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Communications in Mathematical Physics

, Volume 315, Issue 3, pp 643–697 | Cite as

An Example of a Nearly Integrable Hamiltonian System with a Trajectory Dense in a Set of Maximal Hausdorff Dimension

  • Vadim KaloshinEmail author
  • Maria Saprykina
Article

Abstract

The famous ergodic hypothesis suggests that for a typical Hamiltonian on a typical energy surface nearly all trajectories are dense. KAM theory disproves it. Ehrenfest (The Conceptual Foundations of the Statistical Approach in Mechanics. Ithaca, NY: Cornell University Press, 1959) and Birkhoff (Collected Math Papers. Vol 2, New York: Dover, pp 462–465, 1968) stated the quasi-ergodic hypothesis claiming that a typical Hamiltonian on a typical energy surface has a dense orbit. This question is wide open. Herman (Proceedings of the International Congress of Mathematicians, Vol II (Berlin, 1998). Doc Math 1998, Extra Vol II, Berlin: Int Math Union, pp 797–808, 1998) proposed to look for an example of a Hamiltonian near \({H_0(I)= \frac{\langle I, I \rangle}{2}}\) with a dense orbit on the unit energy surface. In this paper we construct a Hamiltonian \({H_0(I)+\varepsilon H_1(\theta , I , \varepsilon)}\) which has an orbit dense in a set of maximal Hausdorff dimension equal to 5 on the unit energy surface.

Keywords

Hamiltonian System Hausdorff Dimension Cohomology Class Double Resonance Rotation Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA
  2. 2.Institutionen för MatematikKTHStockholmSweden

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