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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.

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Author notes

  1. Dalia Terhesiu

    Present address: University of Rome Tor Vergata, via della Ricerca Scientifica, I-00133, Rome, Italy

Authors and Affiliations

  1. Department of Mathematics, University of Surrey, Guildford, Surrey, GU2 7XH, UK

    Ian Melbourne & Dalia Terhesiu

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  1. Ian Melbourne
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  2. Dalia Terhesiu
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Correspondence to Ian Melbourne.

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Melbourne, I., Terhesiu, D. First and higher order uniform dual ergodic theorems for dynamical systems with infinite measure. Isr. J. Math. 194, 793–830 (2013). https://doi.org/10.1007/s11856-012-0154-5

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  • DOI: https://doi.org/10.1007/s11856-012-0154-5

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