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References (21)
-
M. Misiurewicz, K. Ziemian (1991)
Rotation sets and ergodic measures for torus homeomorphismsFundamenta Mathematicae, 137
-
V. I. Arnold (1965)
Small Denominators: 1. On Mappings of a Circle onto ItselfAmer. Math. Soc. Transl., Ser. 2, 46
-
A. Katok, B. Hasselblatt (1995)
10.1017/CBO9780511809187Introduction to the Modern Theory of Dynamical Systems
-
R. Llave (2003)
A Tutorial on Kam Theory -
M. Misiurewicz, K. Ziemian (1989)
Rotation Sets for Maps of ToriJournal of The London Mathematical Society-second Series
-
Danijela Damjanović, A. Katok (2010)
Local rigidity of partially hyperbolic actions I. KAM method and ${\mathbb Z^k}$ actions on the torusAnnals of Mathematics, 172
-
E. Zehnder (1976)
Generalized implicit function theorems with applications to some small divisor problems, ICommunications on Pure and Applied Mathematics, 28
-
F. Hertz (2002)
Stable ergodicity of certain linear automorphisms of the torusAnnals of Mathematics, 162
-
D. Kleinbock (2012)
Ergodic Theory on Homogeneous Spaces and Metric Number TheoryMathematics of Complexity and Dynamical Systems
-
J. Yoccoz (1984)
Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienneAnnales Scientifiques De L Ecole Normale Superieure, 17
-
N. Karaliolios (2018)
Local Rigidity of Diophantine Translations in Higher-Dimensional ToriRegul. Chaotic Dyn., 23
-
J. Moser (1990)
On Commuting Circle Mappings and Simultaneous Diophantine ApproximationsMath. Z., 205
-
D. Kleinbock (2009)
Ergodic Theory on Homogeneous Spaces and Metric Number Theory -
R. S. Hamilton (1982)
65Bull. Amer. Math. Soc. (N. S.), 7
-
V. I. Arnold (1965)
213Amer. Math. Soc. Transl., Ser. 2, 46
-
V. Lazutkin (1993)
Kam Theory and Semiclassical Approximations to Eigenfunctions -
Salvatore Cosentino, L. Flaminio (2015)
Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifoldsJournal of Modern Dynamics, 9
-
D. Castelblanco (2015)
Restrictions on rotation sets for commuting torus homeomorphismsarXiv: Dynamical Systems
-
M. R. Herman (1979)
Sur la conjugaison différentiable des difféomorphismes du cercle à des rotationsInst. Hautes Études Sci. Publ. Math., 49
-
Danijela Damjanović, B. Fayad (2019)
On local rigidity of partially hyperbolic affine ℤk actionsJournal für die reine und angewandte Mathematik (Crelles Journal)
-
B. Fayad, K. Khanin (2006)
Smooth linearization of commuting circle diffeomorphismsAnnals of Mathematics, 170
- Publisher
- Springer Journals
- Copyright
- Copyright © Pleiades Publishing, Ltd. 2021
- ISSN
- 1560-3547
- eISSN
- 1468-4845
- DOI
- 10.1134/s1560354721060083
- Publisher site
- See Article on Publisher Site
Abstract
We generalize results of Moser [17] on the circle to \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{T}^{d}$$\end{document}: we show that a smooth sufficiently small perturbation of a \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{Z}^{m}$$\end{document} action, \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$m\geqslant 2$$\end{document}, on the torus \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{T}^{d}$$\end{document} by simultaneously Diophantine translations, is smoothly conjugate to the unperturbed action under a natural condition on the rotation sets of diffeomorphisms isotopic to identity and we answer the question Moser posed in [17] by proving the existence of a continuum of \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$m$$\end{document}-tuples of simultaneously Diophantine vectors such that every element of the induced \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{Z}^{m}$$\end{document} action is Liouville.
Journal
Regular and Chaotic Dynamics – Springer Journals
Published: Nov 1, 2021
Keywords: KAM theory; simultaneously Diophantine translations; local rigidity; simultaneously Diophantine approximations