Unique ergodicity for foliations in P^2 with an invariant curve

TL;DR

This paper proves a unique ergodicity property for certain foliations in the projective plane with a single invariant line, showing that leaf distributions converge to a specific invariant current, despite most leaves being dense.

Contribution

It establishes a unique ergodic measure for foliations with a single invariant algebraic curve and hyperbolic singularities, extending the theory of densities for currents.

Findings

Unique positive ddc-closed current is the invariant line's current

Leaves' averages converge to the invariant current

Most leaves are dense despite the invariant line presence

Abstract

Consider a foliation in the projective plane admitting a projective line as the unique invariant algebraic curve. Assume that the foliation is generic in the sense that its singular points are hyperbolic. We show that there is a unique positive ddc-closed (1,1)-current of mass 1 which is directed by the foliation and this is the current of integration on the invariant line. A unique ergodicity theorem for the distribution of leaves follows: for any leaf L, appropriate averages of L converge to the current of integration on the invariant line. This property is surprising because for most of such foliations the leaves (except the invariant line) are dense in the projective plane. So one could expect that they spend a significant amount of hyperbolic time in every open set and that there should be a fat ddc-closed non-closed current with support equal to the projective plane. The proof uses an extension of our theory of densities for currents. Foliations on compact Kaehler surfaces with one or several invariant curves are also considered.