Uniformly expanding Markov maps of the real line: exactness and infinite mixing

We give a fairly complete characterization of the exact components of a large class of uniformly expanding Markov maps of $\mathbb{R}$. Using this result, for a class of $\mathbb{Z}$-invariant maps and finite modifications thereof, we prove certain properties of infinite mixing recently introduced by the author.


Introduction
Uniformly expanding Markov maps of the interval represent a paradigm for chaotic dynamical systems.They make up a fairly large class of non-trivial maps, and possess the standard ingredient for chaos, namely, hyperbolicity-insofar as expansivity can be understood as the one-dimensional version of hyperbolicity.On the other hand, they are simple enough to be more or less fully understood via the techniques of the modern theory of dynamical systems; cf. the excellent textbook by Boyarsky and Góra [BG].
In infinite ergodic theory, the analogues of such maps are the uniformly expanding Markov maps of R which preserve an infinite measure.Perhaps not surprisingly, very little is known about them, at least to this author.(The systems studied by Bugiel [B1, B2] are very similar, but they are designed to preserve a finite measure.) In this note we consider a rather large class of uniformly expanding Markov maps of the real line; an example is illustrated in Fig. 1 below.We assume that they preserve the Lebesgue measure, which is no big loss of generality for maps that preserve an absolutely continuous measure with bounded density.We also assume a condition, cf.(A2) below, that rules out periodicity.The remaining assumptions are rather standard and not unduly restrictive.
We are interested in the mixing properties of such dynamical systems.To start with, we consider exactness, which is generally regarded as the strongest form of mixing, and has the advantage of being defined in the same way in both finite and infinite ergodic theory.We prove a theorem (Theorem 2.2) that essentially says that every irreducible component of the map is exact, and gives easy sufficient conditions for exactness.
We then apply the notions of mixing for infinite-measure-preserving dynamical systems recently introduced by the author in [L2].We give these notions, within the present scope, in Section 3 below and refer the reader to [L2, L3, L5] for a more thorough discussion.(The last reference, in particular, uses a more intuitive notation and contains several results that are used in this article.) For this part, we specialize to a much narrower but still nontrivial class of maps.We consider both quasi-lifts of expanding circle maps, i.e., piecewise smooth, translation invariant maps R −→ R, whose quotient on a fundamental domain is an expanding map of the circle (see Fig. 2), and finite modifications thereof, namely, maps that differ form a quasi-lift of an expanding circle map only on a bounded domain (see Fig. 3).In both cases, we prove versions of global-local mixing and global-global mixing.Very loosely, global-local mixing means that any global observable (roughly, a bounded function) and any local observable (an integrable function) decorrelate in time.Global-global mixing means that the same happens for any two global observables.
Of course, we have stronger results for the more specialized class of systems, that is, the quasi-lifts.In particular, we prove a property, called (GLM2) in what follows, that can be recast like this: For any global observable F : R −→ C and any Lebesgue-absolutely continuous probability measure µ, if T n * µ denotes the pushforward of µ via the map T n , then where m(F ) represents, in a sense that is specified below, the average of F over R. Thus, (1.1) can be regarded as a sort of weak convergence of T n * µ, the statistical state of the system at time n, to the "equilibrium state" m, which is independent of the initial condition µ, with the global observables playing the role of test functions.
We are unable to prove this strong property for all finite modifications of quasilifts of circle maps, but we certainly believe it to be true for a large class of such systems.For this reason, we give an example for which a very strong version of (GLM2) can be indeed be shown.As described in Section 4, this example represents a random walk in Z, and in this sense it illustrates a whole family of maps.This is how the paper is organized.In Section 2 we introduce our maps and state their exactness properties.In Section 3 we give our definitions of infinite mixing and apply them to certain subclasses of maps.In Section 4 we give an example of a finite modification of a quasi-lift that verifies a strong form of global-local mixing.Since some proofs are rather cumbersome, we have gathered all the proofs in Section 5, further moving the more standard lemmas to the Appendix.
Acknowledgments.I thank Stefano Isola for a few illuminating discussions and Sara Munday for her help in the preparation of the paper.I also acknowledge the hospitality of the Courant Institute of Mathematical Sciences at New York University, where part of this work was done.

Setup and exactness
We start with a collection of real numbers (a j ) j∈Z , such that, for some c 1 , c 2 > 0 and all j, Let I j := [a j , a j+1 ].We call {I j } j∈Z a partition of R even though formally it is not-the substance of what we discuss in this paper would not change if we made the cleaner yet more cumbersome choice I j := [a j , a j+1 ).
Let us consider T : R −→ R, a Markov map relative to {I j }.More precisely, we assume that there exists κ ∈ Z + such that (A2) T | (a j ,a j+1 ) has a unique extension τ j : I j −→ B j , which is twice differentiable and bijective onto B j := The significance of (A2) is that (safe for the endpoints a j and a j+1 ) T maps I j onto an interval that is made up of at least 2 and at most κ intervals of the Markov partition, including I j .
Let τ j and τ j denote, respectively, the first and second derivatives of τ j , and • ∞ the sup norm in R.There exist constants λ > 1 and η > 0 such that Finally, if m is the Lebesgue measure on R, we require that (A5) T preserves m, i.e., m(T −1 A) = m(A), for any Borel set A ⊂ R.
Observe that (A5) is equivalent to the condition that, for all x ∈ R \ j {a j }, An example of a map satisfying (A1)-(A5) is depicted in Fig. 1.
We now give a couple of definitions that will help us better understand the features of the maps we have just defined.Definition 2.1 The transition matrix associated to the Markov map T is the stochastic matrix P = (p jk ) j,k∈Z , with p jk := m(T −1 I k | I j ).We say that T possesses a certain property pertaining to homogeneous Markov chains if P does.In particular, (a) T is irreducible if the graph whose vertices are the elements of Z and whose bonds are the pairs (j, k) with p jk > 0 is connected.It is easy to see that this occurs if and only if the only T -invariant set that can be written as a union of intervals I j is the whole of R.
(b) T is aperiodic if, ∀j ∈ Z, the greatest common divisor of the numbers n ∈ Z + such that p jk denote the entries of P n .) Notice that (A2) implies p jj > 0, which gives p (n) jj > 0, for all n ≥ 1, and thus aperiodicity.It is important to require that ours maps be aperiodic, because periodicity is a clear obstacle to exactness (see Remark 4.1 below for a detailed example).This is the main result of this section: Theorem 2.2 If B is the Borel σ-algebra of R, the following hold for the dynamical system (R, B, m, T ) defined above: (a) The conservative part C is made up of a countable number of ergodic components X α , each of which is a union of adjacent intervals from {I j } j∈Z and is exact.
(b) The dissipative part D is the disjoint union of at most two ergodic components, X + and X − .Each X ± , if not trivial, has infinite measure, is exact, and is such that, for a.a.x ∈ X ± , lim n→∞ T n (x) = ±∞.
(c) If the dynamical system is not totally dissipative (i.e., m(C) > 0) and T is irreducible, then the system is recurrent and exact.
We can use Theorem 2.2 to find many examples of exact maps of our type.Let us focus on the simplest cases that are not piecewise linear.In what follows, the reference σ-algebra will always be the Borel σ-algebra.

Quasi-lifts of expanding circle maps
Let us consider the case where the elements of the Markov partition have the same size, e.g., I j := [aj, a(j + 1)], for some a > 0, and T acts in the same way on each of them, that is, for x ∈ I j , τ j (x) = τ 0 (x − aj) + aj.This is equivalent to where σ(x) := x + a.In other words, T is translation invariant (by the quantity a).See Fig. 2.
If S a is the circle constructed by identifying the endpoints of [0, a], and T a : S a −→ S a is defined by T a (x) := T (x) mod a, it easy to see that T a is a Lebesguepreserving, uniformly expanding map of the circle, with bounded distortion.It thus possesses all the good mixing properties one expects [BG].T is a sort of lift of S a to R, which we might call quasi-lift.Not that we are interested in the actual lift, which fails to preserve the Lebesgue measure.(Quasi-lifts and similar maps have often been used in nonlinear physics as toy models for normal and anomalous diffusion; see, e.g., [SJ, K, AC, KHK] and references therein.)Proposition 2.3 A quasi-lift of an expanding circle map, as defined above, is irreducible and exact.

Finite modifications of quasi-lifts
Slightly more complex examples than the quasi-lifts of circle maps are the so-called finite modifications of quasi-lifts of circle maps.If T 0 is a quasi-lift as defined earlier, T : R −→ R is a finite modification of T 0 if there exists k ∈ Z + such that T (x) = T 0 (x), for all x ∈ k j=−k I j .We assume that T verifies (A1)-(A5) as well.Constructing maps of this kind is not hard; here's a possibility.Denote τ 0j := T 0 | I j , B 0j := τ 0j (I j ) (compare with the notation at the beginning of Section 2) and set J := {j ∈ Z | B 0j ⊃ I 0 }; notice that, by (A2), either B 0j ⊃ I 0 or B 0j ∩ I 0 consists of at most a point.Now, pick a C 2 function ψ : R −→ R + 0 that is compactly supported in int(I 0 ), the interior of I 0 .For j ∈ J , denote by ϕ 0j : B 0j −→ I j the inverse function of τ 0j , and set ϕ j := ϕ 0j +δ j ψ, which defines a function on B 0j .Here (δ j ) j∈J is a collection of numbers so small, in absolute value, that ϕ j is a monotonic, hence bijective, function B 0j −→ I j .Also, they satisfy j∈J sign(ϕ 0j ) δ j = 0.
T verifies (A1)-(A2) by construction.If the δ j are sufficiently small, (A3)-(A4) Figure 3: A finite modification of a quasi-lift of a circle map, constructed with the procedure given in Section 2.2.are verified as well.As for (A5), for all y ∈ int(I 0 ), j∈Z ϕ j (y) defined (2.4) via (2.3) and the fact that ϕ j and ϕ 0j have the same sign.If y ∈ int(I k ), with k = 0, the same conclusion as (2.4) holds by definition of τ k , namely, ϕ k .In other words, using the notation of (2.1), we have shown that, ∀x ∈ j {a j }, namely, T preserves the Lebesgue measure if and only if T 0 does, which we have assumed.
Coming back to the general case, a useful feature of these maps is that, if T is a finite modification of T 0 , then T n is a finite modification of T n 0 (which is also a quasi-lift of a circle map, by (2.2)).In fact, (A2) shows that, if x ∈ I j , T (x) can land at most κ − 1 intervals away form I j , hence, for all x = k +n(κ−1) j=−k −n(κ−1) I j , T n (x) = T n 0 (x).Finally, a finite modification of a quasi-lift of an expanding circle map need not be irreducible (it is rather easy to think of counterexamples), but, if it is, it is also exact.
Proposition 2.4 If T is a map of the type defined in Section 2.2, and is irreducible, then it is exact.

Infinite mixing
In this section we specify, for the systems at hand, some of the notions of infinite mixing introduced in [L2] (though we use the more recent notation of [L5]).Then we prove some of those properties.
Let us call global observable any function exists uniformly in x 0 (and independently of it, as the notation suggests).Clearly, the class of all global observables forms a linear space, containing, for example, all the functions E γ (x) := e ıγx , (3.2) for γ ∈ R.
Another way to introduce this class, in the general spirit of [L2, L3, L5], would be to define the collection of sets which we call the exhaustive family relative to the partition {I j }.The elements of V , denoted V , play the role of "large boxes in phase space".Since global observables are bounded, it is easy to see that F verifies (3.1) if and only if lim We describe this situation by saying that the average of We also call local observable any function g ∈ L 1 (R; C).For any such g we use the customary notation m(g) := R g dm.Let us consider two (sub)classes G and L of global and local observables, respectively.Relative to G and L, one says that the dynamical system (R, m, T ) is mixing of type (GLM1) if, for all F ∈ G and g ∈ L, with m(g) = 0, It is easy to see that (GLM2) is equivalent to (1.1), and, manifestly, implies (GLM1).As they involve the pairing of a global and a local observable, we say that these are two definitions of global-local mixing.(There exists another definition of global-local mixing, which is a uniform version of (GLM2) and is denoted (GLM3) in [L5].We do not consider it here.) The following is a trivial consequence of a well-known theorem of Lin [Li].
Remark 3.2 One may rephrase the above by saying that an exact dynamical system is (GLM1)-mixing w.r.t.L ∞ and L 1 .This is not formally correct, because L ∞ is not a family of global observables (as it contains functions that do not possess an infinite-volume average), but the meaning is clear so we will use such an expression.
When we consider the "correlation" between two global observables, we study the so-called global-global mixing.We have two definitions for it.The system is called mixing of type (GGM1) if, for all F, G ∈ G, m((F • T n )G) exists for all sufficiently large n, and The above limit, which we call joint infinite-volume and time limit, is defined as The second definition is in essence stronger than the first, as the following proposition shows.
Some of the very maps considered in this paper give a good sense of the relative strength of (GGM2) and (GGM1), as the latter property will be trivially verified while the former will remain an open question; cf.Corollary 3.5.
For an in-depth discussion on the meaning and relevance of the above definitions we refer the reader to [L2].Here we just point out that, in order for them to make sense as indicators of "decorrelation", it must be that, for all F ∈ G and n ∈ N, m(F •T n ) = m(F ).As shown in [L2], this is guaranteed by the following hypothesis: for all n ∈ N, in the sense of the infinite-volume limit, as in (3.4).The above is easily verified for the dynamical systems introduced in Section 2: If V = j=k I j , with −k sufficiently large, it is easy to see that j=k+n(κ−1) Therefore, the numerator of (3.8) is bounded above by (4n(κ − 1) + 2)c 2 , while the denominator is bounded below by ( − k + 1)c 1 ; cf.(A1)-(A2).But the infinitevolume limit here corresponds precisely to the limit − k → +∞ (uniformly in k, ), whence the assertion.
The quasi-lifts of expanding circle maps introduced earlier verify all definitions of infinite mixing, both global-local and global-global, with suitable choices of the global observables.Let us see which.
If ψ is either a global or a local observable, and k ∈ Z + , set By (2.2), this operator commutes with the dynamics, namely (3.13)where the bar denotes closure in the L ∞ -norm.In other words, G 1 is the space of all essentially bounded functions whose aZ-average converges uniformly to an aperiodic function; G 2 is the space generated by the a-quasiperiodic functions.Clearly, To see that all these functions are global observables, we need to verify that every F ∈ G 1 possesses an infinite-volume average m(F ), in the sense of (3.4).Since, in our case, V is of the form [ak, a( + 1)], we see that which, by the hypotheses on F , converges to Examples of elements of G 2 are the functions E γ seen in (3.2).An example of F ∈ G 1 \ G 2 is given by F := j∈Z b j 1 I j , with (b j ) a non-periodic sequence such that {b 2k , b 2k+1 } = {0, 1}, for all k ∈ Z.
We have: Theorem 3.4 A quasi-lift of an expanding circle map, as defined in Section 2.1, is: Proof.See Section 5.
The situation is not as good for finite modifications of quasi-lifts.
Corollary 3.5 If a finite modification of a quasi-lift, cf.Section 2.2, is irreducible, then it is: Proof.See Section 5.

Example: finite modification of a random walk
As seen earlier, the mixing properties of finite modifications of quasi-lifts are harder to prove, in general, than those of quasi-lifts.In particular this holds for the interesting property that we have called (GLM2).However, one expects (GLM2) to hold true for a large class of maps.In this section we present one such case.Even though we pick a specific example, the technique generalizes easily to other maps of the same kind.
We construct a piecewise expanding T : R −→ R which, in a very precise sense that we describe momentarily, represents the (generally inhomogeneous) random walk in Z defined by a transition matrix P = (p jk ) j,k∈Z .(Here p jk ∈ [0, 1] is the probability that the walker jumps from the site j to the site k; thus k p jk = 1, ∀j ∈ Z.) We assume that the walk only admits finite jumps, i.e., there exists κ ∈ Z + such that p jk = 0, for all |k − j| > κ.
For j ∈ Z and k ∈ {j − κ, j − κ + 1, . . ., j + κ}, set I j := [j, j + 1] and with the understanding that, when k = j − κ, the first of the above sums is zero.So {I j } j∈Z is a partition of R and {I jk } j+κ k=j−κ is a partition of I j into intervals of length, respectively, p j,j−κ , p j,j−κ+1 , . . ., p j,j+κ .Notice that, if p jk = 0, I jk reduces to a point.We do not consider such degenerate intervals.Any x ∈ R that is not an endpoint of one of the intervals (4.1), belongs to one, and only one, I jk , with p jk > 0. For such x, define (for all other x, the definition of T (x) is irrelevant).In other words, T maps int(I jk ) affinely onto int(I k ).Observe that m(T −1 I k | I j ) = m(I jk ) = p jk , which shows that P is also the transition matrix of T , in the sense of Definition 2.1.
This explains in what sense T represents the random walk defined by P. In fact, fix j 0 ∈ Z and chose a random x 0 ∈ I j 0 , with respect to m 0 , the Lebesgue measure on I j 0 .Call j 1 the integer such that x 0 ∈ I j 0 ,j 1 (this is unique, save for a negligible set of x 0 ).Then x 1 := T (x 0 ) ∈ I j 1 .Calling j 2 the (almost surely unique) integer such that x 1 ∈ I j 1 ,j 2 ensures that x 2 := T (x 1 ) ∈ I j 2 ; and so on.Let us call (J n ) n∈N the stochastic process defined by J n (x 0 ) := j n .This is the sought random walk because of the Markov property Now, clearly {I j } is a Markov partition for T and verifies (A1).On the other hand, (A2) is generally not true, because τ j is likely to have corners at the endpoints of the intervals I jk .This, however, makes little difference in the proofs, as the lack of regularity is restricted to a countable set of points; cf.Sections 3 and 5. What is more important is the irreducibility of the map, as per Definition 2.1(a).This will be easily checked in the example given below, and is however guaranteed if we make the following assumption: for every j, there exist Notice that, due to the strict inequalities on k 1j , k 2j , this is stronger than (A2).
(A3) holds provided p jk ≤ η −1 , for some η > 1 and all j, k (the lack of regularity of τ j at finitely many points is irrelevant to the L ∞ -norm).(A4) is trivially true.
Before discussing the fundamental assumption (A5), we introduce the Perron-Frobenius operator for T , which is defined by the identity valid for all F ∈ L ∞ and g ∈ L 1 .It is well known that, for a.e.x ∈ R, In our specific case, for all k ∈ Z and x ∈ int(I k ), where τ −1 j is the inverse of the function defined in (A2), which is well defined on x.In view of (2.1) and (4.5), (A5) is verified if and only if P 1 = 1, with 1(x) := 1, i.e., by (4.6), if and only if j p jk = 1 for all k, namely, when the random walk is doubly stochastic; cf.[L1, KLO,L4] and references therein.
Remark 4.1 This family of maps provides easy examples for the importance of the aperiodicity assumption for the exactness of a Markov map.For instance, take P to be the transition matrix of the simple symmetric random walk in Z. Denoting X even := j∈Z I 2j and X odd := j∈Z I 2j+1 , it is clear that T −1 X even = X odd and T −1 X odd = X even .Therefore the tail σ-algebra T contains both X even and X odd , and is not trivial.(It is indeed not hard to show that T = {∅, X even , X odd , R}.) The specific map we consider in this section is the one defined by the transition matrix whose entries are null outside of the shown diagonal strip.Evidently, P is stochastic and symmetric, hence doubly stochastic.Also, its rows (p jk ) k fail to be translations of each other only for j ∈ {−1, 0, 1} (as indicated in (4.7)), which makes T a finite modification of a quasi-lift.More examples of this kind can be constructed using the ideas of [L1, App.A].Given a stochastic vector π := (π j ) j∈Z (this means, π j ∈ [0, 1] and j π j = 1), let us define the local observable Clearly g π is a density w.r.t. the Lebesgue measure, namely, g π ≥ 0 and m(g π ) = 1.
Proposition 4.2 If π is stochastic vector that is symmetric (i.e., π j = π −j , ∀j ∈ Z) and decreasing on N, and g π is its corresponding density via (4.8), then the evolution of g π via the Perron-Frobenius operator is also of the type (4.8).More precisely, P n g π = g π (n) , where π (n) := πP n is the evolution at time n of the initial state π for the random walk described above.Moreover, π (n) is symmetric and decreasing on N.
With this proposition we can prove a very strong instance of (GLM2) for our system.Define This exhaustive family is smaller than the one we have introduced in (3.3), which, in our specific case, reads In a sense, up to inessential variations, V is the smallest collection of sets that make sense as an exhaustive family, because it contains only one increasing sequence of sets that covers the phase space R. Therefore, the class of functions is essentially the largest class of global observables one can imagine for the dynamical system at hand, because m is the infinite-volume average w.r.t.V ; cf.(3.4)-(3.5).
Remark 4.3 It is worthwhile to point out that G is not simply a larger class of global observables than G 1 and G 2 of Section 3. By using V in lieu of V , here, we have changed the notion of infinite-volume average from m, cf.(3.1)-(3.5), to m , cf. (4.9), and therefore extended the very concept of global observable.(Notice that, if m(F ) exists, m (F ) = m(F ).) Proposition 4.4 The map T defined above is irreducible and exact.Also, in addition to the statements of Corollary 3.5, it is (GLM2) relative to G and L 1 .
contradicting both (GGM1) and (GGM2) (recall that, in the latter, the convergence in n is uniform w.r.t. the one in V , and viceversa).
The above says no more and no less than, V is the wrong exhaustive family for the global-global mixing of quasi-lifts, or finite modifications thereof; cf.[L2,Sect. 3].We still expect (GGM1-2) to hold for general classes of global observables in the sense of Section 3, i.e., relative to the exhaustive family V .

Proofs
In this section we give the proofs of all the results stated in the previous sections.Some of more standard lemmas that we use are in turn moved to the Appendix.

By (A1)-(A3), {I
(n) j } j∈Z n is a Markov partition for T n (as always, modulo the endpoints of the intervals) and m(I Given a positive-measure set A and any of its density points x, for n ≥ 1 let n) [x] (in case x belongs to two such elements, being an endpoint of both, we make the convention that is the interval on the right).We have (5.2) For the proof of (a) we posit m(C) > 0 (otherwise there is nothing to prove) and consider any set A ∈ I with m(A) > 0. The m-typical point x ∈ A will be both a density point of A and a recurring point to I[x].This means that there exists a sequence (n k ) such that Since A is invariant (5.3) gives that m(A|I[x]) = 1, which proves that the ergodic decomposition of C is coarser than {I j }.
To prove that each ergodic component X α ⊆ C is a union of adjacent intervals I j , suppose the contrary: there exist j 1 < j 2 < j 3 ∈ Z such that I j 1 , I j 3 ⊆ X α and I j 2 ∩ X α contains at most the endpoints of I j 2 .Now, consider any j < j 2 with I j ⊆ X α .By (A2), T (I j ) contains I j .If it contained I k , for some k > j 2 , then (since T (I j ) is connected) it would contain I j 2 as well, contradicting the definition of j 2 .Thus, X α ∩ (−∞, a j 2 ] is T -invariant, and X α can be further split into ergodic components, contradicting our assumption.Now for the exactness.The hypotheses on T imply that there exists a δ ∈ (0, 1/2) such that, ∀j ∈ Z, if B ⊆ I j with m(B|I j ) > 1 − δ, then m(T B|I j ) > 1/2, entailing that m(T B ∩ B) > 0. (With the help of Section A.1 of the Appendix, we see that a good estimate for δ is c 1 /(2D((κ − 1)c 2 + c 1 )).In fact, in the worst-case scenario, I j is an interval of length c 1 and is mapped onto the union of itself and κ − 1 intervals of length c 2 , cf. (A1)-(A2); T maps I j \ B entirely into I j , expanding it at the maximum rate, which, by Lemma A.1, is smaller than D times the average expansion rate, which is ((κ − 1)c 2 + c 1 )/c 1 .This yields the estimate.)Take any A ∈ T with m(A) > 0 and choose a typical point x as before.By (5.3), there exists n such that m( Thus, Proposition A.5 of the Appendix shows that all the ergodic components are exact. As for (b), let us assume that m(D) > 0 (again, otherwise there would be nothing to prove).Define (5.4)This argument also implies that, for x ∈ X + , lim inf n→∞ T n (x) ≥ R, and, for x ∈ X − , lim sup n→∞ T n (x) ≤ −R.Since R can be arbitrarily large, It remains to prove that X + cannot be split in more than one ergodic component, the argument for X − being analogous.Given A ⊆ D, we say that a set of the type C := j=i I j is A-prevalent if − i + 1 ≥ κ, cf.(A2), and, ∀j = i, . . ., , m(A|I j ) > 1/2.
Assume that A, B ⊂ X + are disjoint, T -invariant and of positive measure.We prove that, for a typical x ∈ A, x n := T n (x) belongs to an A-prevalent set, for all n large enough.In fact, (5.3) and the invariance of A imply that, for n large, m(A|I[x n ]) > 1 − δ , where δ is chosen so small that T κ−1 (A ∩ I[x n ]) has density larger than 1/2 in all the intervals I j that make up the set C = T κ−1 I[x n ]. (Let us explain this point.By (A2), C = j=i I j , for some i < ; we are saying that m(T κ−1 (A ∩ I[x n ])|I j ) > 1/2, ∀j = i, . . ., .)Notice that δ can be chosen to depend only on κ and the other parameters of T , not on x and n.If we show that we have proved that C is an A-prevalent set that contains x n+κ−1 (observe that This would give the assertion made in the previous paragraph, because the above argument holds for all n bigger than a certain value that depends on x.But (5.6) is easily verified: By (A2), (T k I[x n ]) k≥0 is an increasing sequence of sets that are unions of adjacent intervals from I j .The sequence must be strictly increasing, otherwise, for some k, T k I[x n ] would be an invariant set, contradicting that x n+k → +∞, as k → ∞.This implies that C = T κ−1 I[x n ] is made up of at least κ intervals, which is precisely (5.6).So there are infinitely many A-prevalent sets in any right half-line of R. On the other hand, applying the above to a typical y ∈ B, we have that y n := T n (y) belongs to a B-prevalent set, for all large n.But y n → +∞, and the distance between y n and y n−1 , in terms of intervals, is at most κ − 1.This means that, for n big enough, a point y n must fall in an A-prevalent set.But this is a contradiction, since A-prevalent sets and B-prevalent sets cannot overlap.Hence, A and B cannot be disjoint and there is only one ergodic component.
The proof of (c) follows immediately from (a).If m(C) > 0, there is at least one non-trivial ergodic component X α , which is a union of intervals I j .By the irreducibility of T , X α = R.
As to the exactness, if the system has a non-trivial conservative part, Theorem 2.2(c) implies that it is conservative and exact.If the system is dissipative, Theorem 2.2(b) states that there are at most two ergodic components, X + and X − , and each of them is exact.Suppose that neither has measure zero.By (2.2), both components are invariant for the action of σ, thus m(X ± |I j ) is constant in j.But the proof of Theorem 2.2 shows that there exists a (sufficiently large) j such that m(X + |I j ) > 1/2.This, then, holds for all j ∈ Z.The same can be proved for m(X − |I j ).It follows that our assumption was wrong and T is exact.
Proof of Proposition 2.4.We use the notation of Section 2.2.If T has a non-trivial conservative part, then it is conservative and exact by Theorem 2.2(c).So, let us assume that T is dissipative and see which consequences this has on the dynamics of T 0 .
We denote by C 0 the conservative part of T 0 , and by X 0± the sets defined in Theorem 2.2(b), relative to it.The proof of Proposition 2.3 shows that T 0 is exact and R coincides, mod m, with either C 0 , or X 0+ , or X 0− .If first were the case, by ergodicity, the forward T 0 -trajectory of a.e.x ∈ R would intersect B := k j=−k I j .This occurs in particular for x ∈ R \ B, where T 0 and T coincide.So the forward T -trajectory of any such x also intersects B, which contradicts the dissipativity of T (because, by Theorem 2.2(b), if T is dissipative, for any finite-measure B there is a positive-measure set of points whose forward trajectories never see B).
So either X 0+ or X 0− has full measure.Suppose, w.l.g., that it is X 0+ .We want to prove that the same occurs for X + , equivalently, m(X − ) = 0. Assume the contrary and define, for k ∈ N, (5.7) Clearly, X − ⊆ X This yields m(X 0− ) > 0, which is a contradiction.Thus our assumption was wrong and X + = R mod m.Hence, T is exact by Theorem 2.2(b). Q.E.D.
Proof of Theorem 3.4.Assertion (a) comes from Proposition 3.1, because T is exact by Proposition 2.3.
As for (b), fix F ∈ G 1 and k ∈ Z + .Recalling definition (3.11), it is easy to check that m((F , and that T and σ commute and are both minvariant).By definition of G 1 , for every ε > 0, there exists a large enough k such that, uniformly in n, (5.9) Recalling from Section 2.1 the definition of the measure-preserving dynamical system (S a , m, T a ), we see that, since F p is a-periodic, (5.10) as per definition (3.15) (with the slight abuse of notation whereby the projection of F p to S a ∼ = I 0 is still called F p ).
The following lemma is an easy consequence of the exactness of T .
Lemma 5.1 If, for some b ∈ C and ε ≥ 0, the limit holds for some g ∈ L 1 (with m(g) = 0), then it holds for all g ∈ L 1 (with m(g) = 0).
Equations (5.8)-(5.10)show that the lemma can be applied, for any ε > 0, with b = m(F ) and g = 1 [0,ak] , with k depending on ε.Therefore, for all g with m(g) = 0, (5.11) The case m(g) = 0 was already covered when we proved (GLM1).This ends the proof of (b).By simple density arguments, (GGM2) will be verified once the limit (GGM2) is proved for any pair of quasiperiodic observables F, G.More precisely, assume F • σ = e ıaβ F and G • σ = e ıaγ G. Notice that, if β = 0 mod 2π/a, then m(F ) = 0.The analogous implication holds for G.
Set g = a −1 G1 [0,a] .Since V is of the form V = [ak, a( + 1)], let us write (5.12) whence, using (2.2) and the translation-invariance of the Lebesgue measure, (5.13) In the last term above, the factor in front of the integral is bounded by 1 uniformly in k, , namely, in V , while the integral does not depend on it.In fact, the latter term is m((F • T n )g) and, by (GLM2), converges to m(F )m(g), as n → ∞ (since G 2 ⊂ G 1 ).We now have three cases: 1. β = 0 mod 2π/a.In this case m(F ) = 0, therefore (5.13) converges to 0, as n → ∞, uniformly in V .In particular, it converges to 0 in the joint infinitevolume and time limit; cf.(3.6).
2. β = 0 mod 2π/a and γ = 0 mod 2π/a.In this case m(G) = 0 and the factor in front of the integral in (5.13) vanishes when − k → ∞; that is, uniformly in V as m(V ) → ∞, i.e., in the infinite-volume limit.On the other hand, the integral is bounded by F ∞ g 1 , uniformly in n.This implies that, again, in the joint infinite-volume and time limit, (5.13) converges to 0.
3. Both β and γ are 0 mod 2π/a.In this case the factor in front of the integral is identically 1, (5.13) no longer depends on V and, for n → ∞, tends to m(F )m(g) = m(F )m(G), which is the same as the joint infinite-volume and time limit, here.
In all these cases, the limit (GGM2) is verified.In view of Proposition 3.3, (GGM1) will be shown once we have proved that m((F •T n )G) exists for all F, G ∈ G 2 .Once again, since m is a continuous functional in the L ∞ -norm, it is enough to prove the assertion for F, G quasiperiodic.But, in that case, F • T n is quasiperiodic by (2.2), which implies the same for (F • T n )G, which thus has an infinite-volume average.This completes the proof of assertion (c) of Theorem 3.4.
Proof of Corollary 3.5.As in the previous proof, assertion (a) comes from Propositions 2.4 and 3.1.Now for (b).Consider F, G ∈ G 2 and fix n ∈ Z + .Since T n is a finite modification of T n 0 , as emphasized in Section 2.2, the function (F •T n )G differs from (F •T n 0 )G by a compactly supported and bounded function of R.This shows that m((F • T n )G) exists if and only if m((F • T n 0 )G) does, and they are equal.Therefore, Theorem 3.4(c) proves (GGM1) in this case as well. Q.E.D.
Proof of Proposition 4.2.We prove all the assertions for n = 1 and the proposition will follow by induction.For the sake of the notation, let us denote π := π (1) = πP.
Proof of Proposition 4.4.The irreducibility of T is apparent from the expression of the transition matrix P of its associated random walk, see (4.7).The exactness comes from Proposition 2.4.(It is easy to see that T is also conservative, because it represents a finite modification of a recurrent random walk.)Thus, Proposition 4.4 will be proved once the limit (GLM2) is proved for any F ∈ G and some g ∈ L 1 , with m(g) = 0; see Lemma 5.1.We choose g = g π , as defined in (4.8), with π symmetric and decreasing on N (e.g., π j := δ 0j ).Without loss of generality (for (GLM2) is trivial for F = constant), we further assume that m (5.22) Since N is chosen depending on , which in turns depends on ε, (5.19), (5.20) and (5.22) prove the assertion. Q.E.D.

A Appendix: Standard results
In this appendix we collect some standard technical lemmas that are used in the previous section but are not at the core of the proofs.

A.1 Distortion
Lemma A.1 Under the hypotheses of Section 2, there exists D > 1 such that, for all n ≥ 1, j ∈ Z n , x, y ∈ I where z k ∈ (x k ↔ y k ).By (A.1), using (A1) and (A3), we get |x k − y k | ≤ c 2 λ n−k , which, using (A4) as well, gives log Renaming the rightmost term of (A.3) log D yields the assertion. Q.E.D.

A.2 Wandering sets
In this section we (re)prove a few basic results of infinite ergodic theory.(X, µ, T ) denotes a measure-preserving dynamical system with (X, µ) σ-finite and µ(X) = ∞ (T : X −→ X may be invertible or non-invertible).
Let us recall (see, e.g., [A]) that a wandering set for this dynamical system is a (measurable) set W such that the sets {T −n W } ∞ n=0 are disjoint.The main property of W is thus that the forward orbit of any of its points does not intersect W .The dissipative part D of X is a countable disjoint union of wandering sets such that any other wandering set is contained in D mod µ.It can be shown to exist and be unique mod µ.The conservative part C := X \ D is also defined mod µ.Let us fix j ∈ N and x ∈ W j .By the properties of the wandering sets, the orbit (T n x) ∞ n=0 can intersect each A k only once.So, for x to visit A an infinite number of times, it must visit an infinite number of A k , i.e., it must visit B n , for all n ∈ N. Lemma A.4 below shows that the measure of all such x ∈ W j is no larger than µ(B n ), for all n, hence it is zero.We have thus shown that for a.e.x ∈ W j , # {n ≥ 0 | T n x ∈ A} < ∞.Since j is arbitrary, the assertion follows.
Q.E.D. Lemma A.4 Let W be a wandering set and B any measurable set.Then In other words, the measure of the points of W that are in B or will visit it in the future does not exceed the measure of B. The ⊆ inclusion is obvious.As for the opposite inclusion, by thinking recursively in n, one realizes that T −1 B n−1 is the set of all the points in T −n B whose forward orbit up to time n does not intersect W .Therefore, since W is a wandering set, any x ∈ W ∩ T −n B must also belong to T −1 B n−1 , and thus to W n .Now, using (A.6) in (A.5), upon renaming a variable, we obtain ∞ n=0 µ(W ∩ T −n B) ≤ µ(B), (A.7) which implies the sought assertion. Q.E.D.

A.3 The Miernowski-Nogueira exactness criterion
In this section we slightly generalize a criterion by Miernowski and Nogueira [MN] for the exactness of non-singular maps.In other words, under the hypotheses of Proposition A.5, all the non-null ergodic components of the dynamical system (X, A , µ, T ) are exact.The proof that we give below is almost verbatim taken from [MN].

Figure 2 :
Figure 2: An example of a quasi-lift of an expanding circle map.

Remark 4. 5
No form of global-global mixing can hold for T w.r.t.G .This is an instance of a general phenomenon that we have called surface effect in[L2, Sect.3].Very briefly, a counterexample is constructed by choosing, e.g., F = G = Θ, the Heaviside function, which belongs in G , with m(Θ) = 1/2.Since T has bounded dynamics, meaning |T (x) − x| ≤ K, ∀x (in our specific case K < 3), it is clear that Θ • T n (x) = Θ(x), for all |x| > Kn.So, for all n ∈ N, c 2 λ n .Let H := {∅, C, D, R} be the σ-algebra of the Hopf decomposition; I := {A ∈ B | T −1 A = A mod µ} the σ-algebra of the invariant sets; and T := ∞ n=0 T −n B the tail σ-algebra.Clearly, H ⊆ I ⊆ T .
or there would exist R > 0 such that m(D ∩ [−R, R]) > 0 and lim sup n→∞ |T n (x)| < R for a positive measure of x ∈ D. The forward orbit of any such x visits D ∩ [−R, R] an infinite number of times, contradicting Lemma A.3 of the Appendix.Similarly, m(X + ∩ X − ) = 0, because the contrary implies that, for R > κc 2 /2, a positive measure of points x ∈ D visit D ∩ [−R, R] infinitely often in the future (observe that (A1)-(A2) imply that, ∀x ∈ R, |T (x) − x| ≤ κc 2 ).Once again, this is forbidden by Lemma A.3, assuming w.l.g. that m(D ∩ [−R, R]) > 0.

Lemma A. 3
If A ⊂ D and µ(A) < ∞, then, for a.e.x ∈ D, # {n ≥ 0 | T n (x) ∈ A} < ∞.In other words, if A is a finite-measure subset of the dissipative part, almost all of points will visit it at most a finite number of times in the future.Proof.By construction, D = k∈N W k mod µ, where the W k are disjoint wandering sets.Thus, A k := A ∩ W k are also are disjoint wandering sets and A = k∈N A k mod µ.Set B n := k≥n A k .By the hypotheses on A, µ(B n ) → 0, as n → ∞.