Beyond Renormalization in Parabolic Dynamics

Introduction

The ergodic theory of parabolic dynamical systems is an area in smooth ergodic theory that is relevant for its connections to mathematical physics and to analytic number theory. A dynamical system is parabolic whenever its orbits diverge at an intermediate (often polynomial) rate between bounded/logarithmic (called elliptic) and exponential (called hyperbolic) rate.

Proposal Overview

The proposal tackles several outstanding questions in the ergodic theory of parabolic flows with emphasis on quantitative aspects. The main goal is to go beyond renormalization techniques that have proved extremely powerful in several classes of examples:

1. Interval Exchange Transformations and Flows on surfaces
2. Horocycle flows
3. Nilflows on quotients of step-two nilpotent groups
4. Gauss sums

Renormalization methods are not available in other equally fundamental examples of similar nature, such as:

• Billiards in non-rational polygons
• Higher step nilflows
• Non-horospherical unipotent flows in homogeneous dynamics

A unified approach to effective ergodicity is proposed that encompasses all of the above-mentioned examples.

Outstanding Questions

Outstanding questions include:

• Ergodicity and existence of periodic orbits of non-rational billiards in polygons
• Effective ergodicity of higher step nilflows with optimal deviation exponents and applications to bounds on Weyl sums for higher degree polynomials
• Effective ergodicity of non-horospherical unipotent flows on semi-simple finite-volume quotients

Analytical Foundations

The analytical foundations of the method lie in the study of invariant distributions for parabolic flows. In the examples considered, the analysis can be carried out by methods of non-Abelian Fourier analysis (theory of unitary representations).

In general, for non-homogeneous parabolic flows, all questions concerning invariant distributions and their relevance for smooth ergodic theory are wide open. Several problems to probe the question of existence of invariant distributions are proposed.