On a set of numbers arising in the dynamics of unimodal maps

In this paper we initiate the study of the arithmetical properties of a set numbers which encode the dynamics of unimodal maps in a universal way along with that of the corresponding topological zeta function. Here we are concerned in particular with the Feigenbaum bifurcation. 1 Preliminaries. We start by reviewing some basic ideas of (a version of) the kneading theory for unimodal maps. For related approaches and/or more details see [CE], [Dev], [deMvS]. Definition 1.1 A smooth map f : [0, 1] → [0, 1] is called unimodal if it has exactly one critical point 0 < c0 < 1 and moreover f(0) = f(1) = 0. For unimodal maps the orbit of the critical point c0 determines in a sense the complexity of any other orbit. To be more precise, given x ∈ [0, 1] we call itinerary of x with f the sequence i(x) = s1s2s3 . . . where si = 0 or 1 according to f (x) < c0 or f (x) ≥ c0. An important point is that such symbolic representation is in fact ‘faithful’, that is if s(x) = s(x) then x = x. Differently said, the partition of [0, 1] in the two semiintervals P0 = [0, c0) e P1 = [c0, 1) is generating for a unimodal map f with critical point c0. It is clear that if s = i(x) is a sequence obtained as above then i(f(x)) = σ(s) where σ denotes the left-shift: if s = s1s2s3 . . . then σ(s) = s2s3s4 . . .. The itinerary of the point c1 = f(c0) is called kneading sequence K(f) of f . We say moreover that a given sequence s of 0 and 1 is admissibile for f if there is x ∈ [0, 1] such that i(x) = s. A nice way to decide whether or not a given sequence is admissible amounts to establish an ordering on the itineraries which corresponds to ordering of the real line. In this way, the admissible sequences are those which never become greater than the kneading ∗Dipartimento di Matematica e Informatica dell’Università di Camerino and INFM, via Madonna delle Carceri, 62032 Camerino, Italy. e-mail: .