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Analytic nonlinearizable uniquely ergodic diffeomorphisms on \mathbb T^{\mathbf{2}}

Published online by Cambridge University Press:  20 June 2003

MARIA SAPRYKINA
Affiliation:
Department of Mathematics, Royal Institute of Technology, S-10044, Sweden (e-mail: [email protected]) Université Paris 7, Institut de Mathematiques, ‘Géométrie et Dynamique’, 175, rue du Chevaleret, 75013, Paris, France.

Abstract

In this paper we study the behavior of diffeomorphisms, contained in the closure \overline{\mathcal{A}_\alpha} (in the inductive limit topology) of the set \mathcal{A}_\alpha of real-analytic diffeomorphisms of the torus \mathbb T^2, which are conjugated to the rotation R_\alpha:(x,y)\mapsto (x + \alpha , y) by an analytic measure-preserving transformation. We show that for a generic \alpha\in [0,1], \overline{\mathcal{A}_\alpha} contains a dense set of uniquely ergodic diffeomorphisms. We also prove that \overline{\mathcal{A}_\alpha} contains a dense set of diffeomorphisms that are minimal and non-ergodic.

Type
Research Article
Copyright
2003 Cambridge University Press

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