A Canonical Partition of the Periodic Orbits of Chaotic Maps

We show that the periodic orbits of an area-contracting horseshoe map can be partitioned into subsets of orbits of minimum period k, 2k, 4k, 8k, ..., for some positive integer k. This partition is natural in the following sense: for any parametrized area-contracting map which forms a horseshoe, the orbits in one subset of the partition are contained in a single component of orbits in the full parameter space. Furthermore, prior to the formation of the horseshoe, this component contains attracting orbits of minimum period 2mk, for each nonnegative integer m. For parametrized maps of Rn which form horseshoes, there is a rich structure of attracting periodic points for parameter values which precede the existence of the horseshoe. However, once the horseshoe is fully formed, all periodic points are unstable. Let f\ : Rn —> Rn (0 < A < 1) be a C1 map which contracts crosssectional areas. (Precise formulations of hypotheses follow this introduction.) We assume that /o has either no periodic orbits or only attracting ones, and that /i has only unstable periodic orbits (as, for example, in the horseshoe map). A periodic point p of an area-contracting map can have at most one unstable direction. Thus if p is unstable, then Df^(p) (where k is the period of p) has exactly one real eigenvalue /z such that \p\ > 1. These periodic orbits fall into two classes: if p is in (1, oo), we call p a saddle orbit; if p is in (—oo, —1), we call it a Möbius orbit. Franks [F] showed that the existence of an orbit of period k at A = 1 (i.e., a periodic orbit of fi) implies the existence of a sequence {<7¿}¿>o of Möbius orbits at A = 1, where the period of g, is 2lk, for each i > 0. Here we show that the set of all the periodic orbits of /0 and /i can be partitioned into disjoint subsets. Each of these subsets contains an attractor of period k (for some k > 1) at A = 0 or a saddle of period k at A = 1, and a sequence {<7¿}¿>o of Möbius orbits at A = 1, where the period of qt is 2%k, for each i > 0. Hence the existence of an orbit at A = 0 or A — 1 implies the existence of all other orbits in the subset to which it belongs. Furthermore, this partition is a natural one in the sense that an entire subset lies in one connected component of periodic orbits in (x, A)-space (Theorem 2). We also examine how the unstable periodic orbits at A = 1 are related to the attractors which occur in the parameter range 0 < A < 1. In [YA] it was shown that if / has either an attractor of period k at A = 0 or a saddle of period k at A — 1, then it must have a period-doubling cascade of attractors (i.e., / has a sequence {a¿}¿>o of attractors such that the period of a¿ is 2lk, for each i > 0). Received by the editors September 20, 1984. 1980 Mathematics Subject Classification. Primary 58F13, 58F14; Secondary 34C35. 1 Research was partially supported by the National Science Foundation. This paper was written while the author was visiting Michigan State University and the University of Maryland. ©1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page 713 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use