Geometric characterization of Lyapunov exponents for Riemann surface laminations

TL;DR

This paper provides a geometric characterization of Lyapunov exponents for cocycles on hyperbolic Riemann surface laminations, linking them to expansion rates along geodesic rays, advancing understanding in dynamical systems on complex manifolds.

Contribution

It introduces a new geometric framework for understanding Lyapunov exponents in the context of Riemann surface laminations, relating them to geodesic expansion rates.

Findings

Lyapunov exponents are characterized geometrically.

Expansion rates along geodesic rays determine Lyapunov exponents.

Framework applies to cocycles of arbitrary rank.

Abstract

We characterize geometrically the Lyapunov exponents of a cocycle (of arbitrary rank) with respect to a harmonic current defined on a hyperbolic Riemann surface lamination. Our characterizations are formulated in terms of the expansion rates of the cocycle along geodesic rays.