Fixed Points of Certain Area Preserving Diffeomorphisms of D²
by
Shigenori Matsumoto
Nihon University

Let f be a diffeomorphism of the closed unit 2-disc D². A ball B_d(x) centered at a point x of ∂D² and of radius d > 0 is called + (resp. -) -twisted if, writing f as (r, q) → (R, Q) in the polar coordinates of D², dQ/dr is nonnegative (resp. nonpositive) on B_d(x) ∩Int(D²) and nonvanishing on B_d/8(x) ∩Int(D²). Our main result is the following.

Theorem 1. Let f be an Euclidian area preserving diffeomorphism of D² keeping the points of ∂D² fixed. If there exist both + and - -twisted balls of radius d and if f is d/8 near to the identity in the C¹ topology, then the fixed point set of f in Int(D²) is nonempty and nonconnected.

Let us show how Theorem 1 is obtained. Let H be the space of smooth maps H:[0, 1] ×R²n → R such that Supp(H) = Cl( ∪_t Supp(H_t) ) is compact, where we denote H_t(x) = H(t, x). Given H in H, let f^t_H

(with t in [0, 1]) be the one parameter family of diffeomorphisms of R²n determined by the equations df^t_H/dt = J ÑH_t ○f^t_H and f⁰_H = id, where J is the standard almost complex structure and Ñ stands for the gradient. Given a fixed point x of f = f¹_H, define the action A(x;H) by

A(x;H) = (1/2) ó õ 1 0  - J dftH(x)/dt ○ftH(x) dt - ó õ 1 0  Ht(ftH(x)) dt.

It is known that A(x;H) is independent of the choice of H in H satisfying f = f¹_H and we denote A(x;f) = A(x;H). Theorem 1 is obtained as an application of the following fact, which is actually implicit in [HZ].

Theorem 2. If there exists an open ball B_e(y) such that H is nonnegative (resp. nonpositive) on [0, 1] ×B_e(y) and H_t(y) is positive (resp. negative) for some t, and if sup|ÑH| < 4 e, then there exists a fixed point x of f such that A(x;f) is negative (resp. positive).

[HZ] H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, 1994 Birkhäuser, Basel

Date received: January 11, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caab-73.