Generic dynamics for evolution groups and $C_0$-semigroups on Hilbert spaces

Abstract

In this thesis, we study the typical behaviour (from the topological viewpoint) of the decaying rates of the orbits of unitary evolution groups and C0-semigroups on Hilbert spaces. We have found that: (1) some dynamical quantities related to evolution groups have an oscillating behaviour between a polynomially rapid decay and an arbitrarily slowdecay; (2) the decaying rates of each typical orbit, in Baires sense, of C0-semigroups which are stable but not exponentially stable depend on sequences of time going to infinity. The proofs are based on the relations between such decaying rates and some spectral properties of their respective generators