0
2013
Impact Factor

 Douady R., Yoccoz J. Nombre de Rotation des Diffeomorphismes du Cercle et Mesures Automorphes 1999, vol. 4, no. 4, pp.  19-38 Abstract Let $f$ be a $C^1$-diffeomorphism of the circle $\mathbb{T}^1=\mathbb{R}/\mathbb{Z}$ with an irrational rotation number. We show that, for every real number $s$, there exists a probability measure $\mu _s$, unique if $f$ is $C^2$, that satisfies, for any function $\varphi \in C^0(\mathbb{T}^1)$: $$\int\limits_{\mathbb{T}^1}\varphi \,d\mu _s=\int\limits_{\mathbb{T}^1}\varphi \circ f\;(Df)^s\,d\mu _s .$$ This measure continuously depends on the pair $( s,f)$ when one considers the weak topology on measures and the $C^1$-topology on diffeomorphisms. Examples are given where uniqueness fails with $f$ of class $C^1$. These results partially extend to the case of a rational rotation number for non degenerate semi-stable diffeomorphisms of the circle. We then show that the set of diffeomorphisms that have a given irrational rotation number has a tangent hyperplane at any $C^2$-diffeomorphism, the direction of which is the kernel of $\mu _{-1}$. Citation: Douady R., Yoccoz J.,  Nombre de Rotation des Diffeomorphismes du Cercle et Mesures Automorphes, Regular and Chaotic Dynamics, 1999, vol. 4, no. 4, pp. 19-38 DOI:10.1070/RD1999v004n04ABEH000129