Some ergodic and rigidity properties of discrete Heisenberg group actions

TL;DR

This paper investigates ergodic and rigidity properties of smooth actions of the discrete Heisenberg group, revealing conditions under which certain dynamical behaviors occur or are restricted, including rigidity results and limitations on Anosov actions.

Contribution

It establishes the decomposition of tangent spaces for Lyapunov exponents, shows zero exponents for center elements, and proves local rigidity of higher rank actions via a generalized KAM scheme.

Findings

Lyapunov exponents for center elements are zero

No faithful codimension one or two Anosov Heisenberg actions on certain manifolds

Smooth local rigidity for higher rank ergodic actions by toral automorphisms

Abstract

The goal of this paper is to study ergodic and rigidity properties of smooth actions of the discrete Heisenberg group $\H$. We establish the decomposition of the tangent space of any $C^\infty$ compact Riemannian manifold $M$ for Lyapunov exponents, and show that all Lyapunov exponents for the center elements are zero. We obtain that if an $\H$ group action contains an Anosov element, then under certain conditions on the element, the center elements are of finite order. In particular there is no faithful codimensional one Anosov Heisenberg group action on any manifolds, and no faithful codimensional two Anosov Heisenberg group action on tori. In addition, we show smooth local rigidity for higher rank ergodic $\H$ actions by toral automorphisms, using a generalization of the KAM (Kolmogorov-Arnold-Moser) iterative scheme.