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254
FeigenbaumCoulletTresser universality and Milnor's Hairiness Conjecture
, 1999
"... We prove the FeigenbaumCoulletTresser conjecture on the hyperbolicity of the renormalization transformation of bounded type. This gives the first computerfree proof of the original Feigenbaum observation of the universal parameter scaling laws. We use the Hyperbolicity Theorem to prove Milnor’s c ..."
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Cited by 69 (8 self)
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We prove the FeigenbaumCoulletTresser conjecture on the hyperbolicity of the renormalization transformation of bounded type. This gives the first computerfree proof of the original Feigenbaum observation of the universal parameter scaling laws. We use the Hyperbolicity Theorem to prove Milnor’s conjectures on selfsimilarity 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 quadraticlike germs with a complex analytic structure and demonstrate that the hybrid classes form a complex codimensionone foliation of the connectedness locus.
Domains for Computation in Mathematics, Physics and Exact Real Arithmetic
 Bulletin of Symbolic Logic
, 1997
"... 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 dist ..."
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Cited by 58 (13 self)
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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...
ALMOST EVERY REAL QUADRATIC MAP IS EITHER REGULAR OR STOCHASTIC
, 1997
"... 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 “ regul ..."
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Cited by 46 (4 self)
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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.
On the spectrum of Hecke type operators related to some fractal groups
 TRUDY MAT. INST. STEKLOV
, 1999
"... We give the first example of a connected 4regular 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 ..."
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Cited by 46 (20 self)
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We give the first example of a connected 4regular 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 quasiregular 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 selfsimilarity of the groups is reflected in the selfsimilarity of some operators; they are approximated by finite counterparts whose spectrum is computed by an ad hoc factorization process.
Decay of Correlations in OneDimensional Dynamics
, 2002
"... We consider multimodal C³ interval maps f satisfying a summability condition on the derivatives Dn along the critical orbits which implies the existence of an absolutely continuous finvariant probability measure µ. If f is nonrenormalizable, µ is mixing and we show that the speed of mixing (decay ..."
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Cited by 43 (16 self)
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We consider multimodal C³ interval maps f satisfying a summability condition on the derivatives Dn along the critical orbits which implies the existence of an absolutely continuous finvariant probability measure µ. If f is nonrenormalizable, µ 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 → ∞. We also give sufficient conditions for µ to satisfy the Central Limit Theorem. This applies for example to the quadratic Fibonacci map which is shown to have subexponential decay of correlations.
Statistical stability for robust classes of maps with nonuniform expansion, Ergd
 Th. & Dynam. Sys
"... We consider open sets of maps in a manifold M exhibiting nonuniform 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 continuousl ..."
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Cited by 41 (17 self)
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We consider open sets of maps in a manifold M exhibiting nonuniform 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 1norm 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
Regular or stochastic dynamics in real analytic families of unimodal maps
, 2003
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Strong Stochastic Stability and Rate of Mixing for Unimodal Maps
 Ann. Sci. ' Ecole Norm. Sup
, 1994
"... . 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 o ..."
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Cited by 35 (9 self)
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. 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 PerronFrobenius 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 ...
The classification of conformal dynamical systems
 In Current Developments in Mathematics
, 1995
"... Consider the group generated by reflections in a finite collection of disjoint ..."
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Cited by 32 (10 self)
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Consider the group generated by reflections in a finite collection of disjoint