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non-weakly mixing operators

Ergodic Theory and Dynamical Systems (1)

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Disjoint hypercyclicity, Sidon sets and weakly mixing operators
Part of
RODRIGO CARDECCIA
Journal:

Ergodic Theory and Dynamical Systems / Volume 44 / Issue 5 / May 2024

Published online by Cambridge University Press:

22 August 2023, pp. 1315-1329

Print publication:

May 2024
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We prove that a finite set of natural numbers J satisfies that $J\cup \{0\}$ is not Sidon if and only if for any operator T, the disjoint hypercyclicity of $\{T^j:j\in J\}$ implies that T is weakly mixing. As an application we show the existence of a non-weakly mixing operator T such that $T\oplus T^2\oplus\cdots \oplus T^n$ is hypercyclic for every n.