Rotation Vectors and Fixed Points of Area Preserving Surface Diffeomorphisms

Franks, John

doi:10.1090/S0002-9947-96-01502-4

Rotation Vectors and Fixed Points of Area Preserving Surface Diffeomorphisms
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Trans. Amer. Math. Soc. 348 (1996), 2637-2662 Request permission

Abstract:

We consider the (homological) rotation vectors for area preserving diffeomorphisms of compact surfaces which are homotopic to the identity. There are two main results. The first is that if $0$ is in the interior of the convex hull of the rotation vectors for such a diffeomorphism then $f$ has a fixed point of positive index. The second result asserts that if $f$ has a vanishing mean rotation vector then $f$ has a fixed point of positive index. There are several applications including a new proof of the Arnold conjecture for area preserving diffeomorphisms of compact surfaces.

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