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Dynamics of the heat semigroup on symmetric spaces

Published online by Cambridge University Press:  23 June 2009

LIZHEN JI  and
ANDREAS WEBER
LIZHEN JI
Affiliation:
Department of Mathematics, University of Michigan, 1834 East Hall, Ann Arbor, MI 48109-1043, USA (email: [email protected])
ANDREAS WEBER
Affiliation:
Institut für Algebra und Geometrie, Universität Karlsruhe (TH), Englerstrasse 2, 76128 Karlsruhe, Germany (email: [email protected])
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Abstract

The aim of this paper is to show that the dynamics of Lp heat semigroups (p>2) on a symmetric space of non-compact type is very different from the dynamics of the Lp heat semigroups if 1<p≤2. To see this, we show that certain shifts of the Lp heat semigroups have a chaotic behavior if p>2, and that such a behavior is not possible in the cases 1<p≤2. These results are compared with the corresponding situation for Euclidean spaces and symmetric spaces of compact type, where such a behavior is not possible.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 2 , April 2010 , pp. 457 - 468
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Aron, R. M., Seoane-Sepúlveda, J. B. and Weber, A.. Chaos on function spaces. Bull. Aust. Math. Soc. 71(3) (2005), 411–415.CrossRefGoogle Scholar
[2]Banasiak, J. and Moszyński, M.. A generalization of Desch–Schappacher–Webb criteria for chaos. Discrete Contin. Dyn. Syst. 12(5) (2005), 959–972.CrossRefGoogle Scholar
[3]Banks, J., Brooks, J., Cairns, G., Davis, G. and Stacey, P.. On Devaney’s definition of chaos. Amer. Math. Monthly 99(4) (1992), 332–334.CrossRefGoogle Scholar
[4]Bermúdez, T., Bonilla, A., Conejero, J. A. and Peris, A.. Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces. Studia Math. 170(1) (2005), 57–75.CrossRefGoogle Scholar
[5]Bermúdez, T., Bonilla, A. and Martinón, A.. On the existence of chaotic and hypercyclic semigroups on Banach spaces. Proc. Amer. Math. Soc. 131(8) (2003), 2435–2441 (electronic).CrossRefGoogle Scholar
[6]Bermúdez, T., Bonilla, A. and Peris, A.. On hypercyclicity and supercyclicity criteria. Bull. Aust. Math. Soc. 70(1) (2004), 45–54.CrossRefGoogle Scholar
[7]Bès, J. and Peris, A.. Hereditarily hypercyclic operators. J. Funct. Anal. 167(1) (1999), 94–112.CrossRefGoogle Scholar
[8]Conejero, J. A. and Peris, A.. Linear transitivity criteria. Topology Appl. 153(5-6) (2005), 767–773.CrossRefGoogle Scholar
[9]Davies, E. B.. Pointwise bounds on the space and time derivatives of heat kernels. J. Operator Theory 21(2) (1989), 367–378.Google Scholar
[10]Davies, E. B.. Heat Kernels and Spectral Theory (Cambridge Tracts in Mathematics, 92). Cambridge University Press, Cambridge, 1990.Google Scholar
[11]deLaubenfels, R. and Emamirad, H.. Chaos for functions of discrete and continuous weighted shift operators. Ergod. Th. & Dynam. Sys. 21(5) (2001), 1411–1427.CrossRefGoogle Scholar
[12]deLaubenfels, R., Emamirad, H. and Grosse-Erdmann, K.-G.. Chaos for semigroups of unbounded operators. Math. Nachr. 261/262 (2003), 47–59.CrossRefGoogle Scholar
[13]Desch, W., Schappacher, W. and Webb, G. F.. Hypercyclic and chaotic semigroups of linear operators. Ergod. Th. & Dynam. Sys. 17(4) (1997), 793–819.CrossRefGoogle Scholar
[14]Devaney, R. L.. An Introduction to Chaotic Dynamical Systems, 2nd edn(Addison-Wesley Studies in Nonlinearity). Addison-Wesley, Redwood City, CA, 1989.Google Scholar
[15]Mourchid, S. El. On a hypercyclicity criterion for strongly continuous semigroups. Discrete Contin. Dyn. Syst. 13(2) (2005), 271–275.CrossRefGoogle Scholar
[16]Mourchid, S. El. The imaginary point spectrum and hypercyclicity. Semigroup Forum 73(2) (2006), 313–316.CrossRefGoogle Scholar
[17]Engel, K.-J. and Nagel, R.. One-parameter Semigroups for Linear Evolution Equations (Graduate Texts in Mathematics, 194). Springer, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.Google Scholar
[18]Gangolli, R. and Varadarajan, V. S.. Harmonic Analysis of Spherical Functions on Real Reductive Groups (Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 101). Springer, Berlin, 1988.CrossRefGoogle Scholar
[19]Gethner, R. M. and Shapiro, J. H.. Universal vectors for operators on spaces of holomorphic functions. Proc. Amer. Math. Soc. 100(2) (1987), 281–288.CrossRefGoogle Scholar
[20]Godefroy, G. and Shapiro, J. H.. Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 98(2) (1991), 229–269.CrossRefGoogle Scholar
[21]Grosse-Erdmann, K. G.. Recent developments in hypercyclicity. Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003) (Colección Abierta, 64). Univ. Sevilla Secr. Publ, Seville, 2003,pp. 157–175.Google Scholar
[22]Grosse-Erdmann, K.-G.. Universal families and hypercyclic operators. Bull. Amer. Math. Soc. (N.S.) 36(3) (1999), 345–381.CrossRefGoogle Scholar
[23]Helgason, S.. Differential Geometry, Lie Groups, and Symmetric Spaces (Pure and Applied Mathematics, 80). Academic Press, New York, 1978.Google Scholar
[24]Helgason, S.. Groups and Geometric Analysis (Pure and Applied Mathematics, 113). Academic Press, Orlando, FL, 1984.Google Scholar
[25]Herzog, G.. On a universality of the heat equation. Math. Nachr. 188 (1997), 169–171.CrossRefGoogle Scholar
[26]Ji, L. and Weber, A.. L p spectral theory and heat dynamics of locally symmetric spaces. Preprint, 2008, arXiv:0810.0209.Google Scholar
[27]Kalmes, T.. On chaotic C0-semigroups and infinitely regular hypercyclic vectors. Proc. Amer. Math. Soc. 134(10) (2006), 2997–3002 (electronic).CrossRefGoogle Scholar
[28]Kalmes, T.. Hypercyclic, mixing, and chaotic C0-semigroups induced by semiflows. Ergod. Th. & Dynam. Sys. 27(5) (2007), 1599–1631.CrossRefGoogle Scholar
[29]Liskevich, V. A. and Perel’muter, M. A.. Analyticity of sub-Markovian semigroups. Proc. Amer. Math. Soc. 123(4) (1995), 1097–1104.Google Scholar
[30]Reed, M. and Simon, B.. Tensor products of closed operators on Banach spaces. J. Funct. Anal. 13 (1973), 107–124.CrossRefGoogle Scholar
[31]Stanton, R. J. and Tomas, P. A.. Pointwise inversion of the spherical transform on Lp(G/K), 1≤p<2. Proc. Amer. Math. Soc. 73(3) (1979), 398–404.Google Scholar
[32]Strichartz, R. S.. Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal. 52(1) (1983), 48–79.CrossRefGoogle Scholar
[33]Sturm, K.-T.. On the Lp-spectrum of uniformly elliptic operators on Riemannian manifolds. J. Funct. Anal. 118(2) (1993), 442–453.CrossRefGoogle Scholar
[34]Taylor, M. E.. Lp-estimates on functions of the Laplace operator. Duke Math. J. 58(3) (1989), 773–793.CrossRefGoogle Scholar
[35]Varopoulos, N. Th.. Analysis on Lie groups. J. Funct. Anal. 76(2) (1988), 346–410.CrossRefGoogle Scholar
[36]Weber, A.. The Lp spectrum of Riemannian products. Arch. Math. (Basel) 90 (2008), 279–283.CrossRefGoogle Scholar
[37]Weber, A.. Tensor products of recurrent hypercyclic semigroups. J. Math. Anal. Appl. 351(2) (2009), 603–606.CrossRefGoogle Scholar