, Volume 194, Issue 2, pp 793-830
Date: 12 Nov 2012

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

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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 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 second (and higher) order asymptotics. In the process, we derive a higher order Tauberian theorem for scalar power series which, to our knowledge, has not previously been covered.