Prof. Dr. Barney Bramham - Research
Faculties » Faculty of Mathematics » Chairs » Analysis and Geometry

Pseudoholomorphic foliations for area preserving disc maps

Barney Bramham

Abstract

We introduce a new approach to the study of area preserving disc maps. Our main result is the construction, by a deformation argument, of two infinite sequences of 'stable' foliations of the symplectization of a Reeb-like mapping torus, by pseudoholomorphic curves, associated to any smooth, non-degenerate, area preserving diffeomorphism of the closed unit 2-disc that is an irrational rotation on its boundary circle. Using intersection theory we prove a uniqueness statement for such foliations.

We illustrate the potential power of these foliations with a simple proof, making use of the positivity of intersections of holomorphic curves, that their existence alone forces such a disc map to have 1 or infinitely many periodic points, thus reproving, though under more stringent assumptions, a 'remarkable' result of John Franks from '91. More generally, we hope that this will be a powerfull organising tool with which to approach some of the many open questions about area preserving disc maps.

Thesis (Ph.D.)–New York University. ProQuest LLC, Ann Arbor, MI, 2008. 488 pp. ISBN: 978-0549-74518-1