Some operator and spectral criterions for chaoticity of C₀ semigroups in Banach spaces
by
Marcin Moszynski
Warsaw University, Faculty of Mathematics, Informatics and Mechanics, Poland
Coauthors: Jacek Banasiak

We study some kind of chaotic behaviour of C₀ semigroups in Banach spaces. We introduce the notion of subchaos denoting the topological chaos (in the sense of Devaney [Dev]) for the semigroup restricted to an infinite-dimensional, invariant subspace. We show some criterions for subchaos in terms of spectral properties of the generator of the semigroup. The main one is a reformulation of a Desch, Schappacher and Webb criterion for topological chaos [DSW], stating that a C₀ semigroup is topologically chaotic if there exists a selection of eigenvectors of the generator, that is analytic in some open set of a complex plane with non-empty intersection with the imaginary axis, and such that a non-degeneracy condition holds. We prove that without this last condition the semigroup is still subchaotic. We show also some examples and applications (e.g., for generalized birth and death type models).

[Dev] R.L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd edn., Addison-Wesley, New York, 1989.

[DSW] W. Desch, W. Schappacher and G.F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dynam. Systems 17(1997), 793-819.

Date received: June 16, 2004

Copyright © 2004 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # cane-93.