Download e-book for iPad: Communications in Mathematical Physics - Volume 194 by A. Jaffe (Chief Editor)

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We show that for almost all α (in the sense of measure theory), the pair correlation of this sequence is Poissonian. In the quadratic case d = 2, this implies a similar result for the energy levels of the “boxed oscillator” in the high-energy limit. This is a simple integrable system in 2 degrees of freedom studied by Berry and Tabor as an example for their conjecture that the energy levels of generic completely integrable systems have Poisson spacing statistics. 1. Introduction Hermann Weyl [11] proved that for an integer d ≥ 1 and an irrational α, the sequence of fractional parts αnd mod 1, n = 1, 2, .

98, 268–302 (1967) Communicated by D. C. Brydges Commun. Math. Phys. 194, 47 – 60 (1998) Communications in Mathematical Physics © Springer-Verlag 1998 Conformally Symplectic Dynamics and Symmetry of the Lyapunov Spectrum Maciej P. Wojtkowski1 , Carlangelo Liverani2 1 Department of Mathematics, University of Arizona, Tuscon, Arizona 85721, USA. edu 2 Mathematics Department, University of Rome “Tor Vergata”, 00133 Rome, Italy. it Received: 18 March 1997 / Accepted: 5 June 1997 Abstract: A generalization of the Hamiltonian formalism is studied and the symmetry of the Lyapunov spectrum established for the resulting systems.

E. 23). The hypothesis for singular spectrum is precisely β < 21 . Since β < 21 , u1 L ∼ L−β L1/2 while u2 L ∼ Lβ L1/2 , where a ∼ b is shorthand for lim log(a) log(b) = 1. 26) holds in the strong sense that the ratio lies in the interval (5 − √ 24 )). 26) says that |m(E + i )| ∼ −2β √ 24 , 5 + . Since β is continuous, the theory in [3] then says that the local dimension is given by 1 − 2β as claimed. 9. Random Decaying Potentials: The Continuum Case Having done the discrete random case, we will only sketch the continuum case.

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