Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-28T03:50:45.593Z Has data issue: false hasContentIssue false
  • English
  • Français

Recurrence of Cosine Operator Functions on Groups

Published online by Cambridge University Press:  20 November 2018

Chung-Chuan Chen
Chung-Chuan Chen*
Affiliation:
Department of Mathematics Education, National Taichung University of Education, Taiwan e-mail: chungchuan@mail.ntcu.edu.tw
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this note, we study the recurrence and topologically multiple recurrence of a sequence of operators on Banach spaces. In particular, we give a sufficient and necessary condition for a cosine operator function, induced by a sequence of operators on the Lebesgue space of a locally compact group, to be topologically multiply recurrent.

Keywords

47A1654B2043A15topologically multiple recurrencerecurrencetopological transitivityhypercyclicitycosine operator function
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 4 , 01 December 2016 , pp. 693 - 704
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Bayart, F. and Matheron, E., Dynamics of linear operators. Cambridge Tracts in Mathematics, 179, Cambridge University Press, Cambridge, 2009. http://dx.doi.Org/10.1017/CBO9780511581113 Google Scholar
[2] Bayart, F. and Grivaux, S., Invariant Gaussian measures for operators on Banach spaces and linear dynamics. Proc. Lond. Math. Soc. 94(2007), no. 2,181-210. http://dx.doi.Org/10.1112/plms/pdl013 Google Scholar
[3] Bonilla, A. and Miana, P., Hypercyclic and topologically mixing cosine functions on Banach spaces. Proc. Amer. Math. Soc. 136(2008), no. 2, 519–528. http://dx.doi.Org/10.1090/S0002-9939-07-09036-3 Google Scholar
[4] Chang, S-J. and Chen, C-C., Topological mixing for cosine operator functions generated by shifts. Topology Appl. 160(2013), no. 2, 382–386. http://dx.doi.Org/10.101 6/j.topol.2012.11.01 8 Google Scholar
[5] Chen, C-C., Chaos for cosine operator functions generated by shifts. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 24(2014), Article ID 1450108. http://dx.doi.Org/10.1142/S0218127414501089 Google Scholar
[6] Chen, C-C., Chaos for cosine operator functions on groups. Abstr. Appl. Anal. 2014, Article ID 603234. http://dx.doi.Org/10.1155/2014/603234 Google Scholar
[7] Chen, C-C., Topological transitivity for cosine operator functions on groups. Topology Appl. 191(2015), 48–57. http://dx.doi.Org/10.1016/j.topol.2015.05.005 Google Scholar
[8] Chen, C-C., Recurrence for weighted translations on groups. Acta Math. Sci. Ser. B Engl. Ed. 36(2016), no. 2, 443–452. http://dx.doi.Org/10.1016/SO252-96O2(16)30011-X Google Scholar
[9] Chen, C-C. and Chu, C-H., Hypercyclic weighted translations on groups. Proc. Amer. Math. Soc. 139(2011), no. 8, 2839–2846. http://dx.doi.Org/10.1090/S0002-9939-2011-10718-4 Google Scholar
[10] Costakis, G., Manoussos, A., and Parissis, I., Recurrent linear operators. Complex Anal. Oper. Theory 8(2014), no. 8, 1601–1643. http://dx.doi.Org/10.1007/s11785-013-0348-9 Google Scholar
[11] Costakis, G. and Parissis, I., Szemeredi's theorem, frequent hypercyclicity and multiple recurrence. Math. Scand. 110(2012), no. 2, 251–272.Google Scholar
[12] Grosse-Erdmann, K.-G. and Peris Manguillot, A., Linear chaos. Universitext, Springer, London, 2011. http://dx.doi.Org/10.1007/978-1-4471-2170-1 Google Scholar
[13] Kalmes, T., Hypercyclicity and mixing for cosine operator functions generated by second order partial differential operators. J. Math. Anal. Appl. 365(2010), no. 1. 363–375. http://dx.doi.Org/10.1016/j.jmaa.2009.10.063 Google Scholar
[14] Kostic, M., Hypercyclic and chaotic integrated C-cosine functions. Filomat 26(2012), no. 1,1-44. http://dx.doi.Org/10.2298/FIL1201001K Google Scholar
[15] Tian, C. and Chen, G., Chaos of a sequence of maps in a metric space. Chaos Soliton Fractals 28(2006), no. 4, 1067–1075. http://dx.doi.Org/10.1016/j.chaos.2005.08.127 Google Scholar