aa r X i v : . [ m a t h . D S ] J a n Exactness, K-propertyand infinite mixing
Marco Lenci ∗† Version published in
Publicaciones Matem´aticas del Uruguay (2013), 159-170+ minor fixesNovember 2013 Abstract
We explore the consequences of exactness or K-mixing on the notionsof mixing (a.k.a. infinite-volume mixing ) recently devised by the author forinfinite-measure-preserving dynamical systems.Mathematics Subject Classification (2010): 37A40, 37A25.
Currently, in infinite ergodic theory, there is a renewed interest in the issues relatedto mixing for infinite-measure-preserving (or just nonsingular) dynamical systems,in short infinite mixing (see [Z, DS, L1, I3, DR, MT, LP, A2, Ko, T1], and someapplications in [I1, I2, AMPS, L2, T2]).The present author recently introduced some new notions of infinite mixing,based on the concept of global observable and infinite-volume average [L1]. Inessence, a global observable for an infinite, σ -finite, measure space ( X, A , µ ) isfunction in L ∞ ( X, A , µ ) that “looks qualitatively the same” all over X . This is incontrast with a local observable , whose support is essentially localized, so that thefunction is integrable.Postponing the mathematical details to Section 2, the purpose of the globalobservables is basically twofold. First, the past attempts to a general definition ofinfinite mixing involved mainly local observables (equivalently, finite-measure sets),and the problems with such definitions seemed to depend on that. Second, seeking ∗ Dipartimento di Matematica, Universit`a di Bologna, Piazza di Porta S. Donato 5, 40126Bologna, Italy. E-mail: [email protected] † Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy. Marco Lenci inspiration in statistical mechanics (which is the discipline of mathematical physicsthat has successfully dealt with the question of predicting measurements in verylarge, formally infinite, systems), one realizes that many quantities of interest are extensive observables , that is, objects that behave qualitatively in the same way indifferent regions of the phase space. (More detailed discussions about these pointsare found in in [L1, L2].)Extensive observables are “measured” by taking averages over large portionsof the phase space. We import that concept too, by defining the infinite-volumeaverage of a global observable F : X −→ R as µ ( F ) := lim V ր X µ ( V ) Z V F dµ. (1.1)Here V is taken from a family of ever larger but finite-measure sets that somehowcovers, or exhausts the whole of X . The precise meaning of the limit above will begiven in Section 2.Now, let us consider a measure-preserving dynamical system on ( X, A , µ ). Forthe sake of simplicity, let us restrict to the discrete-time case: this means that wehave a measurable map T : X −→ X that preserves µ . Choosing two suitableclasses of global and local observables, respectively denoted G and L , we give fivedefinitions of infinite mixing. These fall in two categories, exemplified as follows.Using the customary (abuse of) notation µ ( g ) := R X g dµ . we say that the systemexhibits: • global-local mixing if, ∀ F ∈ G , ∀ g ∈ L , lim n →∞ µ (( F ◦ T n ) g ) = µ ( F ) µ ( g ); • global-global mixing if, ∀ F, G ∈ G , lim n →∞ µ (( F ◦ T n ) G ) = µ ( F ) µ ( G ).Disregarding for the moment the mathematical issues connected to the abovenotions, we focus on the interpretation of global-local mixing. Restricting, withoutloss of generality, to local observables g ≥ µ ( g ) = 1, and defining dµ g := g dµ ,the above limit reads: lim n →∞ T n ∗ µ g ( F ) = µ ( F ) , (1.2)where the measure T n ∗ µ g is the push-forward of µ g via the dynamics T n (in otherwords, T n ∗ µ g := µ g ◦ T − n = µ P n g , where P is the Perron-Frobenius operator relativeto µ , cf. (3.2)-(3.3)). If (1.2) occurs for all g ∈ L and F ∈ G , the above is a sort of“convergence to equilibrium” for all initial states given by µ -absolutely continuousprobability measures. In this sense the functional µ (not a measure!) plays the roleof the equilibrium state .Exactness and K-mixing (a.k.a. the K-property) are notions that exist and havethe same definition both in finite and infinite ergodic theory. In finite ergodic theorythey are known to be very strong properties, as they imply mixing of all orders, cf.definition (3.1). The purpose of this note is to explore their implications in termsof the notions of infinite mixing introduced in [L1]. nfinite mixing (a) ), the most notable of such implications is aweak form of global-local mixing, whereby any pair of measures µ g , µ h , as introducedearlier, are asymptotically coalescing , in the sense thatlim n →∞ ( T n ∗ µ g ( F ) − T n ∗ µ h ( F )) = 0 , (1.3)for all F ∈ G .In the next section we review the five definitions of global-local and global-globalmixing, together with the already known (though with a different name) definitionof local-local mixing. In Section 3 we prepare, state and prove Theorem 3.5, whichlists some consequences of exactness and the K-property. Finally, in Section 4,we introduce the space of the equilibrium observables , which is a purely ergodic-theoretical construct in which some information about global-local mixing can berecast. Let ( X, A , µ, T ) be a measure-preserving dynamical system, where ( X, A ) is ameasure space, µ an infinite, σ -finite, measure on it, and T a µ -endomorphism, thatis, a measurable surjective map that preserves µ (i.e., µ ( T − A ) = µ ( A ), ∀ A ∈ A ).Denoting by A f := { A ∈ A | µ ( A ) < ∞} the class of finite-measure sets, weassume that the following additional structure is given for the dynamical system: • A class of sets V ⊂ A f , called the exhaustive family . The elements of V will be generally indicated with the letter V . • A subspace
G ⊂ L ∞ ( X, A , µ ; R ), whose elements are called the global ob-servables . These functions are indicated with uppercase Roman letters ( F, G ,etc.). • A subspace
L ⊂ L ( X, A , µ ; R ) whose elements are called the local observ-ables . These functions will be indicated with lowercase Roman letters ( f, g ,etc.).A discussion on the role and the choice of V , G , L is given in [L1], together with theproofs of most assertions made in this section.We assume that V contains at least one sequence ( V j ) j ∈ N , ordered by inclusion,such that S j V j = X . (In actuality, this requirement is never used in the proofs,but, since the elements of V are regarded as large and “representative” regions ofthe phase space X , we keep it to give “physical” meaning to the concept of infinite-volume average, see below.) We also assume that 1 ∈ G (with the obvious notation1( x ) := 1, ∀ x ∈ X ). Marco Lenci
Definition 2.1
Let V be the aforementioned exhaustive family. For φ : V −→ R ,we write lim V ր X φ ( V ) = ℓ when lim M →∞ sup V ∈ V µ ( V ) ≥ M | φ ( V ) − ℓ | = 0 . We call this the ‘ µ -uniform infinite-volume limit w.r.t. the family V ’, or, for short,the infinite-volume limit . We assume that, ∀ n ∈ N , µ ( T − n V △ V ) = o ( µ ( V )) , as V ր X. (2.1)This is reasonable because, if a large V ∈ V is to be considered a finite-measuresubstitute for X , it makes sense to require that a finite-time application of thedynamics does not change it much. Finally, the most crucial assumption is that, ∀ F ∈ G , ∃ µ ( F ) := lim V ր X µ ( V ) Z V F dµ. (2.2) µ ( F ) is called the infinite-volume average of F w.r.t. µ . It easy to check that µ is T -invariant, i.e., for all F ∈ G and n ∈ N , µ ( F ◦ T n ) exists and equals µ ( F ) [L1].With this machinery, we can give a number of definitions of infinite mixing for thedynamical system ( X, A , µ, T ) endowed with the structure of observables ( V , G , L ).The following three definitions will be called global-local mixing , as they in-volve the coupling of a global and a local observable. We say that the system ismixing of type (GLM1) if, ∀ F ∈ G , ∀ g ∈ L with µ ( g ) = 0, lim n →∞ µ (( F ◦ T n ) g ) = 0; (GLM2) if, ∀ F ∈ G , ∀ g ∈ L , lim n →∞ µ (( F ◦ T n ) g ) = µ ( F ) µ ( g ); (GLM3) if, ∀ F ∈ G , lim n →∞ sup g ∈L\ k g k − | µ (( F ◦ T n ) g ) − µ ( F ) µ ( g ) | = 0,where k · k is the norm of L ( X, A , µ ; R ).Clearly, (GLM1–3) are listed in increasing order of strength, with (GLM2) being possibly the most natural definition one can give for the time-decorrelationbetween a global and a local observable (recall that µ ( F ◦ T n ) = µ ( F )). (GLM3) is a uniform version of it, with important implications (cf. Proposition 2.4), while (GLM1) is a much weaker version, as will become apparent in the remainder.Although this note is mostly concerned with global-local mixing, one can alsoconsider the decorrelation of two global observables, namely global-global mixing .For this we need the following terminology: nfinite mixing Definition 2.2
For V as defined above and φ : V × N −→ R , we write lim V ր Xn →∞ φ ( V, n ) = ℓ to mean lim M →∞ sup V ∈ V µ ( V ) ≥ Mn ≥ M | φ ( V, n ) − ℓ | = 0 . As n will take the role of time, we refer to this limit as the ‘joint infinite-volumeand time limit’. For F ∈ L ∞ and V ∈ V , let us also denote µ V ( F ) := µ ( V ) − R V F dµ . We saythat the system is mixing of type (GGM1) if, ∀ F, G ∈ G , lim n →∞ µ (( F ◦ T n ) G ) = µ ( F ) µ ( G ); (GGM2) if, ∀ F, G ∈ G , lim V ր Xn →∞ µ V (( F ◦ T n ) G ) = µ ( F ) µ ( G ).Though (GGM1) seems the cleaner of the two versions, it has the serious draw-back that, for n ∈ N , µ (( F ◦ T n ) G ) might not even exist, for there is no provi-sion in our hypotheses to guarantee the ring property for condition (2.2) (namely, ∃ µ ( F ) , µ ( G ) ⇒ ∃ µ ( F G )). Nor do we want one, if we are to keep our frameworkgeneral enough. (GGM2) solves this question of wellposedness, and is in somesense stronger than (GGM1) : Proposition 2.3 If F, G ∈ G are such that µ (( F ◦ T n ) G ) exists for all n largeenough (depending on F, G ), then lim V ր Xn →∞ µ V (( F ◦ T n ) G ) = ℓ = ⇒ lim n →∞ µ (( F ◦ T n ) G ) = ℓ. (2.3) In particular, if the above hypothesis holds ∀ F, G ∈ G , then (GGM2) implies (GGM1) . Proof.
From Definition 2.2, the left limit of (2.3) implies that, ∀ ε > ∃ M = M ( ε )such that ℓ − ε ≤ µ V (( F ◦ T n ) G ) ≤ ℓ + ε (2.4)for all V ∈ V with µ ( V ) ≥ M and all n ≥ M . By hypothesis, if M is largeenough, the infinite-volume limit of the above middle term exists ∀ n ≥ M andequals µ (( F ◦ T n ) G ). Upon taking such limit, what is left of (2.4) and its conditionsof validity is the very definition of the right limit in (2.3). Q.E.D.
With reasonable hypotheses on the structure of G and L , the strongest versionof global-local mixing implies the “strongest” version of global-global mixing. Thefollowing proposition is a simplified version of a similar result of [L1] (for an intuitiveunderstanding of the hypotheses, see Proposition 3.2 and Remark 3.3 there). Marco Lenci
Proposition 2.4
Suppose there exist a family ( ψ j ) j ∈ N of real-valued functions of X (this will play the role of a partition of unity) and a family ( J V ) V ∈ V of finite subsetsof N such that:(i) ∀ j ∈ N , ψ j ≥ ;(ii) ∀ G ∈ G , ∀ j ∈ N , Gψ j ∈ L ;(iii) in the limit V ր X , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X j ∈ J V ψ j − V (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = o ( µ ( V )) ,where V is the indicator function of V . Then (GLM3) implies (GGM2) . Proof of Proposition 2.4.
Since the limit in (GGM2) is trivial when G isa constant, and since the global observables are bounded functions, it is no loss ofgenerality to prove (GGM2) for the case G ≥ g j := Gψ j verify all thehypotheses of Proposition 3.2 of [L1] (cf. also Remark 3.3). Notice that the identity G = P j g j (which makes sense insofar as ( ψ j ) j is a partition of unity) is illustrativeand not really used in the proof there. Q.E.D.
Since the five definitions presented above deal with the decorrelation of, first, aglobal and a local observable, and then two global observables, symmetry consid-erations would induce one to give a definition of local-local mixing as well. Areasonable possibility would be to call a dynamical system mixing of type (LLM) if, ∀ f ∈ L ∩ G , g ∈ L , lim n →∞ µ (( f ◦ T n ) g ) = 0.In fact, this definition already exists, as it is easy to check that, in the most generalcase (that is, G = L ∞ , L = L ), a dynamical system is (LLM) if and only if, ∀ A, B ∈ A f , lim n →∞ µ ( T − n A ∩ B ) = 0, i.e., if and only if the system is of zero type [HK] (cf. also [DS, Ko]). Incidentally, this is the same definition that Krengel andSucheston call ‘mixing’, for an infinite-measure-preserving dynamical system [KS]. Two of the few definitions that are copied verbatim from finite to infinite ergodictheory are those of exactness and K-mixing. Though they are well known, werepeat them here for completeness. We state the versions for measure-preservingmaps, but they can be given for nonsingular maps as well ( T is nonsingular if µ ( A ) = 0 ⇒ µ ( T − A ) = 0).Let us denote by N the null σ -algebra , i.e., the σ -algebra that only containsthe zero-measure sets and their complements. Also, given two σ -algebras A , B ,we write A = B mod µ if ∀ A ∈ A , ∃ B ∈ B with µ ( A △ B ) = 0, and viceversa;equivalently, the µ -completions of A and B are the same. nfinite mixing Definition 3.1
The measure-preserving dynamical system ( X, A , µ, T ) is called exact if ∞ \ n =0 T − n A = N mod µ. Since exactness implies that T − A = A mod µ , a nontrivial exact T cannotbe an automorphism of the measure space ( X, A , µ )—although in some sense aninvertible map can still be exact, cf. Remark 3.3 below.The counterpart of exactness for automorphisms is the following: Definition 3.2
The invertible measure-preserving dynamical system ( X, A , µ, T ) possesses the K-property (from A. N. Kolmogorov) if ∃ B ⊂ A such that:(i) B ⊂ T B ;(ii) ∞ _ n =0 T n B = A mod µ ;(iii) ∞ \ n =0 T − n B = N mod µ .In this case, one also says that the dynamical system is K-mixing, or that T is aK-automorphism of ( X, A , µ ) . Remark 3.3
Comparing Definition 3.1 with condition (iii) of Definition 3.2, onemight be tempted to say that, if ( X, A , µ, T ) has the K-property, then ( X, B , µ, T )is exact. This is not technically correct because, in all nontrivial cases, the inclusionin Definition 3.2 (i) is strict, thus T is not a self-map of the measure space ( X, B , µ ).That said, if ( X, A , µ ) is a Lebesgue space, ( X, B , µ, T ) is still morally exact, inthe following sense. Assume w.l.g. that B is complete, let X B be the measurablepartition of X that generates B . (In a Lebesgue space there is a one-to-one cor-respondence, modulo null sets, between complete sub- σ -algebras and measurablepartitions [R].) B can be lifted to a σ -algebra for X B , which we keep calling B .Also, defining T B ([ x ]) := [ T ( x )] (where [ x ] denotes the element of X B that contains x ), we verify that T B is well defined as a self-map of ( X B , B , µ ) (in fact, from Defi-nition 3.2 (i) , X T B is a sub-partition of X B ) and T − B A = T − A , ∀ A ∈ B (with theunderstandable abuse of notation whereby A denotes both a subset of X B and asubset of X ). This and Definition 3.2 (iii) show that T B is an exact endomorphismof ( X B , B , µ ). Of course, in all of the above, B can be replaced by B m := T m B ,for all m ∈ Z (because B m can be used in lieu of B in Definition 3.2).In finite ergodic theory, both exactness and the K-property imply mixing of allorders , namely, ∀ k ∈ Z + and A , A , . . . , A k ∈ A , µ ( A ∩ T − n A ∩ · · · T − n k A k ) −→ µ ( A ) µ ( A ) · · · µ ( A k ) , (3.1) Marco Lenci whenever n → ∞ and n i +1 − n → ∞ , ∀ i = 2 , . . . k − µ ( X ) = 1.)One would expect such strong properties to have consequences also in infiniteergodic theory. This is the case, as we describe momentarily. But first we needsome elementary formalism from the functional analysis of dynamical systems. For F ∈ L ∞ and g ∈ L , let us denote h F, g i := µ ( F g ) . (3.2)Define the Koopman operator U : L ∞ −→ L ∞ as U F := F ◦ T . Its adjoint for theabove coupling is called the Perron-Frobenius operator , denoted P : L −→ L . Itsdefining identity is h U F, g i = h F, P g i . (3.3)Let us explain in detail how P is defined through (3.3). Take g ∈ L and assumefor the moment g ≥
0. Take also F = 1 A , with A ∈ A . We see that h U F, g i = R T − A g dµ . Since T preserves µ and is thus nonsingular w.r.t. it, and since themeasure space is σ -finite, the Radon-Nykodim Theorem yields a locally- L , positive,function P g : X −→ R such that R T − A g dµ = R A ( P g ) dµ = h F, P g i . Using F =1 X = 1, we see that P g ∈ L with k P g k = k g k . For a general g ∈ L , we write g = g + − g − , where g + and g − are, respectively, the positive and negative parts of g . Then P g := P g + − P g − is also in L and k P g k ≤ k g k . (3.4)Therefore, through approximations of F via simple functions (in the L ∞ -norm), onecan extend (3.3) to all F ∈ L ∞ .In the process, we have learned that P is a positive operator ( g ≥ ⇒ P g ≥ k P k = 1, whereas, obviously, U is a positive isometry. Moreover, it is easy tosee that P g = g , with g ≥
0, if and only if g is an invariant density, i.e., if µ g definedby dµ g /dµ = g is an invariant measure. (In fact, had we defined (3.2) for F ∈ L and g ∈ L ∞ , (3.3) would have defined a positive operator P : L ∞ −→ L ∞ , with k P k = 1, and such that P Theorem 3.4
The nonsingular dynamical system ( X, A , µ, T ) is exact if and onlyif, ∀ g ∈ L with µ ( g ) = 0 , lim n →∞ k P n g k = 0 . In the rest of the paper we assume to be in one of the following two cases:(H1) ( X, A , µ, T ) is exact. V is any exhaustive family that verifies (2.1). G = L ∞ . L = L . (Given the assumptions of Section 2, this corresponds to the mostgeneral choice of V , G , L .) nfinite mixing X, A , µ, T ) is K-mixing (thus T is an automorphism). V is any exhaustivefamily that verifies (2.1). G is the closure, in L ∞ , of S m> L ∞ ( B m ), where B m = T m B , as defined in Remark 3.3. Lastly, L = L . Theorem 3.5
Under either (H1) or (H2) ,(a) (GLM1) holds true;(b) (LLM) holds true;(c) (GGM2) implies (GLM2) ;(d) If, ∀ F ∈ G , ∃ g F ∈ L , with µ ( g F ) = 0 , such that lim n →∞ µ (( F ◦ T n ) g F ) = µ ( F ) µ ( g F ) , then (GLM2) holds true. As anticipated in the introduction, (GLM1) (which is the most important as-sertion of the theorem) means that the evolutions of two absolutely continuousinitial measures become indistinguishable, as time goes to infinity. We may call thisphenomenon asymptotic coalescence . This implies that they will return the samemeasurements of global observables, but not that this measurements will converge(in which case we would have a sort of convergence to equilibrium). In fact, formany interesting systems, it is not hard to construct F ∈ L ∞ such that h F, P n g i does not converge for all g ∈ L .This is not surprising, for, even in finite ergodic theory, certain proofs of mixing,or decay of correlation, are divided in two parts: asymptotic coalescence and theconvergence of one initial measure. The difference there is that the latter is usuallyeasy.The remainder of this section is devoted to the following: Proof of Theorem 3.5.
Let us start by proving assertion (a) , namely (GLM1) .We use the formalism of functional analysis outlined earlier in the section.If (H1) is the case, the proof is immediate: for F ∈ L ∞ and g ∈ L , with µ ( g ) = 0, | µ (( F ◦ T n ) g ) | = |h F, P n g i| ≤ k F k ∞ k P n g k → , (3.5)as n → ∞ , by Theorem 3.4.In the case (H2), let us observe that, by easy density arguments, all the definitions (GLM1–3) hold true if they are verified w.r.t. G ′ and L ′ which are subspaces of G and L , respectively, in the L ∞ - and L -norms. We can take G ′ := S m> L ∞ ( B m )(which is dense in G by definition) and L ′ := S m> L ( B m ), which is dense in L = L ( A ) by the K-property [A1]. Therefore, it suffices to show (GLM1) for ageneral m > ∀ F ∈ L ∞ ( B m ), ∀ g ∈ L ( B m ) with µ ( g ) = 0.0 Marco Lenci
Using the arguments and the notation of Remark 3.3, we denote by ˆ F the func-tion induced by F on X B m (i.e., ˆ F ([ x ]) := F ( x )), and analogously for all theother B m -measurable functions. We observe that F ◦ T n is B m -measurable and \ F ◦ T n = ˆ F ◦ T n B m . Thus µ (( F ◦ T n ) g ) = µ (( ˆ F ◦ T n B m )ˆ g ) , (3.6)where the r.h.s. is regarded as an integral in X B m . Since ( X B m , B m , µ, T B m ) is exact,and µ (ˆ g ) = µ ( g ) = 0, we use (3.6) in (3.5) to prove that the l.h.s. of (3.6) vanishes,as n → ∞ .The following is a corollary of (GLM1) . Lemma 3.6
Assume either (H1) or (H2) , and fix F ∈ G . If, for some ℓ ∈ R and ε ≥ , the limit lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12) µ (( F ◦ T n ) g ) µ ( g ) − ℓ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε holds for some g ∈ L (with µ ( g ) = 0 ), then it holds for all g ∈ L (with µ ( g ) = 0 ). Proof of Lemma 3.6.
Suppose the above limit holds for g ∈ L . Take any other g ∈ L , with µ ( g ) = 0. We have: (cid:12)(cid:12)(cid:12)(cid:12) µ (( F ◦ T n ) g ) µ ( g ) − ℓ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) µ (cid:18) ( F ◦ T n ) (cid:18) gµ ( g ) − g µ ( g ) (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) µ (( F ◦ T n ) g ) µ ( g ) − ℓ (cid:12)(cid:12)(cid:12)(cid:12) . (3.7)By (GLM1) , the first term of the above r.h.s. vanishes as n → ∞ , whence theassertion. Q.E.D.
Going back to the proof of Theorem 3.5, we see that Lemma 3.6 immediatelyimplies assertion (d) .As for (b) , again we prove it for both cases (H1) and (H2) at the same time.W.l.g., let us assume that A = N mod µ (otherwise L would be trivial). We claimthat sup A ∈ A f µ ( A ) = ∞ . (3.8)In fact, since A is not trivial, the above sup is positive. If it equalled M ∈ R + , itwould be easy to construct an invariant set B with 0 < µ ( B ) ≤ M . But µ ( X ) = ∞ ,therefore T would not be ergodic, contradicting both (H1) and (H2).Now take f ∈ L ∩ G and ε >
0. By (3.8), ∃ A ∈ A f with µ ( A ) ≥ k f k /ε . Set g ε = 1 A /µ ( A ). We have that (cid:12)(cid:12)(cid:12)(cid:12) µ (( f ◦ T n ) g ε ) µ ( g ε ) (cid:12)(cid:12)(cid:12)(cid:12) = | µ (( f ◦ T n ) g ε ) | ≤ k f k k g ε k ∞ ≤ ε. (3.9) nfinite mixing n →∞ (cid:12)(cid:12)(cid:12)(cid:12) µ (( f ◦ T n ) g ) µ ( g ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε (3.10)holds for all g ∈ L with µ ( g ) = 0. Since ε is arbitrary, we get that the above r.h.s.is zero. The case µ ( g ) = 0 is trivial because the same assertion comes directly from (GLM1) . This proves (LLM) , namely, assertion (b) .Finally for (c) . Take a G ∈ G such that µ ( G ) >
0. Since µ V ( G ) → µ ( G ), as V ր X , (GGM2) implies that there exist a large enough M and a V ∈ V , with µ ( V ) ≥ M , such that | µ V (( F ◦ T n ) G ) − µ ( F ) µ ( G ) | ≤ εµ V ( G ) (3.11)for all n ≥ M . Setting g := G V , we can divide (3.11) by µ V ( G ) = µ ( g ) /µ ( V ) andtake the lim sup in n :lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12) µ (( F ◦ T n ) g ) µ ( g ) − µ ( G ) µ V ( G ) µ ( F ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε. (3.12)By Lemma 3.6, the above holds ∀ g ∈ L , with µ ( g ) = 0. Since ε can be takenarbitrarily close to 0 and µ ( G ) /µ V ( G ) arbitrarily close to 1, we have that, for all F ∈ G and g ∈ L , with µ ( g ) = 0,lim n →∞ µ (( F ◦ T n ) g ) = µ ( F ) µ ( g ) . (3.13)The corresponding statement for µ ( g ) = 0 comes from (GLM1) . Q.E.D.
The “pure” ergodic theorist might raise an eyebrow at the constructions of Section2, especially at the ideas of the exhaustive family (which demands that one singlesout some sets as more important than the others) and of the infinite-volume average(which is not a measure, or even guaranteed to always exist).Though these issues (and more) have been addressed in [L1], one might still wantto see if some of the concepts presented here can be viewed from the vantage point oftraditional infinite ergodic theory. For what follows I am indebted to R. Zweim¨uller.As we discussed in the introduction, the definition (GLM2) makes sense as akind of convergence to equilibrium for a large class of initial distributions (see alsothe observation on (GLM1) after the statement of Theorem 3.5). Without worryingtoo much about predetermining good test functions for this convergence (namely,the global observables), and the value of any such limit (namely, the infinite-volumeaverage), one might simply consider the space E = E ( X, A , µ, T ) of all the goodtest functions, in this sense: E := n F ∈ L ∞ (cid:12)(cid:12)(cid:12) ∃ ρ ( F ) ∈ R s.t. lim n →∞ µ (( F ◦ T n ) g ) = ρ ( F ) µ ( g ) , ∀ g ∈ L o . (4.1)2 Marco Lenci (Occasionally, one might want to restrict the space of the initial distributions to somesubspace of L .) Clearly, E is a vector space which contains at least the constantfunctions. ρ ( F ) represents a sort of value at equilibrium of F and, in this context, it neednot have anything to do with µ ( F ) (which might or might not exist), V , or thechoice of G and L . Thus, the elements of the vector space E may be called the equilibrium observables and ρ : E −→ R the equilibrium functional .If we are in either case (H1) or (H2), Theorem 3.5 (d) shows that, for a given F ∈ G , one only need find one local observable that verifies the limit in (4.1). Also,by Theorem 3.5 (b) , any f ∈ G ∩ L belongs to E , with ρ ( f ) = 0. Therefore, in thesecases, it makes sense to introduce ˆ E := E / ( G ∩ L ), and ρ is well defined there. Whentalking about ˆ E , we write F ∈ ˆ E to mean F ∈ E , and F = G to mean [ F ] = [ G ](where [ · ] denotes an equivalence class in E / ( G ∩ L )).Determining ˆ E for a given, say, exact dynamical system appears to be as com-plicated as proving (GLM2) for a truly large class of global observables, thoughoccasionally some information can be obtained quickly. We conclude this note bygiving some examples thereof. Boole transformation.
This is the transformation T : R −→ R defined by T ( x ) := x − /x . This map preserves the Lebesgue measure on R , as it is easy to verify,and is exact [A1]. We can use the fact that T is odd to construct a nonconstantequilibrium observable. Set F ( x ) := sign( x ), and g := 1 [ − , . Clearly, for all n ∈ N , F ◦ T n is odd and µ (( F ◦ T n ) g ) = 0, so F ∈ ˆ E , with F = constant, and ρ ( F ) = 0.Evidently, the same reasoning can be applied to any exact map with an oddsymmetry. Translation-invariant expanding maps of R . Take a C bijection Φ : [0 , −→ [ k , k ], with k , k ∈ Z , and Φ ′ >
1, where Φ ′ denotes the derivative of Φ. (Noticethat these conditions imply Φ(0) = k , Φ(1) = k , and k := k − k ≥ T : R −→ R via T | [ j,j +1) ( x ) := Φ( x − j ) + j, (4.2)for all j ∈ Z . By construction T ( x + 1) = T ( x ) + 1, ∀ x ∈ R , and so T is a k -to-1translation-invariant map, in the sense that it commutes with the natural action of Z in R .Suppose that T preserves the Lebesgue measure, which we denote m R . (One caneasily construct a large class of maps of this kind.) It can be proved that any such T is exact [L3]. Now, define I := [0 ,
1) and T I : I −→ I as T I ( x ) := T ( x ) mod 1.Clearly, ( I, B I , T I , m I ), where B I and m I are, respectively, the Borel σ -algebra andthe Lebesgue measure on I , is a probability-preserving dynamical system. It is easyto see that it is exact, and thus mixing.Now consider a Z -periodic, bounded, F : R −→ R . Evidently, ∀ x ∈ I , ∀ n ∈ N , F ◦ T n ( x ) = F ◦ T nI ( x ). Hence, by the mixing of the quotient dynamical system, for nfinite mixing g supported in I ,lim n →∞ m R (( F ◦ T n ) g ) = lim n →∞ m I (( F ◦ T nI ) g )= m I ( F ) m I ( g )= m I ( F ) m R ( g ) . (4.3)By the exactness of T , the above holds for all g ∈ L ( R ). Hence F ∈ ˆ E , with ρ ( F ) = m I ( F ) = m R ( F ).An analogous procedure (using I j := [0 , j ) instead of I ) can be employed to provethat any ( j Z )-periodic, bounded F belongs in ˆ E , with ρ ( F ) = m R ( F ). In [L3] weextend this result to observables that are quasi-periodic w.r.t. any j Z , and more. Random walks . A special case of the above situation occurs when Φ is linear. Theresult is a piecewise linear Markov map that represents a random walk in Z , in thefollowing sense. Denote by ⌊ x ⌋ the maximum integer not exceeding x ∈ R . If aninitial condition x ∈ I is randomly chosen with law m I , then the stochastic process( ⌊ T n ( x ) ⌋ ) n ∈ N is precisely the random walk starting in 0 ∈ Z , with uniform transitionprobabilities for jumps of k , k + 1 , . . . , k − E contains all L ∞ functions suchthat the limit ρ ( F ) := lim M →∞ Z a + Ma − M F ( x ) dx (4.4)exists independently of and uniformly in a ∈ R . In fact, it is proved in [L1,Thm. 4.6(b)] (see also [L2, Thm. 9]) that, if g ∈ L , F ∈ L ∞ ( A ), where A isthe σ -algebra generated by the partition { [ j, j + 1) } j , and the limitlim j →∞ Z q + jq − j F ( x ) dx =: m R ( F ) (4.5)( j ∈ Z ) exists uniformly in q ∈ Z , then m R (( F ◦ T n ) g ) → m R ( F ) m R ( g ), as n → ∞ .Obviously, comparing (4.4) with (4.5), ρ ( F ) = m R ( F ).Now, for a general F , one can take g = 1 [0 , ∈ L ( A ). It is easy to check that P n g is A -measurable too, thuslim n →∞ m R (( F ◦ T n ) g ) = lim n →∞ h E ( F | A ) , P n g i = m R ( E ( F | A )) m R ( g )= ρ ( F ) m R ( g ) , (4.6)which proves our claim.If the random walk has a drift, say a positive drift, then a.e. orbit will convergeto + ∞ . Therefore, any bounded function G that asymptotically shadows any ofthe above observables—meaning lim x → + ∞ ( G ( x ) − F ( x )) = 0, for some F verifying(4.4)—will also belong to ˆ E , with ρ ( G ) = ρ ( F ).4 Marco Lenci
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