Hyperbolic entropy for harmonic measures on singular holomorphic foliations

TL;DR

This paper investigates the hyperbolic entropy of harmonic measures on certain singular holomorphic foliations, establishing that the entropy is almost everywhere leafwise constant and bounded below by 2.

Contribution

It proves the leafwise constancy of local hyperbolic entropies and provides a lower bound for the entropy of harmonic measures on Brody-hyperbolic foliations with isolated singularities.

Findings

Local hyperbolic entropies are leafwise constant almost everywhere.

The entropy of harmonic measures is at least 2.

Applicable to non-degenerate and saddle-node singularities in dimension 2.

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 $2$.