Download Dynamical Systems and Small Divisors: Lectures given at the by Hakan Eliasson, Sergei Kuksin, Stefano Marmi, PDF

By Hakan Eliasson, Sergei Kuksin, Stefano Marmi, Jean-Christophe Yoccoz

Many difficulties of balance within the concept of dynamical structures face the trouble of small divisors. the main well-known instance is maybe given through Kolmogorov-Arnold-Moser conception within the context of Hamiltonian structures, with many purposes to physics and astronomy. different typical small divisor difficulties come up contemplating circle diffeomorphisms or quasiperiodic Schroedinger operators. during this quantity Hakan Eliasson, Sergei Kuksin and Jean-Christophe Yoccoz illustrate the newest advancements of this conception either in finite and limitless measurement. an inventory of open difficulties (including a few difficulties contributed by means of John Mather and Michel Herman) has been integrated.

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Extra resources for Dynamical Systems and Small Divisors: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 13-20, 1998

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Let now F = Q-l FQ. Then F will be smooth on the same partition as jj and satisfy the same estimate but with the first factor 13 replaced by €. The equation becomes jjg -a-b -b-b_ -b -b DaKa - KaDb - Fa - Ga' jjg lf any two eigenvalues of jj~ and respectively differ by ::; p, then the equation reduces to -b -b-b Ka=O and Ga=Fa, and if at least two eigenvalues differ by > p, then the equation becomes -a-b -b-b -b DaKa - KaDb = Fa -b and G a = O. The estimate of G is clear so we only need to estimate i< from the second equation.

For some K, II nw II 2': r;F \:In E Z \ T o. 4) Theorem 13. 1) has a solution uk E l2(Z) with E = Eoo(8 + kw). The set {uk: k E Z} is an orthogonal basis for l2(Z). 1) as an operator He, then the statement says that He is pure point for almost all O. The spectrum of He is the closure of Range(Eoo) and it is easy to verify that the Lebesgue measure of [inf V, sup Vl \ Range( Eoo) goes to 0 when e goes to o. 1) with a matrix D matrix D is diagonal and D E N:F(a + eF as in the introduction. The = 1, {3,"(,>" = 1,J,L = I,ll = 1,p = 1).

1) where V is a real valued function on the one-dimensional torus T = lR/(27rZ) and w is a real number. We assume that V is piecewise Gevrey smooth on a partition with p many pieces and \:Ik 2': The functions V (8 conditions { o. 2) + y) - V (8)) satisfies for all 8, y the two transversality maxo

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