Parabolic dynamics and Anisotropic Banach spaces

TL;DR

This paper explores the connection between distributions in parabolic flow ergodic averages and hyperbolic system statistical properties, showing that growth rates are governed by transfer operator eigenvalues in a simple torus flow.

Contribution

It establishes a conceptual link between ergodic averages of parabolic flows and transfer operator eigenvalues, suggesting broader applicability.

Findings

Growth of ergodic averages is controlled by transfer operator eigenvalues.

The connection between parabolic and hyperbolic dynamics is demonstrated in a simple flow.

The approach simplifies technical issues to highlight core ideas.

Abstract

We investigate the relation between the distributions appearing in the study of ergodic averages of parabolic flows (e.g. in the work of Flaminio-Forni) and the ones appearing in the study of the statistical properties of hyperbolic dynamical systems (i.e. the eigendistributions of the transfer operator). In order to avoid, as much as possible, technical issues that would cloud the basic idea, we limit ourselves to a simple flow on the torus. Our main result is that, roughly, the growth of ergodic averages of a parabolic flows is controlled by the eigenvalues of a suitable transfer operator associated to the renormalising dynamics. The conceptual connection that we illustrate is expected to hold in considerable generality.