By P. Collet (auth.), E. Tirapegui, W. Zeller (eds.)
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Additional info for Instabilities and Nonequilibrium Structures III
I. INTB OPUCTION. Consider both the unique quadratic map at the boundary between zero and positive entropy, and any other smooth enough generic unimodal mapping with the same topological dynamics. e. you can guess where to look for the various ratios describing the fine structure of the Cantor sets. It will take a fair amount of this paper to transform this single long statement to a long sequence of shorter and (hopefully) understandable ones. On the way, I shall give crude numbers and some remarks and theorems, some of which refer to deep questions but all of which are elementary.
Eckmannn and H. Koch: "Period doubling bifurcations for families of maps on IR n ", J. Stat. Phys. 25 (1980)1-15. M] H. EI Hamouly and C. Paris,293 (1981 ),525-528. M. Gambaudo " Linked fixed points of a C t orientation preserving embedding of D 2 ", to appear in Proc. of the Cambridge Phil. Soc .. E. A. , 6, (1982), 427-434. [Kat] A. Katok, "Lyapunov exponents, entropy and periodic orbits for diffeomorphims", Pub. Math. S. 51 (1980), 137-174. [Kol] B. Kolev " Point fixe lie a une orbite periodique d'un diffeomorphisme de IR2", to appear in the Comptes rendus Acad.
TRESSER R(~) Figure 4. ). ) to the boundary of R I . Of course, here the map A is not uniquely defined. We would have to add some supplementary condition to determine it completly. K) and its geometrical content has been described in [GST). ) and in a neighborhood of codimension 1 manifold Wl~cC ¢*), >*, a local containing ¢*, on which the maps have the following real dynamics:they have a periodic orbit with period 2 n for all n, no other periodic orbits HOW HORSESHOES ARE CREATED 23 and an invariant Cantor set.