2025/03/12 by Deroin, Bertrand, Dupont, Christophe, Kleptsyn, Victor
#Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2503.09152
Let F be a singular holomorphic foliation on an algebraic complex surface S, with hyperbolic singularities and no foliated cycle. We prove a formula for the transverse Hausdorff dimension of the unique harmonic current, involving the Furstenberg entropy and the Lyapunov exponent. In particular, we extend Brunella's inequality to every holomorphic foliation F on \mathbb P2: if F has degree d ≥ 2, then the Hausdorff dimension of its harmonic current is smaller than or equal to d-1 \over d+2, in particular the harmonic current is singular with respect to the Lebesgue measure. We also show that the Hausdorff dimension of the harmonic current of the Jouanolou foliation of degree 2 is equal to 1/4, and that the same property holds for topologically conjugate foliations on \mathbb P2.