Cr - closing lemma problems for vector fields on surfaces.

Abstract

The Cr Closing Lemma, r>l, is an important open problem in Dynamics, of which flows on surfaces provide a vital test case. The aim of this project is to generalize to higher genus surfaces several partial results that are already known for the torus. Firstly, we will determine whether there are counterexamples to local C2 closing for higher genus flows, as have been constructed on the torus. This will be done by analyzing the positions of blocking singularities through a sequence of renormalizations of the surface. The resolution of the Cr Closing Lemma problem rests on finding closing methods for flows exhibiting recurrence of bounded type. This assumption restricts the number and types of wandering domains possible in flows on the torus, and we shall determine whether there is a corresponding restriction in the higher genus case. Finally, distortion estimates will be used to seek sufficient conditions for Cr closing in terms of local properties at the saddle points. (AU)