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


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.


Hamiltonian System Hausdorff Dimension Cohomology Class Double Resonance Rotation Vector 
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© Springer-Verlag 2012

Authors and Affiliations

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

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