Expansive flows

Doctoral thesis English OPEN
Flinn, Laurence William
  • Subject: QA
    arxiv: Physics::Fluid Dynamics

In this paper we attempt to develop a theory of expansive continuous flows on compact metric spaces analogous to the theory for discrete expansive flows.\ud \ud In Section 1, we develop the concept of expansiveness for continuous flows, and consider some examples of this concept.\ud \ud In Section 2, we use the ideas of local cross-sections and flow boxes for a continuous flow to derive a special type of open cover for a space admitting a fixed point free flow. We define what it means for such an open cover to be a generator and show that a flow is expansive if and only if it admits a generator.\ud \ud In Section 3, we prove some elementary theorems about expansive flows, including the result that, for am expansive flow, there is an exponential growth rate on the number of periodic orbits.\ud \ud In Section 4, we consider asymptotic properties of expansive flows, and show 1n particular that the nonisolated closed orbits of an expansive flow are topologically sources, saddles, or sinks.\ud \ud In Section 5, we prove that every expansive fixed point free flow 1S a factor of the flow obtained by suspending the shift map on a subspace of the space of sequences modelled on a finite set.\ud \ud In Section 6, we consider the question of the existence of expansive flows on compact 2-manifolds, and prove that there are no such.\ud \ud In the Appendix we give a proof (due essentially to DoVo Anosov) that Anosov flows are expansive.\ud
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