Infinite-Volume Mixing for Dynamical Systems Preserving an Infinite Measure

Infinite Measure Preserving Transformations with "mixing"

On infinite-volume mixing

In the context of the long-standing issue of mixing in infinite ergodic theory, we introduce the idea of mixing for observables possessing an infinitevolume average. The idea is borrowed from statistical mechanics and appears to be relevant, at least for extended systems with a direct physical interpretation. We discuss the pros and cons of a few mathematical definitions that can be devised, te...

Quasi-factors for Infinite-measure Preserving Transformations

This paper is a study of Glasner’s definition of quasi-factors in the setting of infinite-measure preserving system. The existence of a system with zero Krengel entropy and a quasi-factor with positive entropy is obtained. On the other hand, relative zero-entropy for conservative systems implies relative zero-entropy of any quasi-factor with respect to its natural projection onto the factor. Th...

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

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 Ln 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) (...

Extensions and Multiple Recurrence of infinite measure preserving systems

We prove that an extension of an invertible, multiply-recurrent infinite measure preserving transformation is also multiply-recurrent.