Generalizations of the Poincaré-Birkhoff Theorem

TLDR: In this article, a generalization of the Poincar&Birkhoff theorem on area-preserving homeomorphisms of the annulus which satisfy a boundary twist condition is presented.

Abstract: In this article we prove some generalizations of the classical Poincar&Birkhoff theorem on area-preserving homeomorphisms of the annulus which satisfy a boundary twist condition. The work of G. D. Birkhoff on this theorem and its applications can be found in [B1], [B2], and Chapter V of [B3]. A more modern treatment can be found in [B-N]. We prove a theorem for the open annulus A = S1 x (0, 1), since, as we will see, the theorem for the closed annulus can easily be obtained from this result. For the open annulus, however, it is not immediately obvious what should be the analogue of the twist condition. It turns out that the most general hypothesis, and the most natural from the point of view of our proof, involves the notion of positively and negatively returning disks for some lift of f to the covering space A = R x (0, 1). More precisely, if f: A -, A is a lift of f: A -, A we will say that the e is a positively returning disk for f if there is an open disk U c A such that f(U) n U = 0 and fP(U) n (U + k) # 0 forsome n, k > 0(here U + k denotes the set {(x + k, t) I(x, t) e U }). Thus U is disjoint from its image but under iteration by f returns to a positive translate of itself. A negatively returning disk is defined similarly but with k 0 such that fl(U) n U # 0. In Section 2 we prove the following.