A NOTE ON OPERATOR SEMIGROUPS ASSOCIATED TO CHAOTIC FLOWS

The transfer operator associated to a ow (continuous time dy- namical system) is a one-parameter operator semigroup. We consider the operator-valued Laplace transform of this one-parameter semigroup. Esti- mates on the Laplace transform have been used in various settings in order to show the rate at which the ow mixes. Here we consider the case of exponen- tial mixing and the case of rapid mixing (super polynomial). We develop the operator theory framework amenable to this setting and show that the same estimates may be used to produce results, in terms of the operators, which go beyond the results for the rate of mixing.