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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 27 (10 self)
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Consider the group generated by reflections in a finite collection of disjoint
Rigidity of critical circle mappings I
 J. Eur. Math. Soc. (JEMS
"... Abstract. Let f be a smooth homeomorphism of the circle having one cubicexponent critical point and irrational rotation number of bounded combinatorial type. Using certain pullback and quasiconformal surgery techniques, we prove that the scaling ratios of f about the critical point are asymptotic ..."
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Cited by 24 (3 self)
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Abstract. Let f be a smooth homeomorphism of the circle having one cubicexponent critical point and irrational rotation number of bounded combinatorial type. Using certain pullback and quasiconformal surgery techniques, we prove that the scaling ratios of f about the critical point are asymptotically independent of f. This settles in particular the golden mean universality conjecture. We introduce the notion of holomorphic commuting pair, a complex dynamical system that, in the analytic case, represents an extension of f to the complex plane and behaves somewhat as a quadraticlike mapping. We define a suitable renormalization operator that acts on such objects. Through careful analysis of the family of entire mappings given by z ↦ → z+θ − 1 sin 2πz, θ real, we construct examples of holomorphic commuting 2π pairs, from which certain necessary limit set prerigidity results are extracted. The rigidity problem for f is thereby reduced to one of renormalization convergence. We handle this last problem by means of Teichmüller extremal methods made available through the recent work of Sullivan on Riemann surface laminations and renormalization of unimodal mappings.
Autonomous Agents, AI and Chaos Theory
"... Agent theory in AI and related disciplines deals with the structure and behaviour of autonomous, intelligent systems, capable of adaptive action to pursue their interests. In this paper it is proposed that a natural reinterpretation of agenttheoretic intentional concepts like knowing, wanting, liki ..."
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Cited by 12 (1 self)
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Agent theory in AI and related disciplines deals with the structure and behaviour of autonomous, intelligent systems, capable of adaptive action to pursue their interests. In this paper it is proposed that a natural reinterpretation of agenttheoretic intentional concepts like knowing, wanting, liking, etc., can be found in process dynamics. This reinterpretation of agent theory serves two purposes. On the one hand we gain a well established mathematical theory which can be used as the formal mathematical interpretation (semantics) of the abstract agent theory. On the other hand, since process dynamics is a theory that can also be applied to physical systems of various kinds, we gain an implementation route for the construction of artificial agents as bundles of processes in machines. The paper is intended as a basis for dialogue with workers in dynamics, AI, ethology and cognitive science. 1 Introduction Agent theory is a branch of artificial intelligence (Kiss, 1988). Its domain is...
Rational maps and Kleinian groups
 In Proceedings of the International Congress of Mathematicians Kyoto
, 1990
"... this paper we will survey three chapters of this developing theory, and the Riemann surface techniques they employ: ..."
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Cited by 11 (8 self)
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this paper we will survey three chapters of this developing theory, and the Riemann surface techniques they employ:
Frontiers in complex dynamics
 Bull. of Amer. Math. Soc
"... Rational maps on the Riemann sphere occupy a distinguished niche in the general theory of smooth dynamical systems. First, rational maps are complexanalytic, so a broad spectrum of techniques can contribute to their study (quasiconformal mappings, potential theory, algebraic geometry, etc.). The ra ..."
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Cited by 7 (1 self)
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Rational maps on the Riemann sphere occupy a distinguished niche in the general theory of smooth dynamical systems. First, rational maps are complexanalytic, so a broad spectrum of techniques can contribute to their study (quasiconformal mappings, potential theory, algebraic geometry, etc.). The rational maps of a given
Causes and effects of chaos
 MIT Artificial Intelligence Lab
, 1990
"... Most of the recent literature on chaos and nonlinear dynamics is written either for popular science magazine readers or for advanced mathematicians. This paper gives a broad introduction to this interesting and rapidly growing field at a level that is between the two. The graphical and analytical to ..."
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Cited by 4 (3 self)
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Most of the recent literature on chaos and nonlinear dynamics is written either for popular science magazine readers or for advanced mathematicians. This paper gives a broad introduction to this interesting and rapidly growing field at a level that is between the two. The graphical and analytical tools used 'in the literature are explained and demonstrated, the rudiments of the current theory are outlined and that theory is discussed in the context of several examples: an electronic crcuit a chemical reaction and a system of satellites in the solar system.
Random Logistic Maps and Lyapunov Exponents
"... . We prove that under certain basic regularity conditions, a random iteration of logistic maps converges to a random point attractor when the Lyapunov exponent is negative, and does not converge to a point when the Lyapunov exponent is positive. 1. ..."
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Cited by 3 (0 self)
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. We prove that under certain basic regularity conditions, a random iteration of logistic maps converges to a random point attractor when the Lyapunov exponent is negative, and does not converge to a point when the Lyapunov exponent is positive. 1.
Contents
, 2005
"... Abstract. In this paper geometric properties of infinitely renormalizable real Hénonlike maps F in R 2 are studied. It is shown that the appropriately defined renormalizations R n F converge exponentially to the onedimensional renormalization fixed point. The convergence to onedimensional systems ..."
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Abstract. In this paper geometric properties of infinitely renormalizable real Hénonlike maps F in R 2 are studied. It is shown that the appropriately defined renormalizations R n F converge exponentially to the onedimensional renormalization fixed point. The convergence to onedimensional systems is at a superexponential rate controlled by the average Jacobian and a universal function a(x). It is also shown that the attracting Cantor set of such a map has Hausdorff dimension less than 1, but contrary to the onedimensional intuition, it is not rigid, does not lie on a smooth curve, and generically has unbounded geometry.
Nonextensive Random Matrix Theory A Bridge Connecting Chaotic and Regular Dynamics
, 2008
"... We consider a possible generalization of the random matrix theory, which involves the maximization of Tsallis ’ qparametrized entropy. We discuss the dependence of the spacing distribution on q using a nonextensive generalization of Wigner’s surmises for ensembles belonging to the orthogonal, unita ..."
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We consider a possible generalization of the random matrix theory, which involves the maximization of Tsallis ’ qparametrized entropy. We discuss the dependence of the spacing distribution on q using a nonextensive generalization of Wigner’s surmises for ensembles belonging to the orthogonal, unitary and symplectic symmetry universal classes. PACS numbers: 03.65.w, 05.45.Mt, 05.30.Ch 1