On the Quasisymmetrical Classification of Infinitely Renormalizable Maps I. Maps with Feigenbaum's Topology

We begin by considering the set of infinitely renormalizable unimodal maps on the interval [−1, 1]. A function f defined on [−1, 1] is said to be unimodal if it is continuous, increasing on [−1, 0], decreasing on [0, 1] and symmetric about 0, and if it fixes −1 and maps 1 to −1 . Moreover, it is said to be renormalizable if there is an integer n > 1 and a subinterval I 6= [−1, 1] containing 0 such that f (I) ⊆ I. We will assume that one endpoint q of I is fixed by f ◦n and I is symmetric about 0. If we normalize I to [−1, 1] by the linear map α, which maps q to −1, then R(f) = α ◦ f ◦n ◦α is unimodal, too. This map R will be called the renormalization operator. To fix our notation, we will assume that n is the minimum such integer and call it the return time. A unimodal map f is infinitely renormalizable if every R(f) is renormalizable, say with return time nk. Furthermore, f is of bounded type if all the return times are less than a constant integer, otherwise, f is of unbounded type. In particular, we call f a Feigenbaum map if all the return times are 2. A well-known example f = qλ∞ of a Feigenbaum map [4, 6, 18] is obtained by period-doubling cascade in the family {qλ(z) = −(1+λ)z +λ}0≤λ≤1. Let U be the space of unimodal maps f = h◦Qt where Qt(x) = −|x| t for some t > 1 and h is a diffeomorphism (a unimodal map can always be written in this form by some smooth change of coordinate [12]). We may assume either h is a C-diffeomorphism with nonpositive Schwarzian derivative [3] or h is a C-diffeomorphism [17]. However, the smoothness is not important in this paper as long as h satisfies the distortion properties discussed in [17]. To avoid many technical notations, henceforth, we will assume that h is a C-diffeomorphism with nonpositive Schwarzian derivative. We note that the Schwarzian derivative S(h) of a C-diffeomorphism is S(h) = h h′ − 3 2 (h h′ )2 .