Random dynamical systems and transfer operator cycles
Random dynamical systems are generated by recursively applied sequences of maps, where the choice of map at each timestep is determined by a stochastic process. Such systems are usually formulated as a skew-product, in which the \base" dynamics is autonomous and the \ bre-to- bre" mappings are determined by the base. An observer watching only the bres sees non-autonomous dynamics, or \random" orbits. Typical questions of interest relate to the long-term distribution of orbits, mass- transport, rates of mixing and so on, and there are numerous real-world applications. This talk will introduce the important ideas for studying random dynamics from an ergodic theory viewpoint. Questions of \stochastic stability" can be formulated (and answered) in this way, and much of the theory works as one might expect when the base process is IID. In such cases, insight can even be gained via a transfer operator obtained by averaging over all bres. When the base process is not IID (for example, an ergodic dynamical system), averaging may yield irrelevant objects, and one must study cocycles of transfer operators and the important dynamical structures on bres become random variables. This picture will be outlined, and some positive results on accessing the distribution of orbits of certain random interval maps will be given.
SubjectsField of Research::01 - Mathematical Sciences::0101 - Pure Mathematics::010109 - Ordinary Differential Equations, Difference Equations and Dynamical Systems
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