First and higher order uniform dual ergodic theorems for dynamical systems with infinite measure

TL;DR

This paper extends dual ergodic theorems to infinite measure dynamical systems, providing new methods for higher order asymptotics and improved convergence rate bounds.

Contribution

It introduces a higher order Tauberian theorem and a novel approach for uniform dual ergodicity in infinite measure systems, including Pomeau-Manneville maps.

Findings

Established higher order asymptotics for ergodic theorems.

Developed a method for better convergence rate estimates.

Derived a new higher order Tauberian theorem.

Abstract

We generalize the proof of Karamata's Theorem by the method of approximation by polynomials to the operator case. As a consequence, we offer a simple proof of \emph{uniform dual ergodicity} for a very large class of dynamical systems with infinite measure, and we obtain bounds on the convergence rate. In many cases of interest, including the Pomeau-Manneville family of intermittency maps, the estimates obtained through real Tauberian remainder theory are very weak. Building on the techniques of complex Tauberian remainder theory, we develop a method that provides \emph{second (and higher) order asymptotics}. In the process, we derive a \emph{higher order Tauberian theorem} for scalar power series, which to our knowledge, has not previously been covered.