Home » Pseudoholomorphic foliations for area preserving disc maps. by Barney Bramham
Pseudoholomorphic foliations for area preserving disc maps. Barney Bramham

Pseudoholomorphic foliations for area preserving disc maps.

Barney Bramham

Published
ISBN : 9780549745181
NOOKstudy eTextbook
488 pages
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 About the Book 

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, byMoreWe 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.