On the struture of homeomorphims of the open annulus
Let $h$ be a without fixed point lift to the plane of a homeomorphism of the open annulus isotopic to the identity and without wandering point. We show that $h$ admits a $h$-invariant dense open set $O$ on which it is conjugate to a translation and we study the action of $h$ on the compactly connected components of the closed and without interior set ${\bf R}^2 \setminus O$.
Bounded recurrent sets for planar homeomorphisms
We answer a question proposed by Barge and Franks in 1993 by showing that a bounded connected set in the plane which is closed or open and meets an infinite number of its iterates under an orientation-preserving fixed point free homeomorphism must meet all of them.
An infinite class of periodic solutions of periodically perturbed Duffing equations at resonance
In this paper, by using a generalized form of the PoincareBirkhoff Theorem, we demonstrate that the Duffing equation d2 + g(x) = p(t) (= p(t + 27r)) may also admit an infinite number of 27r-periodic solutions even in a resonance case. 1. In this paper we study the existence problem of periodic solutions for the Duffing equation (1.1) dt2 + g(x) = p(t) where g(x), p(t) E C(R, R) and p(t) is 2Xr-periodic. In the super-linear case, that is, lim [x-lg(x)] = oo, jxj-> oo it has been proved in a recent paper by W. Y. Ding that (1.1) has infinitely many 2ir-periodic solutions [1]. D. E. Leach proved the existence and uniqueness of 27r-periodic solution of (1.1) under the following Loud condition (1.2) m2 0 and two such constants X and ,t [2]. R. Reissig proved the existence of periodic solutions of (1.1) under a weaker condition [3] m 2 a > 0. The crucial point of the conditions above is to exclude the resonance cases. At resonance, (1.1) may have no bounded solution [6]; therefore, the existence problem of periodic solutions challenges attention. A. C. Lazer and D. E. Leach, and L. Cesari succeeded to prove the existence of 2ir-periodic solutions of (1.1) at a point of resonance provided that h(x) = g(x) m2x is bounded and some other additional conditions are assumed [4, 5]. In a recent paper, the present author resolved the existence problem of 2ir-periodic solutions of (1.1) under a weakened version of (1.2) [6]: m2 < g'(x) < (m + 1)2 (g(0) 0). Received by the editors February 25, 1981. 1980 Mathematics Subject Classification. Primary 34C25; Secondary 70K20.