Hyperbolic entropy for harmonic measures on singular holomorphic foliations

Abstract

Let $\mathcal{F}=(M,\mathcal{L},E)$ be a Brody-hyperbolic singular holomorphic foliation on a compact complex manifold $M$. Suppose that $\mathcal{F}$ has isolated singularities and that its Poincaré 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 $\mathcal{F}$. 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 $2$.

arXiv commentMinor changes, some details added to a proof. To appear in Advances in Mathematics

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• Poincaré metric of holomorphic foliations with non-degenerate singularities
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