Raphael Douady

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 $\mu _{-1}$.