Interconnections between parabolic and hyperbolic dynamics have been recently studied regarding renormalization techniques. In particular, they were observed in the deviation of ergodic integrals and solving cohomological equations for some parabolic flows (cf. Liverani-Giulietti, Adam-Baladi, Faure-Gouëzel-Lanneau, and Butterley-Simonelli, etc.).
In this talk, we introduce the geometric anisotropic Banach spaces on Heisenberg nilmanifolds. This construction shows how the Ruelle resonances for the transfer operator associated with the renormalization map (partially hyperbolic nil-automorphism) are related to the invariant distributions for cohomological equations of Heisenberg nilflows. This is joint work with Oliver Butterley.
