Surface group representations and entropy degeneration

For surface group representations into rank one Lie groups, or more generally the isometry group
of a CAT(-1)-metric space, the connection between the asymptotic growth rate of orbits (entropy)
and the Hausdorff dimension of the limit set yields a universal positive lower bound on the entropy.  Allowing higher rank Lie groups, or more generally isometry groups of CAT(0)-metric spaces, this connection evaporates, and many interesting examples are known of surface group representations
whose entropy is arbitrarily close to zero.

In this talk, we will discuss some Riemannian geometric methods to study some fine scale properties
of such representations.  As an application, we will give a dynamical interpretation of the asymptotic decoupling behavior for solutions of Hitchin's self-duality equations studied in the context of harmonic
flat bundles by Collier-Li and Mochizuki.