Hyperbolic entropy for harmonic measures on singular holomorphic foliations

Abstract

Let $\mathscr{F}=(M,\mathscr{L},E)$ be a Brody-hyperbolic singular holomorphic foliation on a compact complex manifold $M$. Suppose that $\mathscr{F}$ has isolated singularities and that its Poincar\'e metric is complete. This is the case for a very large class of singularities, namely, non-degenerate and saddle-nodes in dimension $2$. Let $\mu$ be an ergodic harmonic measure on $\mathscr{F}$. We show that the upper and lower local hyperbolic entropies of $\mu$ are leafwise constant almost everywhere. Moreover, we show that the entropy of $\mu$ is at least $2$.