Rotation Vectors for Surface Diffeomorphisms

John Franks
Additional contact information
John Franks: Northwestern University, Department of Mathematics

A chapter in Proceedings of the International Congress of Mathematicians, 1995, pp 1179-1186 from Springer

Abstract: Abstract We consider the concepts of rotation number and rotation vector for area preserving diffeomorphisms of surfaces and their applications. In the case that the surface is an annulus A the rotation number for a point x ∈ A represents an average rate at which the iterates of x rotate around the annulus. More generally the rotation vector takes values in the one-dimensional homology of the surface and represents the average “homological motion” of an orbit. There are two main results. The first is that if 0 is in the interior of the convex hull of the recurrent rotation vectors for an area preserving diffeomorphism ƒ isotopic to the identity, then ƒ has a fixed point of positive index. The second result asserts that if ƒ has a vanishing mean rotation vector, then ƒ has a fixed point of positive index. Applications include the result that an area preserving diffeomorphism of A that has at least one periodic point must in fact have infinitely many interior periodic points. This is a key step in the proof of the theorem that every smooth Riemannian metric on S2 has infinitely many distinct closed geodesies. Another application is a new proof of the Arnold conjecture for area preserving diffeomorphisms of closed oriented surfaces.

Date: 1995
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-9078-6_111

Ordering information: This item can be ordered from
http://www.springer.com/9783034890786

DOI: 10.1007/978-3-0348-9078-6_111

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().