## A variation on the Poincaré-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

### Keywords:

chain recurrence; complete Lyapunov functions

### Citations:

Zbl 0665.00009; Zbl 0397.34056; Zbl 0507.55002