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Distributionally \(n\)-scrambled set for weighted shift operators. (English) Zbl 06793137

Summary: The aim of the present paper is to investigate distributionally \(n\)-scrambled sets for weighted shift operators. We prove that the unilateral weighted shift operator admits densely invariant distributionally \(n\)-\(\epsilon\)-scrambled linear manifolds for any \(\epsilon \in (0,1)\) and any integer \(n \geq 2\), showing that this operator can exhibit maximal distributional \(n\)-chaos on a dense invariant linear manifold. Analogous results for the bilateral weighted shift operator are also obtained.

MSC:

47A16 Cyclic vectors, hypercyclic and chaotic operators
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
37B10 Symbolic dynamics
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