2 results
Disjoint hypercyclicity, Sidon sets and weakly mixing operators
- Part of
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- 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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- Article
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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.
Disjoint Hypercyclicity and Weighted Translations on Discrete Groups
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- Journal:
- Canadian Mathematical Bulletin / Volume 60 / Issue 4 / 01 December 2017
- Published online by Cambridge University Press:
- 20 November 2018, pp. 712-720
- Print publication:
- 01 December 2017
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Let
$1\le p<\infty $, and let 
$G$ be a discrete group. We give a sufficient and necessary condition for weighted translation operators on the Lebesgue space 
${{\ell }^{p}}(G)$ to be densely disjoint hypercyclic. The characterization for the dual of a weighted translation to be densely disjoint hypercyclic is also obtained.
