No abstract

No abstract

No abstract

Since Smale [7] described the complicated dynamical behavior of horseshoe maps, many dynamical systems depending on a parameter have been shown to develop horseshoes. In the horseshoe, there are 2 k fixed points for the kth. iterate of the map. Numerical studies of parametrized maps and the investigations of Newhouse [5] and Robinson [6] indicate there is a rich structure of attractors for parameters preceding the existence of the horseshoe. Once the horseshoe is formed, however, all periodic points are unstable and almost any trajectory starting in the horseshoe eventually leaves it. Assuming that cross-sectional areas are contracting, we prove that infinitely many cascades of period doublings must occur in the process of forming a horseshoe. Each such cascade need not evolve regularly or monotonically, but it must contain attracting periodic points of all the periods k,2k,4k,8k,..., for some k. For the area preserving case with n = 2, we have an analogous result, where elliptic periodic points replace attracting periodic points.We would like to thank L. Tedeschini-Lalli and S. Pelikan for their helpful suggestions.First, we introduce notation and give hypotheses for the formation of horseshoes. Let

No abstract