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

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.

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*Journal Article*|

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*English*

Author

Kaloshin, Vadim

^{ISTA}^{}; Saprykina, MariaAbstract

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

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### Cite this

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-xKaloshin, 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*. Springer Nature. https://doi.org/10.1007/s00220-012-1532-xKaloshin, 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*. Springer Nature, 2012. 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. Springer Nature, pp. 643–697, 2012.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.