Complutense University Library

The set of periods of chaotic operators and semigroups

Muñoz-Fernández, Gustavo Adolfo and Seoane Sepúlveda, Juan Benigno and Weber, Andreas (2011) The set of periods of chaotic operators and semigroups. Revista de la Real Academia de Ciencias Exactas Fisicas y Naturales Serie A: Matemáticas, 105 (2). pp. 397-402. ISSN 1578-7303

Official URL: http://www.springerlink.com/content/g512831qv5028833/

View download statistics for this eprint

==>>> Export to other formats

Abstract

This expository paper is devoted to the review of some very recent results concerning the set of periods of a chaotic operator T or a chaotic semigroup {T (t): t a parts per thousand yen 0} acting on a complex Banach space. We obtain information about the structure of the set of periods and we give techniques to construct (chaotic) strongly continuous semigroups with prescribed periods.

Item Type:Article
Uncontrolled Keywords:Periods of strongly continuous semigroups; Hypercyclic semigroups; Chaotic semigroups
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:16894
References:

Ali Akbar, K., Kannan, V., Gopal, S., Chiranjeevi, P.: The set of periods of periodic points of a linear operator. Linear Algebra Appl. 431(1–2), 241–246 (2009) MR2522571 (2010f:47032)

Aron, R.M., Seoane-Sep´ulveda, J.B.,Weber, A.: Chaos on function spaces. Bull.Austral. Math. Soc. 71(3), 411–415 (2005) MR2150930 (2006a:47013)

Banks, J., Brooks, J., Cairns, G., Davis, G., Stacey, P.: On Devaney’s definition of chaos. Amer. Math. Monthly 99(4), 332–334 (1992) MR1157223 (93d:54059)

Bayart, F., Berm´udez, T.: Semigroups of chaotic operators. Bull. London Math. Soc. 41(5), 823–830 (2009) MR2557463 (2011a:47024)

Bayart, F., Grivaux, S.: Hypercyclicity and unimodular point spectrum. J. Funct. Anal. 226(2), 281–300 (2005) MR2159459 (2006i:47014)

Bayart, F., Matheron, ´ E: Dynamics of linear operators, Cambridge Tracts in Mathematics, vol. 179. Cambridge University Press, Cambridge (2009) MR2533318 (2010m:47001)

Block, L.S., Coppel, W.A.: Dynamics in one dimension, Lecture Notes in Mathematics, vol. 1513. Springer, Berlin (1992) MR1176513 (93g:58091)

Conejero, J.A, Martínez-Giménez, F., Peris, A.: Sets of periods for chaotic linear operators (2011) (preprint)

DeLaubenfels, R., Emamirad, H.: Chaos for functions of discrete and continuous weighted shift operators. Ergod. Theory Dynam. Syst. 21(5), 1411–1427 (2001) MR1855839 (2002j:47030)

Devaney, R.L.: An introduction to chaotic dynamical systems, second ed., In: Addison-Wesley studies in nonlinearity. Addison-Wesley Publishing Company Advanced Book Program, Redwood City (1989) MR1046376 (91a:58114)

Elaydi, S.: On a converse of Sharkovsky’s theorem. Am. Math. Mon. 103(5), 386–392 (1996) MR1400719 (97g:58049)

Godefroy, G., Shapiro, J.H.: Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 98(2), 229–269 (1991) MR1111569 (92d:47029)

Grosse-Erdmann, K.-G.: Recent developments in hypercyclicity, Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), In: Colecc. Abierta, vol. 64, Univ. Sevilla Secr. Publ., Seville, 2003, pp. 157–175 MR2041336 (2004m:47019)

Grosse-Erdmann, K.-G.: Universal families and hypercyclic operators. Bull. Amer. Math. Soc.(N.S.) 36(3), 345–381 (1999) MR1685272 (2000c:47001)

Kalish, G.: On operators on separable Banach spaces with arbitrary prescribed point spectrum. Proc. Amer. Math. Soc. 34(1), 207–208 (1972) MR0315474 (47 #4023)

Muñoz-Fernández, G.A., Seoane-Sepúlveda, J.B., Weber, A.: Periods of strongly continuous semigroups (2010) (preprint)

Pazy, A.: Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44. Springer, New York (1983) MR0710486 (85g:47061)

Sarkovskii, A.N.: Co-existence of cycles of a continuous mapping of a line onto itself. Ukranian Math. Z. 16, 61–71 (1964) MR0159905 (28 #3121)

Smolyanov, O.G., Shkarin, S.A.: On the structure of the spectra of linear operators in Banach spaces. Mat. Sb. 192(4), 99–114 (2001) MR1834094 (2002a:47004)

Deposited On:26 Oct 2012 08:12
Last Modified:16 May 2013 18:05

Repository Staff Only: item control page