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DCDS

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 chaos
of 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 of
Poincaré recurrence. Using a result of Kříž an example of a proximal spacing shift
with 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 on
difference sets.

keywords:
Spacing shift
,
distributional chaos.
,
topological entropy
,
hereditary shift
,
beta shift

DCDS-B

We survey the connections between entropy, chaos, and independence in topological dynamics.
We present extensions of two classical results placing the following notions in the context of symbolic dynamics:

1. Equivalence of positive entropy and the existence of a large (in terms of asymptotic and Shnirelman densities) set of combinatorial independence for shift spaces.

2. Existence of a mixing shift space with a dense set of periodic points with topological entropy zero and without ergodic measure with full support, nor any distributionally chaotic pair.

Our proofs are new and yield conclusions stronger than what was known before.

1. Equivalence of positive entropy and the existence of a large (in terms of asymptotic and Shnirelman densities) set of combinatorial independence for shift spaces.

2. Existence of a mixing shift space with a dense set of periodic points with topological entropy zero and without ergodic measure with full support, nor any distributionally chaotic pair.

Our proofs are new and yield conclusions stronger than what was known before.

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