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A variation on the PoincarĂ©-Birkhoff theorem. (English) Zbl 0679.58026

Hamiltonian dynamical systems, Proc. AMS-IMS-SIAM Jt. Summer Res. Conf., Boulder/Color. 1987, Contemp. Math. 81, 111-117 (1988).
[For the entire collection see Zbl 0665.00009.]
The author gives a brief exposition of C. Conley’s not widely known theory on chain recurrence and complete Lyapunov functions [Isolated invariant sets and the Morse index (Regional Conf. Ser. Math. 38) (1978; Zbl 0397.34056)] and uses this theory to prove the following special case of a result of P. Carter [Trans. Am. Math. Soc. 269, 285-299 (1982; Zbl 0507.55002)]:
Theorem. If f: \(A\to B\) is a homeomorphism of the annulus which is homotopic to the identity and satisfies a boundary twist condition, then either f has at least one fixed point or there is a smoothly embedded essential curve C in A with \(f(C)\cap C=\emptyset\).
Reviewer: J.Szilasi

MSC:

37B99 Topological dynamics
55M20 Fixed points and coincidences in algebraic topology
54H25 Fixed-point and coincidence theorems (topological aspects)
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
55M25 Degree, winding number