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Estimates of the conjugacy to rotations of circle diffeomorphisms. Successives conjugacies and smooth realizations

Abstract : This thesis deals with some questions on differentiable dynamical systems. It comprises two relatively independent parts, with two chapters each. The first part deals with estimates of the conjugacy to rotations of circle diffeomorphisms and their applications. The second part deals with successive conjugacies and the smooth realization problem. The first chapter is based on a celebrated theorem by Herman and Yoccoz, which asserts that if the rotation number $\alpha$ of a $C^\infty$-diffeomorphism of the circle $f$ satisfies a Diophantine condition, then $f$ is $C^\infty$-conjugated to a rotation. We establish explicit relationships between the $C^k$ norms of this conjugacy and the Diophantine condition on $\alpha$. To obtain these estimates, we follow a suitably modified version of Yoccoz's proof. In the second chapter, we use some of these estimates to show two related results. The first is on quasi-reducibility: for a Baire-dense set of $\alpha$, for any diffeomorphism $f$ of rotation number $\alpha$, it is possible to accumulate $R_\alpha$ with a sequence $h_n f h_n^{-1}$, $h_n$ being a diffeomorphism. The second result of this chapter is: for a Baire-dense set of $\alpha$, given two commuting diffeomorphisms $f$ and $g$, such that $f$ has $\alpha$ for rotation number, it is possible to approach each of them by commuting diffeomorphisms $f_n$ and $g_n$ that are differentiably conjugated to rotations. The third chapter deals with the problem of non-standard smooth realization of translations of the torus. On some manifolds admitting a circle action, we construct volume-preserving diffeomorphisms that are metrically isomorphic to ergodic translations on the torus, where one given coordinate of the translation is an arbitrary Liouville number. To obtain this result, we determine sufficient conditions on translation vectors of the torus that allow to explicitly construct the sequence of successive conjugacies in Anosov-Katok's method, with suitable estimates of their norm. %we introduce and study the problem of non-standard couples of angles. W In the fourth chapter, on the same manifolds as previously, we show that the smooth closure of the smooth volume-preserving conjugation class of some Liouville rotations $S_\alpha$ of angle $\alpha$ contains a smooth volume-preserving diffeomorphism $T$ that is metrically isomorphic to an irrational rotation $R_\beta$ on the circle, with $\alpha \not\eq \pm \beta$, and with $\alpha$ and $\beta$ chosen either rationally dependent or rationally independent. In particular, the closed annulus $[0,1] \times \varmathbb{T}^1$ admits a smooth ergodic pseudo-rotation $T$ of angle $\alpha$ that is metrically isomorphic to the rotation $R_\beta$.
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Submitted on : Saturday, October 13, 2012 - 3:11:43 PM
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Mostapha Benhenda. Estimates of the conjugacy to rotations of circle diffeomorphisms. Successives conjugacies and smooth realizations. Dynamical Systems [math.DS]. Université Paris-Nord - Paris XIII, 2012. English. ⟨tel-00741531⟩

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