An Example of a Nearly Integrable Hamiltonian System with a Trajectory Dense in a Set of Maximal Hausdorff Dimension
2012 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 315, no 3, 643-697 p.Article in journal (Refereed) Published
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 with a dense orbit on the unit energy surface. In this paper we construct a Hamiltonian which has an orbit dense in a set of maximal Hausdorff dimension equal to 5 on the unit energy surface.
Place, publisher, year, edition, pages
2012. Vol. 315, no 3, 643-697 p.
Arnold Diffusion, Lagrangian Systems, Instability, Stability, Points
Mathematics Physical Sciences
IdentifiersURN: urn:nbn:se:kth:diva-104994DOI: 10.1007/s00220-012-1532-xISI: 000309718600003ScopusID: 2-s2.0-84867441458OAI: oai:DiVA.org:kth-104994DiVA: diva2:570060
QC 201211162012-11-162012-11-152012-11-16Bibliographically approved