We prove the Feigenbaum-Coullet-Tresser conjecture on the hyperbolicity of the renormalization transformation of bounded type. This gives the first computer-free proof of the original Feigenbaum observation of the universal parameter scaling laws. We use the Hyperbolicity Theorem to prove Milnor’s conjectures on self-similarity and “hairiness ” of the Mandelbrot set near the corresponding parameter values. We also conclude that the set of real infinitely renormalizable quadratics of type bounded by some N> 1 has Hausdorff dimension strictly between 0 and 1. In the course of getting these results we supply the space of quadratic-like germs with a complex analytic structure and demonstrate that the hybrid classes form a complex codimension-one foliation of the connectedness locus.

We prove uniform hyperbolicity of the renormalization operator for all possible real combinatorial types. We derive from it that the set of infinitely renormalizable parameter values in the real quadratic family Pc: x ↦ → x² + c has zero measure. This yields the statement in the title (where “ regular ” means to have an attracting cycle and “stochastic” means to have an absolutely continuous invariant measure). An application to the MLC problem is given.

We prove that the period doubling operator has an expanding direction at the fixed point. We use the induced operator, a “Perron-Frobenius type operator”, to study the linearization of the period doubling operator at its fixed point. We then use a sequence of linear operators with finite ranks to study this induced operator. The proof 1 is constructive. One can calculate the expanding direction and the rate of expansion of the period doubling operator at the fixed point. Contents §1 Introduction. §2 The Period Doubling Operator and the Induced Operator. §2.1 From the period doubling operator to the induced operator. §2.2 The induced operator Lϕ.

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. We show that, if the linearization of a map at a fixed point leaves invariant a spectral subspace which satisfies certain non-resonance conditions, the map leaves invariant a smooth manifold tangent to this subspace. This manifold is as smooth as the map, but is unique among C L invariant manifolds, where L depends only on the spectrum of the linearization. We show that if the non-resonance conditions are not satisfied, a smooth invariant manifold need not exist and also establish smooth dependence on parameters. We also discuss some applications of these invariant manifolds and briefly survey related work. 1. Introduction Besides their intrinsic appeal, invariant manifold theorems are interesting in Dynamics because they provide landmarks which organize the long--time behavior. ?From this point of view, having more invariant manifolds is quite desirable, since it means having more tools for the analysis of the dynamical system. In particular, it is often the case, that associate...

This paper is linked to McMullen's paper [McM1] in this volume: concepts and results discussed in [McM1] may be used here without extra comments. 2. Combinatorics of complex unimodal maps

One of the predominant themes of nonlinear dynamics during the past twenty years has been the characterization of the "routes to chaos". Within generic families of vector fields, there are ubiquitous patterns that describe how qualitative properties of vector fields can change with a varying parameter. Bifurcation theory classifies these patterns, and they form the substrate for determining the routes to chaos that one expects to see in physical systems. The presence of symmetry typically alters the patterns that one observes, and it has been well understood that symmetry can lead to the persistence of new types of dynamical behavior and bifurcation. This paper gives another, more extreme, example of how symmetry can affect the routes to chaos. In the systems that we describe below, there are persistent bifurcations that lead directly from a "trivial" steady state to chaotic attractors of small amplitude. These bifurcations are "supercritical" in the sense that the attractors emerge fr...

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

Abstract. In the paper we discuss two questions about smooth expanding dynamical systems on the circle. (i) We characterize the sequences of asymptotic length ratios which occur for systems with Hölder continuous derivative. The sequence of asymptotic length ratios are precisely those given by a positive Hölder continuous function s (solenoid function) on the Cantor set C of 2-adic integers satisfying a functional equation called the matching condition. The functional equation for the 2-adic integer Cantor set is s(2x + 1) = s(x) s(2x)

In this paper we extend M. Lyubich's recent results on the global hyperbolicity of renormalization of quadratic-like germs to the space U r of C r unimodal maps with quadratic critical point. We show that in U r the bounded-type limit sets of the renormalization operator have an invariant hyperbolic structure provided r 2+ with close to one. As an intermediate step between Lyubich's results and ours, we prove that the renormalization operator is hyperbolic in a Banach space of real analytic maps. We construct the local stable manifolds and prove that they form a continuous lamination whose leaves are C 1 codimension one Banach submanifolds of U r , and whose holonomy is C 1+ for some > 0. We also prove that the global stable sets are C 1 immersed (codimension one) submanifolds as well, provided r 3+ with close to one. As a corollary, we deduce that in generic one parameter families of C r unimodal maps, the set of parameters corresponding to innitely r...