Chaotic operators on hypergroups

Abstract

In this paper, we initiate a study of chaos, in the sense of Devaney, and topological transitivity on the L p space of hypergroups, and give some sufficient and necessary conditions for weighted translation operators on hypergroups to be chaotic and transitive in terms of the Haar measure, weight functions and center elements of hypergroups. A characterization of topologically mixing weighted translations on hypergroups is also given. Mathematics subject classification (2010): 47A16, 43A62, 47B38.

Dynamics of weighted translations generated by group actions

“…Also, the existence of hypercyclic weighted translations on L p (G) was discussed in [15]. We refer to papers [13,14] for readers interested in the hypergroup case and vector-valued version. Besides, linear dynamics for weighted translations on the Orlicz space L Φ (G) were first investigated by [2,12] where Φ is a Young function.…”

Linear dynamics of operators on B_O(H)

“…This concept is related to some other notions such as topological transitivity, topological mixing and chaos. Especially, hypercyclicity of weighted translation operators on Lebesgue spaces in the context of locally compact groups and hypergroups have been studied; see [13,10,8,4]. Recently, the focus was on studying hypercyclicity of operators on other special function spaces such as Orlicz spaces and solid Banach function spaces; see [7,9,11].…”

Chaotic translations on weighted Orlicz spaces

“…Also, the existence of hypercyclic weighted translations on L p (G) was discussed in [14]. For readers interested in the hypergroup case and vector-valued version, we refer to papers [12,13]. Besides, the study of linear dynamics for weighted translations on the Orlicz space L Φ (G) was initialed by [3,11] where Φ is a Young function.…”

“…Also, the existence of hypercyclic weighted translations on L p (G) was discussed in [14]. For readers interested in the hypergroup case and vector-valued version, we refer to papers [12,13]. Besides, the study of linear dynamics for weighted translations on the Orlicz space L Φ (G) was initialed by [3,11] where Φ is a Young function.From another view, Abakumov and Kuznetsova in [1] focused on the density of translates in the weighted Lebesgue space L p w (G), and observed some different phenomenon from that in [10], where w is a weight on G. Indeed, a translation cannot have a dense orbit on the unweighted space L p (G).…”