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 $\mu{-1}$.
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