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

article An example of a nearly integrable Hamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension published yes Vadim Kaloshin author FE553552-CDE8-11E9-B324-C0EBE56974250000-0002-6051-2628 Maria Saprykina author 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. Springer Nature2012 eng Mathematical PhysicsStatistical and Nonlinear Physics Communications in Mathematical Physics 0010-3616 1432-091610.1007/s00220-012-1532-x 3153643-697 yes 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 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:<a href="https://doi.org/10.1007/s00220-012-1532-x">10.1007/s00220-012-1532-x</a> 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. Springer Nature, 2012. <a href="https://doi.org/10.1007/s00220-012-1532-x">https://doi.org/10.1007/s00220-012-1532-x</a>. 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:<a href="https://doi.org/10.1007/s00220-012-1532-x">10.1007/s00220-012-1532-x</a>. V. Kaloshin, M. Saprykina, Communications in Mathematical Physics 315 (2012) 643–697. 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, 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. <a href="https://doi.org/10.1007/s00220-012-1532-x">https://doi.org/10.1007/s00220-012-1532-x</a> 85022020-09-18T10:47:16Z2021-01-12T08:19:44Z