---
_id: '8502'
abstract:
- lang: eng
  text: '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.'
article_processing_charge: No
article_type: original
author:
- first_name: Vadim
  full_name: Kaloshin, Vadim
  id: FE553552-CDE8-11E9-B324-C0EBE5697425
  last_name: Kaloshin
  orcid: 0000-0002-6051-2628
- first_name: Maria
  full_name: Saprykina, Maria
  last_name: Saprykina
citation:
  ama: Kaloshin V, Saprykina M. An example of a nearly integrable Hamiltonian system
    with a trajectory dense in a set of maximal Hausdorff dimension. <i>Communications
    in Mathematical Physics</i>. 2012;315(3):643-697. doi:<a href="https://doi.org/10.1007/s00220-012-1532-x">10.1007/s00220-012-1532-x</a>
  apa: Kaloshin, V., &#38; Saprykina, M. (2012). An example of a nearly integrable
    Hamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension.
    <i>Communications in Mathematical Physics</i>. Springer Nature. <a href="https://doi.org/10.1007/s00220-012-1532-x">https://doi.org/10.1007/s00220-012-1532-x</a>
  chicago: Kaloshin, Vadim, and Maria Saprykina. “An Example of a Nearly Integrable
    Hamiltonian System with a Trajectory Dense in a Set of Maximal Hausdorff Dimension.”
    <i>Communications in Mathematical Physics</i>. Springer Nature, 2012. <a href="https://doi.org/10.1007/s00220-012-1532-x">https://doi.org/10.1007/s00220-012-1532-x</a>.
  ieee: V. Kaloshin and M. Saprykina, “An example of a nearly integrable Hamiltonian
    system with a trajectory dense in a set of maximal Hausdorff dimension,” <i>Communications
    in Mathematical Physics</i>, vol. 315, no. 3. Springer Nature, pp. 643–697, 2012.
  ista: 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.
  mla: Kaloshin, Vadim, and Maria Saprykina. “An Example of a Nearly Integrable Hamiltonian
    System with a Trajectory Dense in a Set of Maximal Hausdorff Dimension.” <i>Communications
    in Mathematical Physics</i>, vol. 315, no. 3, Springer Nature, 2012, pp. 643–97,
    doi:<a href="https://doi.org/10.1007/s00220-012-1532-x">10.1007/s00220-012-1532-x</a>.
  short: V. Kaloshin, M. Saprykina, Communications in Mathematical Physics 315 (2012)
    643–697.
date_created: 2020-09-18T10:47:16Z
date_published: 2012-11-01T00:00:00Z
date_updated: 2021-01-12T08:19:44Z
day: '01'
doi: 10.1007/s00220-012-1532-x
extern: '1'
fulldoi: https://doi.org/10.1007/s00220-012-1532-x
intvolume: '       315'
issue: '3'
keyword:
- Mathematical Physics
- Statistical and Nonlinear Physics
language:
- iso: eng
month: '11'
oa_version: None
page: 643-697
publication: Communications in Mathematical Physics
publication_identifier:
  issn:
  - 0010-3616
  - 1432-0916
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
status: public
title: An example of a nearly integrable Hamiltonian system with a trajectory dense
  in a set of maximal Hausdorff dimension
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 315
year: '2012'
...