Impact Factor

Raphael Douady

61 av. du Pdt. Wilson, 94235 Cachan, France
C.N.R.S. et C.M.L.A., Ecole Normale Superieure de Cachan


Douady R., Yoccoz J.
Nombre de Rotation des Diffeomorphismes du Cercle et Mesures Automorphes
1999, vol. 4, no. 4, pp.  19-38
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

Back to the list