Abstract
We consider nilflows on the Heisenberg nilmanifolds which are renormalized by partially hyperbolic automorphisms, i.e., parabolic flows on 3-dimensional manifolds which are renormalized by circle extensions of Anosov diffeomorphisms. The transfer operators associated to the renormalization maps, acting on anisotropic Sobolev spaces, are known to have good spectral properties (this relies on ideas which have some resemblance to representation theory but also apply to non-algebraic systems). The spectral information is used to describe the deviation of ergodic averages and solutions of the cohomological equation for the parabolic flow.
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