Operator renewal theory and mixing rates for dynamical systems with infinite measure

TL;DR

This paper develops operator renewal theory for infinite measure dynamical systems, enabling the determination of asymptotic behaviour of transfer operator iterates and establishing mixing rates for complex systems like Pomeau-Manneville maps.

Contribution

It introduces a new operator renewal framework for infinite ergodic theory, providing asymptotic analysis and mixing rates for a broad class of systems previously intractable.

Findings

Asymptotic behaviour of transfer operator iterates determined

Higher order expansions and mixing rates obtained for specific maps

Error estimates established for arcsine laws in these systems

Abstract

We develop a theory of operator renewal sequences in the context of infinite ergodic theory. For large classes of dynamical systems preserving an infinite measure, we determine the asymptotic behaviour of iterates $L^n$ of the transfer operator. This was previously an intractable problem. Examples of systems covered by our results include (i) parabolic rational maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly expanding interval maps with indifferent fixed points. In addition, we give a particularly simple proof of pointwise dual ergodicity (asymptotic behaviour of $\sum_{j=1}^nL^j$) for the class of systems under consideration. In certain situations, including Pomeau-Manneville intermittency maps, we obtain higher order expansions for $L^n$ and rates of mixing. Also, we obtain error estimates in the associated Dynkin-Lamperti arcsine laws.