This post intends to treat the classical results on infinite ergodic theory, specifically Hopf’s ratio ergodic theorem, the non-existence of Birkhoff’s theorem, Aaronson’s theorem, rational ergodicity and law of large numbers. We won’t provide proofs for these results and for that we refer the reader to the book An Introduction to Infinite Ergodic Theory of Jon Aaronson.
Let be a measure-preserving system: is a measure space, a -finite measure and is a measurable transformation on that is invariant under . Whenever , we have Poincaré’s recurrence theorem. If , this is not true in general and one has to assume an additional condition, described by the
Definition 1 We say is conservative if whenever is such that are pairwise disjoint.
Exercise 1 Prove that a conservative measure-preserving system satisfies Poincaré’s recurrence theorem.
Conservativity is also a necessary condition: the translation , , is invariant for the counting measure on but the orbit of every point scapes to infinity. This is indeed, if one assumes ergodicity and invertibility, the only example.
Lemma 2 An invertible ergodic measure-preserving system is non-conservative if and only if it is isomorphic to the translation , , with the counting measure on .
Proof: Assume is an invertible ergodic non-conservative measure-preserving system and let such that are pairwise disjoint. Because is invertible, are pairwise disjoint and thus, by ergodicity,
Again by ergodicity, is a -atom (non trivial subsets of of give rise to non trivial -invariant sets ). This guarantees the map is an isomorphism from to .
From now on, we always assume is ergodic and conservative. Let be a measurable function. A successful area in ergodic theory deals with the convergence of the averages , , when goes to infinity. The well known Birkhoff’s theorem states that, if , such limit exists for almost every whenever is a -function. This is not the case when is infinite. Indeed, if , these averages converge to zero for almost every . We give an idea why this is so: if is a measure-preserving system with , Birkhoff’s theorem asserts that
for -almost every . Heuristically, if we take in the right hand side of the above convergence, then the Birkhoff averages converge to zero. This argument demands a little care, and indeed is more convenient to conclude it as a corollary of Hopf’s ratio ergodic theorem (see below). At this point, it is natural to ask if there exists some “appropriate” rate of convergence: is there a normalizing sequence of constants such that converges almost surely? And the answer is no!
Theorem 3 (Aaronson) Let be a measure-preserving system with , and let be a sequence of positive real numbers. Then, for every such that and ,
This means that any attempt of normalization will under or overestimate the behavior of Birkhoff sums. Nevertheless, the Birkhoff sums fluctuate in the same proportional rate, according to the following result.
Theorem 4 (Hopf’s ratio ergodic theorem) Let be a measure-preserving system. Then, for every such that and ,
for -almost every .
Exercise 2 Prove, using Hopf’s ratio ergodic theorem, that if is a measure-preserving system with then, for every ,
for -almost every .
Hopf’s ratio ergodic theorem is an indication that some sort of regularity might exist and it might still be possible, for a specific sequence , that the averages oscillate without converging to zero or infinity and so one can hope for a summability method that smooths out the fluctuations and forces convergence.
1. Law of large numbers
More generally, one can hope for a law of large numbers.
Definition 5 A law of large numbersfor a measure-preserv\-ing system is a function such that, for any , the equality
holds for -almost every .
One can see the function as a sort of blackbox: given the input of hittings of a generic point to a fixed set , the output is the measure of . For example, if , the function defined by
is a law of large numbers. The infinite measure situation is quite different: there are systems with no law of large numbers. For example, let be squashable: there is , commuting with , such that
for some . If had a law of large numbers, say , then for -almost every we would have
contradicting the assumption of squashability. See An Introduction to Infinite Ergodic Theoryfor more on squashable systems.
There are, fortunately, some conditions that guarantee the existence of law of large numbers.
2. Rational ergodicity
Given , let be the Birkhoff sum of the characteristic function with respect to .
Definition 6 A measure-preserving system is called rationally ergodic if there is a sweep-out set with satisfying a Renyi inequality: there is such that
for every . We say is rationally ergodic along a subsequence of iterates if the above inequality holds for a subsequence of positive integers.
By sweep-out we mean that every point of eventually enters in , that is
is called a return sequence of and it is unique up to asymptotic equality.
Theorem 8 (Aaronson) Every measure-preserving system that is rationally ergodic along a subsequence of iterates is weakly homogeneous. More specifically, every subsequence can be refined to a further subsequence such that (2) holds for -almost every .
Theorem 8 also gives the explicit description of the normalizing constants:
Observe that weak homogeneity defines an explicit law of large numbers by
The next post will focus on a result in collaboration with P. Cirilo and E. Pujals in which we construct a class of cylinder flows that are rationally ergodic along a subsequence of iterates (and thus have explicit law of large numbers).