The Essential Spectral Radius and Asymptoti Properties of Transfer Operators'

Abs t r ac t The statistical behavior of deterministic and stochastic dynamical sys-tems may be described using transfer operators, which generalize the no-tion of Frobenius-Perron and Koopman operators. Since numerical tech-niques to analyse dynamical systems based on eigenvalues problems for the corresponding transfer operator have emerged, bounds on its essential spectral radius became of interest. This article shows that they are also of great theoretical interest. We give an analytical representation of the essential spectral radius in L\fj,), which then is exploited to analyse the asymptotical properties of transfer operators by combining results from functional analysis, Markov operators and Markov chain theory. In par ticular, it is shown that an essential spectral radius less than 1, uniform constrictiveness and some "weak form " of the so-called Doeblin condition are equivalent. Finally, we apply the results to study three main prob-lem classes: deterministic systems stochastically perturbed deterministic systems and stochastic systems K e y w o r d s, uniformly constrictive, asymptotically stable, exact, asymptotically pe