# Parabolic Flows Renormalized by Partially Hyperbolic Maps

## TL;DR

This paper studies parabolic flows on 3D manifolds that are renormalized by partially hyperbolic maps, analyzing their spectral properties to understand ergodic averages and cohomological equations.

## Contribution

It introduces a framework for analyzing parabolic flows renormalized by partially hyperbolic automorphisms using anisotropic Sobolev spaces and spectral theory.

## Key findings

- Spectral properties of transfer operators are established.
- Deviation of ergodic averages is characterized.
- Solutions to the cohomological equation are described.

## Abstract

We consider parabolic flows on 3-dimensional manifolds which are renormalized by circle extensions of Anosov diffeormorphisms. This class of flows includes nilflows on the Heisenberg nilmanifold which are renormalized by partially hyperbolic automorphisms. 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.