Equilibrium measures for certain isometric extensions of Anosov systems

TL;DR

This paper establishes the uniqueness of equilibrium measures for specific isometric extensions of Anosov systems, including frame flows on negatively curved manifolds and automorphisms of the Heisenberg manifold, with implications for ergodicity.

Contribution

It provides new proofs and extends the understanding of equilibrium measures and ergodic properties for certain isometric extensions of Anosov systems.

Findings

Unique equilibrium measure for frame flow on negatively curved manifolds (excluding dimension 7)

Unique equilibrium measure for automorphisms of the Heisenberg manifold

Alternative proof of ergodicity for these flows

Abstract

We prove that for the frame flow on a negatively curved, closed manifold of odd dimension other than 7, and a Holder continuous potential that is constant on fibers, there is a unique equilibrium measure. We prove a similar result for automorphisms of the Heisenberg manifold fibering over the torus. Our methods also give an alternate proof of Brin and Gromov's result on the ergodicity of these frame flows.