Lyapunov Exponents and Rigidity in Elliptic and Hyperbolic Settings

This thesis studies a pair of problems relating rigidity and Lyapunov exponents. In Chapter 2, we study Anosov automorphisms of nilmanifolds. More precisely, we obtain necessary and sufficient conditions for an Anosov automorphism of a nilmanifold with simple Lyapunov spectrum to be locally Lyapunov spectrum rigid. In Chapter 3, we study perturbations of random walks on isotropic manifolds. Our main result in this section is a necessary and sufficient criterion for this random walk to be isometric with respect to some metric. This criterion is a generalization of work of Dolgopyat and Krikorian.