Fine Structure of Universal Cantor Sets

Abstract

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. If you look well enough at the invariant Cantor sets of these maps, not only do you see more and more similar geometries when looking at finer and finer scales, which is the main fact in this context, but it seems that you can also understand the way the small scales are organized, i.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. The very existence of some elementary mathematics in this context has been my main surprise these years; I “have known” (quotation marks often mean numerical evidence) the facts reported here (and some more details) since 1977, at the time Pierre Coullet and I discovered on our part universality and the role of the renormalization group in dynamics [CT] (i.e. before we learned of Mitchell Feigenbaum and of his slightly anterior but in any case, by then unpublished similar findings [Fe]). I shall be quite brief on generalizations to other renormalizable settings: I talked about these in Chile, but I shall only give very few hints here.