Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T15:37:35.319Z Has data issue: false hasContentIssue false
  • English
  • Français

Topological weak mixing and diffusion at all times for a class of Hamiltonian systems

Part of: Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems

Published online by Cambridge University Press:  25 May 2021

BASSAM FAYAD  and
MARIA SAPRYKINA
BASSAM FAYAD
Affiliation:
IMJ-PRG CNRS, UP7D, 58–56, avenue de France, Boite Courrier 7012, 75205Paris Cedex 13, France (e-mail: bassam.fayad@imj-prg.fr)
MARIA SAPRYKINA*
Affiliation:
Department of Mathematics, KTH, Royal Institute of Technology, SE-100 44Stockholm, Sweden
*
e-mail: masha@kth.se
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present examples of nearly integrable analytic Hamiltonian systems with several strong diffusion properties: topological weak mixing and diffusion at all times. These examples are obtained by AbC constructions with several frequencies.

Keywords

Hamiltonian systemsdiffusiontopological weak mixingAnosov–Katok methodAbC method

MSC classification

Primary: 37J40: Perturbations, normal forms, small divisors, KAM theory, Arnol'd diffusion
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 2: Anatole Katok Memorial Issue Part 1: Special Issue of Ergodic Theory and Dynamical Systems , February 2022 , pp. 777 - 791
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Footnotes

To our great friend and mentor Anatoly Katok

References

Arnold, V. I.. Mathematical problems in classical physics. Trends and Perspectives in Applied Mathematics (Applied Mathematical Sciences, 100). Springer, New York, 1994, pp. 1–20.Google Scholar
Anosov, D. V. and Katok, A. B.. New examples in smooth ergodic theory. Ergodic diffeomorphisms. Trans. Moscow Math. Soc. 23 (1970), 1–35.Google Scholar
Bernard, P., Kaloshin, V. and Zhang, K.. Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders. Acta Math. 217(1) (2016), 1–79.CrossRefGoogle Scholar
Eliasson, H., Fayad, B. and Krikorian, R.. On the stability of KAM tori. Duke Math. J. 164(9) (2015), 1733–1775.CrossRefGoogle Scholar
Fayad, B.. Analytic mixing reparametrizations of irrational flows ETDS. Ergod. Th. & Dynam. Sys. 22(2) (2002), 437–468.CrossRefGoogle Scholar
Farré, G. and Fayad, B.. Instabilities for analytic quasi-periodic invariant tori. Preprint, 2019, arXiv:1912.01575. J. Eur. Math. Soc., to appear.Google Scholar
Fayad, B. and Saprykina, M.. Isolated elliptic fixed points for smooth Hamiltonians. Modern Theory of Dynamical Systems (Contemporary Mathematics, 692). American Mathematical Society, Providence, RI, 2017, pp. 67–82.CrossRefGoogle Scholar
Herman, M.. Some Open Problems in Dynamical Systems (Proc. Int. Congress of Mathematicians, Vol. II, Berlin, 1998). Doc. Math. Extra Vol II (1998), 797–808.Google Scholar
Halmos, P.. Lectures on Ergodic Theory. Chelsea Publishing, New York, 1956.Google Scholar
Katok, A. B.. Spectral properties of dynamical systems with an integral invariant on the torus. Funct. Anal. Appl. 1 (1967), 296–305.CrossRefGoogle Scholar
Katok, A. B.. Ergodic perturbations of degenerate integrable Hamiltonian systems. Math. USSR Izv. 7(3) (1973), 535–571.CrossRefGoogle Scholar
Kochergin, A. V.. On the absence of mixing in special flows over the rotation of a circle and in flows on a two-dimensional torus. Dokl. Akad. Nauk 205(3) (1972), 515–518.Google Scholar
Kaloshin, V., Zhang, K. and Zheng, Y.. Almost dense orbit on energy surface. Proc. XVIth Int. Congress on Mathematical Physics (Prague, Czech Republic, 3–8 August 2009). Ed. Exner, P.. World Scientific Publishing Co., Hackensack, NJ, 2010, pp. 314–322.CrossRefGoogle Scholar
Yoccoz, J.-C.. Petits diviseurs en dimension 1 (Astérisque, 231). Société mathématique de France, Paris, 1995.Google Scholar