An example of a nearly integrable Hamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension

V. Kaloshin, M. Saprykina, Communications in Mathematical Physics 315 (2012) 643–697.

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Journal Article | Published | English
Author
Kaloshin, VadimIST Austria; Saprykina, Maria
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 H0(I)=⟨I,I⟩2 with a dense orbit on the unit energy surface. In this paper we construct a Hamiltonian H0(I)+εH1(θ,I,ε) which has an orbit dense in a set of maximal Hausdorff dimension equal to 5 on the unit energy surface.
Publishing Year
Date Published
2012-11-01
Journal Title
Communications in Mathematical Physics
Volume
315
Issue
3
Page
643-697
IST-REx-ID

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Kaloshin V, Saprykina M. An example of a nearly integrable Hamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension. Communications in Mathematical Physics. 2012;315(3):643-697. doi:10.1007/s00220-012-1532-x
Kaloshin, V., & Saprykina, M. (2012). An example of a nearly integrable Hamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension. Communications in Mathematical Physics, 315(3), 643–697. https://doi.org/10.1007/s00220-012-1532-x
Kaloshin, Vadim, and Maria Saprykina. “An Example of a Nearly Integrable Hamiltonian System with a Trajectory Dense in a Set of Maximal Hausdorff Dimension.” Communications in Mathematical Physics 315, no. 3 (2012): 643–97. https://doi.org/10.1007/s00220-012-1532-x.
V. Kaloshin and M. Saprykina, “An example of a nearly integrable Hamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension,” Communications in Mathematical Physics, vol. 315, no. 3, pp. 643–697, 2012.
Kaloshin V, Saprykina M. 2012. An example of a nearly integrable Hamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension. Communications in Mathematical Physics. 315(3), 643–697.
Kaloshin, Vadim, and Maria Saprykina. “An Example of a Nearly Integrable Hamiltonian System with a Trajectory Dense in a Set of Maximal Hausdorff Dimension.” Communications in Mathematical Physics, vol. 315, no. 3, Springer Nature, 2012, pp. 643–97, doi:10.1007/s00220-012-1532-x.

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