We use the generization of Weyl's equidistribution theorem to characterize several necessary conditions of hypercyclic weighted translation operators with periodic element.

Let G be a locally compact group, w be a weight on G and Φ be a Young function. We give some characterizations for translation operators to be topologically transitive and chaotic on the weighted Orlicz space L Φ w (G). In particular, transitivity is equivalent to the blow-up/collapse property in our case. Moreover, the dense set of periodic elements implies transitivity automatically.

No abstract

In this paper, we initiate the study of distributional chaos for weighted translations on locally compact groups, and give a sufficient condition for such operators to be distributionally chaotic. We also investigate the set of distributionally irregular vectors of weighted translations from the views of modulus, cone, equivalent class and atom. In particular, we show that the set of distributionally irregular vectors is residual if the group is the integer. Besides, the equivalent class of distributionally irregular vectors is path connected if the field is complex. introductionIn the past several decades, the study of linear dynamics has attracted a lot of attention. At this present stage, there are some excellent books (for instance, see [3,13,15]) on this topic. Hypercyclicity and linear chaos play important roles in this investigation. A linear operator T on a separable Banach space X is called hypercyclic if there is a vector x ∈ X such that the set {x, T x, T 2 x, • • •} is dense in X. It is well known that hypercyclicity is equivalent to topological transitivity on separable Banach spaces. If T is hypercyclic together with the dense set of periodic points, then T is said to be Devaney chaotic.Devaney chaos is highly related to distributional chaos which was introduced by Schweizer and Smítal in [19] and can be viewed as an extension of Li-Yorke chaos. Recently, the notion of distributional chaos was considered for linear operators on Banach spaces and Fréchet spaces in [4,5,12,16,17,18]. Inspired by these, in this paper, we initiate the investigation of distributional chaos for a class of specific linear operators, namely, weighted translations on locally compact groups, and give some characterizations for their properties.

No abstract