Dimension theory of group actions by circle diffeomorphisms II: Minimal sets

Abstract

We establish a dimension theory for smooth group actions on the circle. For finitely generated groups of real-analytic circle diffeomorphisms preserving a Cantor minimal set, we introduce a dynamically defined critical exponent and prove the following properties of that set: (1) The box dimension exists and coincides with its Hausdorff dimension; (2) The dimension is given by a formula involving the dynamical critical exponent; (3) The dimension lies strictly between $0$ and $1$; (4) A parabolic fixed point in this Cantor minimal set yields a stronger lower bound for its dimension. These results substantially generalize the classical dimension theory of limit sets of Fuchsian groups to a broader smooth setting. Moreover, item (3) strengthens a recent breakthrough of Deroin--Kleptsyn--Navas, who showed that such sets have zero Lebesgue measure. A key novelty of this work is the use of the dynamical critical exponent to link the group action with stationary and conformal measures. This provides a framework combining methods from nonuniform hyperbolic dynamics and Patterson-Sullivan theory.