Dimension theory of group actions by circle diffeomorphisms I: Stationary measures

We establish a structure theorem for smooth random walks on the circle and study the dimensional properties of their stationary measures, including exact dimensionality and a Ledrappier-Young-type dimension formula. These results provide the technical tools needed for the main theorems in Part II “Dimension theory of group actions by circle diffeomorphisms II: Minimal sets”. They also yield an approximation theorem in the spirit of Katok, Avila-Crovisier-Wilkinson, and Morris-Shmerkin, together with a time-reversal symmetry for circle actions: every semigroup $T$ contained in ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{+}^{1+\alpha }({\mathbb{S}}^1)$ without finite orbits has the same number of minimal sets as its inverse semigroup $T^{-1}$.