Number Theory and Dynamical Systems
M. M. Dodson, J. A. G. Vickers, N. J. Hitchin
Cambridge University Press, Nov 9, 1989 - Mathematics - 172 pages
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.
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algebraic Anosov ﬂows badly approximable numbers bounded orbits Cantor set coefficients Con(N condition consider constant contains continued fractal continued fractions converges corresponding cusp form Dani deﬁned deﬁnition denote differential equation dimensional Diophantine approximation Dragon curves dynamical systems eigenvalues element endpoint entropy Euclidean exists ﬁnd ﬁnite order ﬁrst ﬁxed point Fourier fractal fundamental region G-stability geodesic geodesic ray geometry given Hamiltonian Hausdorff dimension hence holomorphic horocycle hyperbolic incompressible subset inﬁnite inﬁnitesimally inner product integer intersection invariant tori iteration lattice Lemma line segment linear manifolds Math Mathematical matrix measure Mendes France metric neighbourhood non-degenerate normal form notation Number Theory obtain perturbation Poincaré map positive problem proof Proposition prove quadratic form quadratic irrational radius Re(p real number result Riemann Hypothesis Riemannian satisﬁes set of points Siegel simple geodesic small divisors space subgroup Suppose symbolic dynamics Theorem tion vector ﬁeld veriﬁed winning set zero