Skip to main content
×
×
Home

Dynamics of the heat semigroup on symmetric spaces

  • LIZHEN JI (a1) and ANDREAS WEBER (a2)
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.

Copyright
References
Hide All
[1]Aron, R. M., Seoane-Sepúlveda, J. B. and Weber, A.. Chaos on function spaces. Bull. Aust. Math. Soc. 71(3) (2005), 411415.
[2]Banasiak, J. and Moszyński, M.. A generalization of Desch–Schappacher–Webb criteria for chaos. Discrete Contin. Dyn. Syst. 12(5) (2005), 959972.
[3]Banks, J., Brooks, J., Cairns, G., Davis, G. and Stacey, P.. On Devaney’s definition of chaos. Amer. Math. Monthly 99(4) (1992), 332334.
[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), 5775.
[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), 24352441 (electronic).
[6]Bermúdez, T., Bonilla, A. and Peris, A.. On hypercyclicity and supercyclicity criteria. Bull. Aust. Math. Soc. 70(1) (2004), 4554.
[7]Bès, J. and Peris, A.. Hereditarily hypercyclic operators. J. Funct. Anal. 167(1) (1999), 94112.
[8]Conejero, J. A. and Peris, A.. Linear transitivity criteria. Topology Appl. 153(5-6) (2005), 767773.
[9]Davies, E. B.. Pointwise bounds on the space and time derivatives of heat kernels. J. Operator Theory 21(2) (1989), 367378.
[10]Davies, E. B.. Heat Kernels and Spectral Theory (Cambridge Tracts in Mathematics, 92). Cambridge University Press, Cambridge, 1990.
[11]deLaubenfels, R. and Emamirad, H.. Chaos for functions of discrete and continuous weighted shift operators. Ergod. Th. & Dynam. Sys. 21(5) (2001), 14111427.
[12]deLaubenfels, R., Emamirad, H. and Grosse-Erdmann, K.-G.. Chaos for semigroups of unbounded operators. Math. Nachr. 261/262 (2003), 4759.
[13]Desch, W., Schappacher, W. and Webb, G. F.. Hypercyclic and chaotic semigroups of linear operators. Ergod. Th. & Dynam. Sys. 17(4) (1997), 793819.
[14]Devaney, R. L.. An Introduction to Chaotic Dynamical Systems, 2nd edn(Addison-Wesley Studies in Nonlinearity). Addison-Wesley, Redwood City, CA, 1989.
[15]Mourchid, S. El. On a hypercyclicity criterion for strongly continuous semigroups. Discrete Contin. Dyn. Syst. 13(2) (2005), 271275.
[16]Mourchid, S. El. The imaginary point spectrum and hypercyclicity. Semigroup Forum 73(2) (2006), 313316.
[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.
[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.
[19]Gethner, R. M. and Shapiro, J. H.. Universal vectors for operators on spaces of holomorphic functions. Proc. Amer. Math. Soc. 100(2) (1987), 281288.
[20]Godefroy, G. and Shapiro, J. H.. Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 98(2) (1991), 229269.
[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. 157175.
[22]Grosse-Erdmann, K.-G.. Universal families and hypercyclic operators. Bull. Amer. Math. Soc. (N.S.) 36(3) (1999), 345381.
[23]Helgason, S.. Differential Geometry, Lie Groups, and Symmetric Spaces (Pure and Applied Mathematics, 80). Academic Press, New York, 1978.
[24]Helgason, S.. Groups and Geometric Analysis (Pure and Applied Mathematics, 113). Academic Press, Orlando, FL, 1984.
[25]Herzog, G.. On a universality of the heat equation. Math. Nachr. 188 (1997), 169171.
[26]Ji, L. and Weber, A.. L p spectral theory and heat dynamics of locally symmetric spaces. Preprint, 2008, arXiv:0810.0209.
[27]Kalmes, T.. On chaotic C0-semigroups and infinitely regular hypercyclic vectors. Proc. Amer. Math. Soc. 134(10) (2006), 29973002 (electronic).
[28]Kalmes, T.. Hypercyclic, mixing, and chaotic C0-semigroups induced by semiflows. Ergod. Th. & Dynam. Sys. 27(5) (2007), 15991631.
[29]Liskevich, V. A. and Perel’muter, M. A.. Analyticity of sub-Markovian semigroups. Proc. Amer. Math. Soc. 123(4) (1995), 10971104.
[30]Reed, M. and Simon, B.. Tensor products of closed operators on Banach spaces. J. Funct. Anal. 13 (1973), 107124.
[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), 398404.
[32]Strichartz, R. S.. Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal. 52(1) (1983), 4879.
[33]Sturm, K.-T.. On the Lp-spectrum of uniformly elliptic operators on Riemannian manifolds. J. Funct. Anal. 118(2) (1993), 442453.
[34]Taylor, M. E.. Lp-estimates on functions of the Laplace operator. Duke Math. J. 58(3) (1989), 773793.
[35]Varopoulos, N. Th.. Analysis on Lie groups. J. Funct. Anal. 76(2) (1988), 346410.
[36]Weber, A.. The Lp spectrum of Riemannian products. Arch. Math. (Basel) 90 (2008), 279283.
[37]Weber, A.. Tensor products of recurrent hypercyclic semigroups. J. Math. Anal. Appl. 351(2) (2009), 603606.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed