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Turkish Journal of Mathematics

DOI

10.3906/mat-1403-41

Abstract

The aim of this paper is to study distributional chaos for bounded linear operators. We show that distributional chaos of type k \in {1,2} is an invariant of topological conjugacy between two bounded linear operators. We give a necessary condition for distributional chaos of type 2 where it is possible to distinguish distributional chaos and Li--Yorke chaos. Following this condition, we compare distributional chaos with other well-studied notions of chaos for backward weighted shift operators and give an alternative proof to the one where strong mixing does not imply distributional chaos of type 2 (Martínez-Giménez F, Oprocha P, Peris A. Distributional chaos for operators with full scrambled sets. Math Z 2013; 274: 603--612.). Moreover, we also prove that there exists an invertible bilateral forward weighted shift operator such that it is DC1 but its inverse is not DC2.

Keywords

Distributional chaos, operators, weighted shifts, topological conjugacy, strong mixing

First Page

251

Last Page

258

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