Spencer Durham
University of Maryland
Title
KAM Rigidity for affine Z^2 actions on the torus
Abstract
When is a dynamical system conjugate to a perturbation of itself? This rigidity phenomenon is rare in the case of Z-actions, but among higher-rank actions, KAM methods can be applied to obtain rigidity under small, mean-zero, volume-preserving perturbations. Moser treated the case of Diophantine torus translations while Katok and Spatzier and Damjanovic and Katok dealt with the case of hyperbolic and partially hyperbolic toral automorphisms respectively. Most recently, Damjanovic, Fayad, and Saprykina, showed a dichotomy between rigidity and an algebraic property known as lockedness for affine parabolic actions under the assumption that one of the generators was step 2. In this work, we remove the step 2 assumption. The crucial element of the proof is showing that, for an unlocked action, the cohomological equation can be solved without losing half of the regularity.