AbstractParabolic flows are slowly chaotic flows for which nearby trajectories diverge polynomially in time. Examples of smooth parabolic flows are unipotent flows on semisimple Lie groups and nilflows on nilmanifolds, which are both well-understood. Beyond the homogeneous set-up, however, very little is known for generic smooth parabolic flows and a general theory about their ergodic properties is missing. In this thesis, we study three classes of smooth, non-homogeneous parabolic flows and we show how a common geometric shearing mechanism can be exploited to prove mixing.
We first establish a quantitative mixing result in the setting of locally Hamiltonian flows on compact surfaces. More precisely, given a compact surface with a smooth area form, we consider an open and dense set of locally Hamiltonian flows which admit at least one saddle loop homologous to zero and we prove that the restriction to any minimal component of typical such flows is mixing. We provide an estimate of the speed of the decay of correlations for a class of smooth observables.
We then focus on perturbations of homogeneous flows.
We study time-changes of quasi-abelian filiform nilflows, which are nilflows on a class of higher dimensional nilmanifolds. We prove that, within a dense set of time-changes of any uniquely ergodic quasi-abelian filiform nilflow, mixing occurs for any time-change which is not cohomologous to a constant, and we exhibit a dense set of explicit mixing examples.
Finally, we construct a new class of perturbations of unipotent flows in compact quotients of SL(3,R) which are not time-changes and we prove that, if they preserve a measure equivalent to Haar, then they are ergodic and, in fact, mixing.
|Date of Award||25 Sep 2018|
|Supervisor||Corinna Ulcigrai (Supervisor)|