ergodic. A hereditary shift is DC1-chaotic if and only if it isnot proximal (has more than one minimal set). As every spacing shift and everybeta shift is hereditary the results apply to those classes of shifts. Two openproblems on topological entropy and distributional chaos of spacing shifts froman article of Banks et al. are solved thanks to this characterization.Moreover, it is shown that a spacing shift $\Omega_P$ has positive topologicalentropy if and only if $\mathbb{N}\setminus P$ is a set of Poincar\'{e}recurrence. Using a result of K\v{r}\'{\i}\v{z} an example of a proximalspacing shift with positive entropy is constructed. Connections between spacingshifts and difference sets are revealed and the methods of this paper are usedto obtain new proofs of some results on difference sets.

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