Number theory and dynamical systems / edited by M.M. Dodson, J.A.G. Vickers.

Contributions from a meeting held at the University of York, March 30-April 15, 1987.

Includes bibliographical references.

Print version record.

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.

Cover; Title; Copyright; Contents; Contributors; Introduction; References; 1 Non-degeneracy in the perturbation theory of integrable dynamical systems; 1. The Problem; 2. A general non-degeneracy condition; 3. Formulation of the existence theorem; 4. Lower dimensional invariant tori; 5. The twist mapping theorem; References; 2 Infinite dimensional inverse function theorems and small divisors; 1. Introduction.; 2. Stability under Group Actions; 3. Linearisation and Newton's tangent method.; 4. The infinite dimensional case: finite orderand G-stability

5. Finite order, small divisors and exceptionalsets6. Coflnite G-stability; 7. Normal forms and Siegel's Theorem; 8. References; 3 Metric Diophantine approximation of quadratic forms; 1. Introduction; References; 4 Symbolic dynamics and Diophantine equations; 1. The problems; 2. The Proofs; References; 5 On badly approximable numbers, Schmidt games and bounded orbits of flows; 1. Introduction; 2. Bounded geodesies and horocycles; 3. Anosov flows; 4. Flows on SL(n, R)/SL(n, Z); 5. The Schmidt game; 6, Examples of winning sets in Rn; 7, Back to bounded geodesies

8. Comments on the proofs of other results9. Bounded orbits and simultaneous Diophantine approximation; 10. Miscellaneous comments and questions; 11. References; 6 Estimates for Fourier coefficients of cusp forms; 1. Introduction; 2. Estimation of Satake parameters; 3. Modified Rankin-Selberg method; 4. References; 7 The integral geometry of fractals; 1. Fractals; 2. Integral Geometry; 3. Towards Inequality A; 4. Towards inequality B; 5. Applications to Brownian Motion; 6. Sets with large intersection; 7. References; 8 Geometry of algebraiccontinued fractals; 1. Introduction

2. Quadratic continued fractals. 3. Applications to dynamical systems.; 4. Appendix; 5. References; 9 Chaos implies confusion; 1. A dynamical system and transcendental numbers; 2. Dragon curves; 3. The dimension of a planar curve [2], [10]; 4. Resolvable curves; 5. Geometric probability; 6. Entropy of a finite curve [4], [7], [8].; 7. Thermodynamics [4], [7].; 8. Entropy of unbounded curves; 9. Entropy and dimension; 10. References; 10 The Riemann hypothesis and the Hamiltonian of a quantum mechanical system; 1. Introduction

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