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Elliptic islands in generalized Sinai billiards

Published online by Cambridge University Press:  14 October 2010

Victor J. Donnay
Victor J. Donnay
Affiliation:
Department of Mathematics, Bryn Mawr College, Bryn Mawr, Pa, 19010, USA
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Abstract

We find necessary conditions for a generalized Sinai billiard to be ergodic, independent of the geometry of the disks; i.e. if the conditions are not satisfied, then we can always construct special geometries for which the system is not ergodic. We prove non-ergodicity by constructing an elliptic periodic orbit and showing that this orbit satisfies the non-degeneracy condition of KAM theory. By a combination of theory and numerical calculation involving a computer algebra system (Mathematica), we show that the first Birkhoff invariant is non-zero. We apply this result to produce non-ergodic Hamiltonian flows that are generated by smooth potentials of finite range (repelling potentials and Lennard–Jones potentials) and by geodesic flows on the torus.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 5 , October 1996 , pp. 975 - 1010
Copyright
Copyright © Cambridge University Press 1996

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