Applications and optimality of filling rates for KAM-like flow on the torus
by
H. Scott Dumas
Department of Mathematical Sciences, University of Cincinnati

Given a unit vector \omega in Rⁿ, consider the Kronecker flow on Tⁿ with direction (or frequency) \omega (i.e., rectilinear flow winding around the flat n-torus with unit speed in the direction \omega). If \omega is nonresonant (i.e., k in Zⁿ & k·\omega = 0  ===>  k=0), then orbits of the flow fill Tⁿ densely. In Hamiltonian systems, interest focuses on ``KAM-like flow, '' or Kronecker flow with frequency that is not only nonresonant, but ``highly nonresonant, '' satisfying Diophantine conditions. KAM-like flow on Tⁿ not only fills Tⁿ densely, but does so quickly, and in many applications it is important to know how quickly filling occurs. In dimension n=2, it's not hard to get a good estimate of the filling rate using continued fractions, but this method apparently does not generalize to higher dimensions n >= 3. Solution of the higher dimensional problem is interesting because it involves ideas from several areas of mathematics (dynamical systems, combinatorics, harmonic analysis), and because it has a number of applications. New interest in these applications has recently led to higher dimensional estimates that are probably optimal. As time permits, I will briefly discuss these estimates, their optimality, and applications (to particle-solid interactions, Lorentz gas-billiard problems, Aubry-Mather theory, and Arnold diffusion in nearly integrable Hamiltonian systems).

Date received: May 1, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calh-07.