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20
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 51 (5 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.
On computational complexity of Siegel Julia sets
 Commun. Math. Physics
"... Abstract. It has been previously shown by two of the authors that some polynomial Julia sets are algorithmically impossible to draw with arbitrary magnification. On the other hand, for a large class of examples the problem of drawing a picture has polynomial complexity. In this paper we demonstrate ..."
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Cited by 10 (4 self)
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Abstract. It has been previously shown by two of the authors that some polynomial Julia sets are algorithmically impossible to draw with arbitrary magnification. On the other hand, for a large class of examples the problem of drawing a picture has polynomial complexity. In this paper we demonstrate the existence of computable quadratic Julia sets whose computational complexity is arbitrarily high. 1. Foreword Let us informally say that a compact set in the plane is computable if one can program a computer to draw a picture of this set on the screen, with an arbitrary desired magnification. It was recently shown by the second and third authors, that some Julia sets are not computable [BY]. This in itself is quite surprising to dynamicists – Julia sets are among the “most drawn ” objects in contemporary mathematics, and numerous algorithms exist to produce their pictures. In the cases when one has not been able to produce informative pictures (the dynamically pathological cases, like maps with a Cremer or a highly Liouville Siegel point) the feeling had been that this was due to the immense computational resources required by the known algorithms.
Constructing NonComputable Julia Sets
 Proc. of STOC 2007
"... While most polynomial Julia sets are computable, it has been recently shown [12] that there exist noncomputable Julia sets. The proof was nonconstructive, and indeed there were doubts as to whether specific examples of parameters with noncomputable Julia sets could be constructed. It was also unk ..."
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Cited by 4 (0 self)
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While most polynomial Julia sets are computable, it has been recently shown [12] that there exist noncomputable Julia sets. The proof was nonconstructive, and indeed there were doubts as to whether specific examples of parameters with noncomputable Julia sets could be constructed. It was also unknown whether the noncomputability proof can be extended to the filled Julia sets. In this paper we give an answer to both of these questions, which were the main open problems concerning the computability of polynomial Julia sets. We show how to construct a specific polynomial with a noncomputable Julia set. In fact, in the case of Julia sets of quadratic polynomials we give a precise characterization of Julia sets with computable parameters. Moreover, assuming a widely believed conjecture in Complex Dynamics, we give a polytime algorithm for computing a number c such that the Julia set J z 2 +cz is noncomputable. In contrast with these results, we show that the filled Julia set of a polynomial is always computable.
Computability of Julia Sets
, 2008
"... In this paper we settle most of the open questions on algorithmic computability of Julia sets. In particular, we present an algorithm for constructing quadratics whose Julia sets are uncomputable. We also show that a filled Julia set of a polynomial is always computable. ..."
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Cited by 4 (0 self)
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In this paper we settle most of the open questions on algorithmic computability of Julia sets. In particular, we present an algorithm for constructing quadratics whose Julia sets are uncomputable. We also show that a filled Julia set of a polynomial is always computable.
Rigidity Of C² Infinitely Renormalizable Quadratic Maps
"... Given C² infinitely renormalizable quadratic unimodal maps f and g with the same bounded combinatorial type, we prove that they are C 1+ff conjugate along the closure of the corresponding forward orbits of the critical points, for some ff ? 0. ..."
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Cited by 1 (1 self)
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Given C² infinitely renormalizable quadratic unimodal maps f and g with the same bounded combinatorial type, we prove that they are C 1+ff conjugate along the closure of the corresponding forward orbits of the critical points, for some ff ? 0.
LOCAL CONJUGACY CLASSES FOR ANALYTIC TORUS FLOWS
"... Abstract. If a realanalytic flow on the multidimensional torus close enough to linear has a unique rotation vector which satisfies an arithmetical condition Y, then it is analytically conjugate to linear. We show this by proving that the orbit under renormalization of a constant Y vector field attr ..."
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Cited by 1 (1 self)
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Abstract. If a realanalytic flow on the multidimensional torus close enough to linear has a unique rotation vector which satisfies an arithmetical condition Y, then it is analytically conjugate to linear. We show this by proving that the orbit under renormalization of a constant Y vector field attracts all nearby orbits with the same rotation vector. 1.
Global Hyperbolicity of Renormalization for C^r Unimodal Mappings
"... In this paper we extend M. Lyubich's recent results on the global hyperbolicity of renormalization of quadraticlike germs to the space U r of C r unimodal maps with quadratic critical point. We show that in U r the boundedtype limit sets of the renormalization operator have an invariant hype ..."
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In this paper we extend M. Lyubich's recent results on the global hyperbolicity of renormalization of quadraticlike germs to the space U r of C r unimodal maps with quadratic critical point. We show that in U r the boundedtype 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...
Cylinder renormalization for Siegel disks and a constructive Measurable Riemann Mapping Theorem
, 2006
"... The boundary of the Siegel disk of a quadratic polynomial with an irrationally indifferent fixed point with the golden mean rotation number has been observed to be selfsimilar. The geometry of this selfsimilarity is universal for a large class of holomorphic maps. A renormalization explanation of ..."
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The boundary of the Siegel disk of a quadratic polynomial with an irrationally indifferent fixed point with the golden mean rotation number has been observed to be selfsimilar. The geometry of this selfsimilarity is universal for a large class of holomorphic maps. A renormalization explanation of this universality has been proposed in the literature. However, one of the ingredients of this explanation, the hyperbolicity of renormalization, has not been proved yet. The present work considers a cylinder renormalization a novel type of renormalization for holomorphic maps with a Siegel disk which is better suited for a hyperbolicity proof. A key element of a cylinder renormalization of a holomorphic map is a conformal isomorphism of a dynamical quotient of a subset of C to a biinfinite cylinder C/Z. A construction of this conformal isomorphism is an implicit procedure which can be performed using the Measurable Riemann Mapping Theorem. We present a constructive proof of the Measurable Riemann Mapping Theorem, and obtain rigorous bounds on a numerical approximation of the desired conformal isomorphism. Such control of the uniformizing conformal coordinate is of key importance for a rigorous computerassisted study of cylinder renormalization. Renormalization for Siegel disks and Measurable Riemann Mapping Theorem 1 1.
Regularity of Conjugacies Between Critical Circle Maps: An Experimental Study
, 2001
"... We develop numerical implementations of several criteria to asses the regularity of functions. The criteria are based on finite difference method and harmonic analysis: LittlewoodPaley theory and wavelet analysis. ..."
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Cited by 1 (0 self)
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We develop numerical implementations of several criteria to asses the regularity of functions. The criteria are based on finite difference method and harmonic analysis: LittlewoodPaley theory and wavelet analysis.