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

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.