On ergodic averages for parabolic product flows

TL;DR

This paper investigates the convergence rates of ergodic averages in a combined system of suspension flows over substitution systems and arbitrary ergodic flows, providing new spectral estimates.

Contribution

It introduces novel uniform spectral measure estimates for suspension flows over substitution systems, addressing a question posed by Jon Chaika.

Findings

Quantitative convergence speed estimates for ergodic integrals.

New uniform spectral measure bounds for suspension flows.

Application to product systems combining substitution and ergodic flows.

Abstract

We consider a direct product of a suspension flow over a substitution dynamical system and an arbitrary ergodic flow and give quantitative estimates for the speed of convergence for ergodic integrals of such systems. Our argument relies on new uniform estimates of the spectral measure for suspension flows over substitution dynamical systems. The paper answers a question by Jon Chaika.