Directed harmonic currents near hyperbolic singularities

Let $\mathscr{F}$ be a holomorphic foliation by curves defined in a neighborhood of $0$ in $\mathbb{C}^{2}$ having $0$ as a hyperbolic singularity. Let $T$ be a harmonic current directed by $\mathscr{F}$ which does not give mass to any of the two separatrices. We show that the Lelong number of $T$ at $0$ vanishes. Then we apply this local result to investigate the global mass distribution for directed harmonic currents on singular holomorphic foliations living on compact complex surfaces. Finally, we apply this global result to study the recurrence phenomenon of a generic leaf.

Symmetric and reversible families of vector fields near a partially hyperbolic singularity

We study smooth local models of families of symmetric and reversible vector fields near a partially hyperbolic singularity. Special attention is given to the question of whether the involved changes of variables commute with the symmetry.