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 present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chao...

. We consider small random perturbations of a large class of nonuniformly hyperbolic unimodal maps and prove stochastic stability in the strong sense (L 1 -convergence of invariant densities) and uniform bounds for the exponential rate of decay of correlations. Our method is based on an analysis of the spectrum of a modified Perron-Frobenius operator for a tower extension of the Markov chain. 1. Introduction Let I ae R be a compact interval and f : I ! I be a smooth unimodal map with f(I) ae int (I). The prototype we have in mind are the quadratic maps f(x) = \Gammax 2 + a but our arguments and conclusions hold in the general context of maps with negative Schwarzian derivative and nondegenerate critical point. Let c 2 I be the critical point of f and c k = f k (c) for k 0. Throughout this paper we assume that (A1) jf k (c) \Gamma cj e \Gammaffk for all k H 0 , (A2) j(f k ) 0 (c 1 )j k c for all k H 0 , (A3) f is topologically mixing on the interval bounded by c ...

We consider multimodal C 2 interval maps f satisfying a summability condition on the derivatives Dn along the critical orbits which implies the existence of an absolutely continuous f - invariant probability measure . If f is non-renormalizable, is mixing and we show that the speed of mixing (decay of correlations) is strongly related to the rate of growth of the sequence (Dn ) as n ! 1. We also give sucient conditions for to satisfy the Central Limit Theorem. This applies for example to the quadratic Fibonacci map which is shown to have subexponential decay of correlations.

Abstract. We give the first example of a connected 4-regular graph whose Laplace operator’s spectrum is a Cantor set, as well as several other computations of spectra following a common “finite approximation ” method. These spectra are simple transforms of the Julia sets associated to some quadratic maps. The graphs involved are Schreier graphs of fractal groups of intermediate growth, and are also “substitutional graphs”. We also formulate our results in terms of Hecke type operators related to some irreducible quasi-regular representations of fractal groups and in terms of the Markovian operator associated to noncommutative dynamical systems via which these fractal groups were originally defined in [Gri80]. In the computations we performed, the self-similarity of the groups is reflected in the self-similarity of some operators; they are approximated by finite counterparts whose spectrum is computed by an ad hoc factorization process. 1.

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.

this paper we explore a construction which attempts to provide an element of the dictionary that has so far been missing: an explicit object that plays for a rational map the role played by the hyperbolic 3-orbifold quotient of a Kleinian group. To build this object we replace the notion of manifold by \lamination", Date: June 25, 1998. Appeared in J. Dierential Geom. 47 (1997) 17-94.

Consider the group generated by reflections in a finite collection of disjoint

We consider open sets of maps in a manifold M exhibiting non-uniform expanding behaviour in some domain S ⊂ M. Assuming that there is a forward invariant region containing S where each map has a unique SRB measure, we prove that under general uniformity conditions, the SRB measure varies continuously in the L 1-norm with the map. As a main application we show that the open class of maps introduced in [V] fits to this situation, thus proving that the SRB measures constructed in [A] vary continuously with the map. 1

In this paper we prove that in any non-trivial real analytic family of unimodal maps, almost any map is either regular (i.e., it has an attracting cycle) or stochastic (i.e., it has an absolutely continuous invariant measure). To this end we show that the space of analytic maps is foliated by codimensionone analytic submanifolds, \hybrid classes". This allows us to transfer the regular or stochastic property of the quadratic family to any non-trivial real analytic family.