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, testing them on a prototypical class of infinite measure-preserving dynamical systems, namely, the random walks. Mathematics Subject Classification: 37A40, 37A25, 82B41, 60G50. 1 Historical introduction The textboox definition of mixing for a transformation T : M −→ M preserving a probability measure μ is lim n→∞ μ(TA ∩ B) = μ(A)μ(B) (1.1) for all measurable sets A,B ⊂ M [W]. Extending this definition to the case where μ is a σ-finite measure with μ(M) = ∞ is a fundamental issue in infinite ergodic theory. References to this problem can be found in the literature at least as far back as 1937, when Hopf devoted a section of his famous Ergodentheorie [H] to an example of a dynamical system that he calls ‘mixing’. It consists of a set M ⊂ R, of infinite Lebesgue measure, and a map T : M −→ M preserving μ, the Lebesgue measure on M. He proved a property that is equivalent to this one: there exists a sequence {ρn}n∈N of positive numbers such that lim n→∞ ρn μ(T A ∩ B) = μ(A)μ(B) (1.2) ∗Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy. E-mail: [email protected]