PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 9, September 1998, Pages 2805–2810 S 0002-9939(98)04267-1 ROTATION INTERVALS FOR CHAOTIC SETS KATHLEEN T. ALLIGOOD AND JAMES A. YORKE (Communicated by Linda Keen) Abstract. Chaotic invariant sets for planar maps typically contain periodic orbits whose stable and unstable manifolds cross in grid-like fashion. Consider the rotation of orbits around a central fixed point. The intersections of the invariant manifolds of two periodic points with distinct rotation numbers can imply complicated rotational behavior. We show, in particular, that when the unstable manifold of one of these periodic points crosses the stable manifold of the other, and, similarly, the unstable manifold of the second crosses the stable manifold of the first, so that the segments of these invariant manifolds form a topological rectangle, then all rotation numbers between those of the two given orbits are represented. The result follows from a horseshoe-like construction. Let f be an orientation-preserving diffeomorphism of the annulus A = S 1 × I , where we parametrize S 1 by x in the unit interval, identifying endpoints. The average rotation of an orbit under the action of f is given by the rotation number, which is an asymptotic average of the rate of rotation (i.e., angle per iterate) along an orbit of f . See, for example, [4] for background material on rotation numbers. Formally, let U = R × I be the universal cover of A, let ˜ f : U → U be a lift of f , and let π x : U → R be the projection onto the first coordinate. Then the forward and backward rotation numbers, ρ + and ρ - , respectively, are defined for a point (x, y) of A as ρ + (x, y) = lim n→∞ 1 n [π x ˜ f n (x, y) - x], if this limit exists, and ρ - (x, y) = lim n→∞ 1 n [π x ˜ f -n (x, y) - x], if this limit exists. In one dimension the situation is simple: for an orientation-preserving homeo- morphism of the circle, the rotation number ρ + = ρ - always exists and is indepen- dent of the choice of point on the circle. For annulus maps, however, ρ + need not equal ρ - or even exist, for any given point, and different points can have different Received by the editors January 24, 1997. 1991 Mathematics Subject Classification. Primary 58Fxx. The authors’ research was partially supported by the National Science Foundation. The second author’s research was also supported by the Department of Energy (Office of Scientific Computing). c 1998 American Mathematical Society 2805 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use