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This volume contains selected contributions from a very successful meeting on Number Theory and Dynamical Systems held at the University of York in 1987. There are close and surprising connections between number theory and dynamical systems. One emerged last century from the study of the stability of the solar system where problems of small divisors associated with the near resonance of planetary frequencies arose. Previously the question of the stability of the solar system was answered in more general terms by the celebrated KAM theorem, in which the relationship between near resonance (and so Diophantine approximation) and stability is of central importance. Other examples of the connections involve the work of Szemeredi and Furstenberg, and Sprindzuk. As well as containing results on the relationship between number theory and dynamical systems, the book also includes some more speculative and exploratory work which should stimulate interest in different approaches to old problems.

Number theory --- Differentiable dynamical systems

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Differential geometry. Global analysis --- Differentiable dynamical systems --- Bifurcation theory --- 517.987 --- 531.3 --- Dynamica --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Differential equations, Nonlinear --- Stability --- Measures. Representations of Boolean algebras. Metric theory of dynamic systems --- Numerical solutions --- 517.987 Measures. Representations of Boolean algebras. Metric theory of dynamic systems --- Differentiable dynamical systems.

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This book, based on lectures given at the Accademia dei Lincei, is an accessible and leisurely account of systems that display a chaotic time evolution. This behaviour, though deterministic, has features more characteristic of stochastic systems. The analysis here is based on a statistical technique known as time series analysis and so avoids complex mathematics, yet provides a good understanding of the fundamentals. Professor Ruelle is one of the world's authorities on chaos and dynamical systems and his account here will be welcomed by scientists in physics, engineering, biology, chemistry and economics who encounter nonlinear systems in their research.

Differentiable dynamical systems. --- Ergodic theory. --- Chaotic behavior in systems. --- Attractors (Mathematics)

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Classical mechanics. --- Dynamical systems. --- Equations of motion. --- Finite element method. --- Hamiltonian functions. --- Optimal control. --- Time marching. --- Trajectory optimization. --- Variational principles.

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Group theory --- Partial differential equations --- Semigroups of operators --- Differential equations, Partial --- Nonlinear operators --- Differentiable dynamical systems --- Semi-groupes d'opérateurs --- Equations aux dérivées partielles --- Opérateurs non linéaires --- Dynamique différentiable --- Congresses --- Congrès --- 51 --- Mathematics --- 51 Mathematics --- Semi-groupes d'opérateurs --- Equations aux dérivées partielles --- Opérateurs non linéaires --- Dynamique différentiable --- Congrès