The aim of this paper is to study distributional chaos for bounded linear operators. We show that distributional chaos of type k ∈ {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.
In this paper, we construct a homeomorphism on the unit closed disk to show that an invertible mapping on a compact metric space is Li-Yorke chaotic does not imply its inverse being Li-Yorke chaotic.2000 Mathematics Subject Classification. Primary 37B99, 54H20; Secondary 37C15.
In order to study Li-Yorke chaos by the scalar perturbation for a given bounded linear operator T on Banach spaces X, we introduce the Li-Yorke chaos translation set of T , which is defined by SLY (T ) = {λ ∈ C; λ + T is Li-Yorke chaotic}. In this paper, some operator classes are considered, such as normal operator, compact operator, shift and so on. In particular, we show that the Li-Yorke chaos translation set of Kalisch operator on Hilbert space L 2 [0, 2π] is a simple point set {0}.Mathematics Subject Classification (2010). Primary 47A16; Secondary 37D45.
In this paper, we give two examples to show that an invertible mapping being Li-Yorke chaotic does not imply its inverse being Li-Yorke chaotic, in which one is an invertible bounded linear operator on an infinite dimensional Hilbert space and the other is a homeomorphism on the unit open disk. Moreover, we use the last example to prove that Li-Yorke chaos is not preserved under topological conjugacy.
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