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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 28 (11 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 23 (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 12 (9 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 8 (2 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.
Portfolio Optimisation of Mediumsize Stochastic Systems with Nonlinear Constraints: Application to Equity and Bond markets
"... This paper presents results of multistage numerical optimisation of medium sized portfolios to both bond and equity markets using High Performance Computing. Constraints applied are both linear and nonlinear and the optimisation process is designed to handle multiple markets in multiple countries, ..."
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This paper presents results of multistage numerical optimisation of medium sized portfolios to both bond and equity markets using High Performance Computing. Constraints applied are both linear and nonlinear and the optimisation process is designed to handle multiple markets in multiple countries, producing a series of alternative scenarios dependent upon a number of riskreturn requirements. Portfolios are allowed to contain instruments for which stochastic or historic data (or some combination) is available. Comparative performance for both real and artificial data sets is discussed and extrapolation to very large datasets will be presented. The comparative benefits of the deployment of large scale High Performance Computing in this class of problem will be made. Some discussion of the problem of portfolio stability in the context of this model is also included.
Physics © SpringerVerlag 1989 Complex Universality
"... Abstract. The theory of period doublings for oneparameter families of iterated real mappings is generalized to period ntuplings for complex mappings. An ntupling occurs when the eigenvalue of a stable periodic orbit passes through the value ω = exp(2τπm/n) as the parameter value is changed. Each ..."
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Abstract. The theory of period doublings for oneparameter families of iterated real mappings is generalized to period ntuplings for complex mappings. An ntupling occurs when the eigenvalue of a stable periodic orbit passes through the value ω = exp(2τπm/n) as the parameter value is changed. Each choice of m defines a different sequence of rctuplings, for which we construct a period fttupling renormalization operator with a universal fixpoint function, a universal unstable manifold and universal scaling numbers. These scaling numbers can be organized by Farey trees. The present paper gives a general description and numerical support for the universality conjectured above.