Ergodic Functions That are not Almost Periodic Plus L 1 − Mean Zero
aa r X i v : . [ m a t h . A P ] F e b ERGODIC FUNCTIONS THAT ARE NOT ALMOST PERIODIC PLUS L − MEAN ZERO
JEAN SILVA
Abstract.
Ergodic Functions are bounded uniformly continuous (BUC) functions thatare typical realizations of continuous stationary ergodic process. A natural question iswhether such functions are always the sum of an almost periodic with an L − mean zeroBUC function. The paper answers this question presenting a framework that can provideinfinitely many ergodic functions that are not almost periodic plus L − mean zero. Introduction
Let (Ω , A , µ ) be a probability space and T : R n × Ω → Ω a family of mappings (which weshall call dynamical system) with the followings properties:( T ) (Group Property) T (0) = Id , T ( x + y ) = T ( x ) ◦ T ( y ), where Id : Ω → Ω is theidentity mapping.( T ) (Invariance) For every x ∈ R n and every set E ∈ A , we have T ( x ) E ∈ A and µ ( T ( x ) E ) = µ ( E ) . ( T ) (Measurability) For any measurable function f : Ω → R , the function f ( T ( x ) ω )defined on the cartesian product R n × Ω is also measurable.We shall say that the dynamical system T is ergodic if for any measurable function f : Ω → R satisfying f ( T ( x ) ω ) = f ( ω ) for any x ∈ R n and µ − almost everywhere ω ∈ Ω we must have f is constant µ − almost everywhere in Ω. It is well known that this notion of ergodicity isequivalent to the property: If E ∈ A satisfies T ( x ) E = E for all x ∈ R n , then µ ( E ) ∈ { , } .A measurable function F : R n × Ω → R is a stationary ergodic process if, for somemeasurable function f : Ω → R and some ergodic dynamical system T : R n × Ω → Ω, wehave F ( x, ω ) = f ( T ( x ) ω ) . For each fixed ω ∈ Ω, we call f ( T ( x ) ω ) a realization of the process F ( x, ω ) = f ( T ( x ) ω ).In the case where Ω is a compact topological space endowed with a probability measuredefined in the Borel subsets of Ω, and the dynamical system T : R n × Ω → Ω is a continuousmapping, and moreover f : Ω → R is a continuous function, it was proven in [2] that, Mathematics Subject Classification.
Primary: 35B40, 35B35, 74Q10; Secondary: 35L65, 35K55,35B27.
Key words and phrases. ergodic functions, almost-periodic functions, weakly almost periodic functions,algebra with mean value, homogenization. for a.a. fixed ω ∈ Ω, the realization f ( T ( · ) ω ) belongs to an ergodic algebra, a conceptwhose definition we recall subsequently. The validity of this fact for general stationaryergodic process was asserted in [12] without proof and precise hypotheses. Nevertheless,in [9], it is shown that for a general stationary ergodic process, if the family of realizations n f ( T ( · ) ω ); ω ∈ Ω o is equicontinuous, then it is possible to reduce this case to the justmentioned case addressed in [2], thus proving the validity of assertion in [12] also in thismore general case, that is, for almost all fixed ω ∈ Ω, the realization f ( T ( · ) ω ) belongs toan ergodic algebra. The latter is a concept introduced by Zhikov and Krivenko in [18] (seealso [12]). In order to recall its definition, we first need to recall the definition of algebrawith mean value (w.m.v. for short). The latter is a closed linear subspace A of the space ofbounded uniformly continuous function in R n such that: a) A is an algebra of functions; b)if f ∈ A , then f ( · + k ) ∈ A for all k ∈ R n ; c) all elements of A has a mean value, that is, if f ∈ A , the sequence { f ( · /ε ) } ε> converges, in the duality with L ∞ and compactly supportedfunctions, to the constant M ( f ), and, in particular, M ( f ) = lim R →∞ | B ( x , R ) | Z B ( x ,R ) f ( y ) dy, for x ∈ R n , where B ( x , R ) is the open ball centred at x and | B ( x , R ) | denotes its n − dimensional Lebesgue measure. Given an algebra w.m.v. A , we consider the semi-norm[ f ] := M ( f ) / , take the quotient with respect to the equivalence relation f ∼ g ⇐⇒ [ f − g ] = 0, and take the completion of the quotient space and denote it by B , the Besi-covitch space with exponent 2 associated with A . An algebra w.m.v. A is said to be ergodicif whenever f ∈ B satisfies f ( · + k ) = f ( · ) in the sense of B for all k ∈ R n , then f isequivalent in B to a constant.A bounded uniformly continuous function over R n is said to be an ergodic function if itbelongs to some ergodic algebra w.m.v. A . It is worth mentioning that as far as the authorcould verify in the literature, all examples of ergodic functions presented in the literatureare:(1) The continuous periodic functions.(2) The almost periodic functions. Given a continuous function f : R n → R and ǫ > p ∈ R n is a ǫ − almost period of f if (cid:12)(cid:12)(cid:12) f ( x + p ) − f ( x ) (cid:12)(cid:12)(cid:12) < ǫ for all x ∈ R n . We shall denote the set of ǫ − almost periods of f by T ( ǫ, f ). A continuous functions f : R n → R is said to be almost periodic if for any ǫ >
0, the set T ( ǫ, f ) is relativelydense in R n , that is, there exists l = l ( ǫ ) > l contains at least one ǫ − period. The set of almost periodic functions on R n is denotedby AP( R n ). The following important characterizations of an almost periodic functionare classical and can be found in [4, 13]. A continuous function f is in AP( R n ) ⇐⇒ the family of its translates { f ( · + t ) : t ∈ R n } is pre-compact in the norm of sup ⇐⇒ RGODIC FUNCTIONS 3 f may be uniformly approximated by finite linear combinations of functions in theset { sin( λ · x ) , cos( λ · x ) } λ ∈ R n .(3) The continuous functions with limit at infinite.(4) The Fourier-Stieltjes functions, which are the uniform approximations of the boundedcontinuous functions f which satisfy f ( x ) = R R n e ix · y dµ ( y ), for some complex-valueRadon measure in R n (cf. also [3]). This space is denoted by FS( R n ).(5) The weakly almost periodic functions, whose space is denoted by W AP( R n ), is thespace of the bounded continuous functions f in R n , such that de family { f ( · + t ) : t ∈ R n } is pre-compact in the weak topology of C ( R n )(the space of the boundedcontinuous functions). The main properties of this space was studied by Erberlein in[7, 8]. In [15], Rudin proved that the inclusion FS( R n ) ⊆ WAP( R n ) is strict, showingan example of a weakly almost periodic function that cannot be approximated in thesup norm by Fourier-Stieltjes transforms.(6) The weakly* almost periodic functions over R n . In [8], Eberlein established thefollowing important decomposition for functions f ∈ W AP( R n ), which allows towrite any function as f = f ap + f , where f ap ∈ AP( R n ) and M ( | f | ) = 0. This property satisfied by the weakly almostperiodic functions served as defining property to a natural broader class of functionsconsidered by H. Frid in [9]. This class is denoted by W ∗ AP( R n ) and it is defined asthe algebraic sum W ∗ AP( R n ) := AP( R n ) + N ( R n ) , where N ( R n ) is the subspace of the bounded uniformly continuous function f suchthat M ( | f | ) = 0. Thus, it is clear that W AP( R n ) ⊆ W ∗ AP( R n ).Therefore, all known examples of ergodic functions over R n are weakly* almost periodicfunctions. Therefore, an important question is whether there exist ergodic functions whichare not weakly* almost periodic functions. In this article, we shall give an answer to thisquestion by constructing a probability compact space and a continuous function such thatthe majority of its realizations are ergodic functions that are beyond of the weakly* almostperiodic settings. This will be done through the introduction of a family of BUC − functionsΩ that is equivalent, topologically and measure theoretically to Ω × R n / Z n , where Ω := {− , } Z n , and a dynamical system over Ω that is equivalent to a natural shift mapping onΩ × R n / Z n . This shows not only the existence of ergodic functions with different behaviorfrom those discovered by Eberlein in [7] but also brings out a curious statistical evidence:The weakly almost periodic oscillations may not be so often. From the practical point ofview, what happens is that in many situations, the stochastic homogenization problems ofdifferential type, can be reduced to a homogenization problem of the differential operatorswhose coefficients are ergodic functions. Thus, the afore-knowledge of the nature of the ”self-averaging” behavior of these ergodic functions can dictate the complexity of the solution.For example, if you know that almost all realizations are of weakly* almost periodic type, an JEAN SILVA application of the Birkhoff’s theorem allows us to conclude that in fact almost all realizationsare of almost periodic type. The reduction of the stationary ergodic settings to the almostperiodic one can simplify a lot the solution of the homogenization’s problem(compare forinstance the paper [9] with [5] and [11] with [17]).2.
The construction of the Example
We construct the example in dimension 1 in order to simplify notations. However, as thereader will see, the construction may be easily extended to the multi-dimensional case.Denote the set of integers numbers by Z . On the compact set {− , } , we define anelementary radon probability measure λ = λ q as follows: The measure of the one-point set {− } is equal to q , and the measure of the set { } is equal to 1 − q . Here, we consider0 < q <
1. Let Ω be the space productΩ := {− , } Z , that is a compact space by Tychonoff’s theorem. The elements of Ω are sequences whichassume values in the set {− , } . Denote by ν = ν q the product of the elementary measures λ and the function τ : Z n × Ω → Ω by τ ( k, x ) := { x j + k } j ∈ Z (The shift operator). Forsimplicity, we shall use τ k x to denote τ ( k, x ). It is well known that the function τ is anergodic discrete dynamical system over the compact probability space (Ω , ν ) (see, e.g., [14]pag. 101) The followings concepts will be useful for us here. • A real sequence { x k } k ∈ Z is said to be a periodic sequence if there exists a integer p > x k + p = x k , for all k ∈ Z . • A real sequence { x k } k ∈ Z is said to be an almost periodic sequence if to any ǫ > k ( ǫ ), such that among any k consecutive integersthere exists an integer p (called ǫ − almost period) with the property | x k + p − x k | < ǫ, for all k ∈ Z . The next lemma states the fact the almost periodic elements of Ω are periodic. Lemma 2.1.
Every almost periodic sequence x ∈ Ω is a periodic sequence.Proof. First, suppose that x ∈ Ω is an almost periodic element. Then, let ǫ ∈ (0 ,
1) andchoose a ǫ − almost period p >
0. Hence, by definition | x k + p − x k | < ǫ, for all k ∈ Z . Now, we have only two possibilities: • x k + p = x k for all k ∈ Z . • x k + p = x k for some k ∈ Z . Since our sequence assumes its values only in the set {− , } , we must have 2 = | x k + p − x k | < ǫ , which is a contradiction with the choiseof ǫ . Thus, only the first possibility must happen. This proves our lemma. (cid:3) RGODIC FUNCTIONS 5
Now, take ϕ ∈ C ( R ) such that • supp ϕ ⊆ ( − / , / • ≤ ϕ ( x ) ≤ x ∈ R . • ϕ ( x ) = 1 if and only if x = 0.The set Ω can be naturally associated with the set ( Λ x : R → R ; Λ x ( · ) := X m ∈ Z x m ϕ ( · − m ) , x ∈ Ω ) . The following lemma will be important for our proposals.
Lemma 2.2.
There exists a bijection between the set
Ω := (cid:26) Λ x ( · + δ ); ( x , δ ) ∈ Ω × R (cid:27) and the set Ω × R / Z .Proof.
1. First, it is important to establish the following remark: If Λ x ( t + δ ) = Λ y ( t + δ )for all t ∈ R , then, the properties below hold: • τ ⌊ δ ⌋ x = τ ⌊ δ ⌋ y . • δ − ⌊ δ ⌋ = δ − ⌊ δ ⌋ ,where ⌊ x ⌋ denotes the unique number in Z such that x − ⌊ x ⌋ ∈ [0 , − dimensional torus R / Z with the interval [0 , δ − δ ∈ Z .Indeed, define θ := δ − δ − ⌊ δ − δ ⌋ ∈ [0 , x ( t + δ ) = Λ y ( t + δ ) for all t ∈ R ,changing t by t − δ and using the definition of Λ x and Λ y , we get: X m ∈ Z x m ϕ ( t − m ) = X m ∈ Z y m ϕ ( t − m + δ − δ ) = X m ∈ Z y m ϕ ( t − m + θ + ⌊ δ − δ ⌋ )= X m ∈ Z ( τ ⌊ δ − δ ⌋ y ) m ϕ ( t − m + θ ) , for all t ∈ R . Taking t = 0, we have that there exists an unique m ∈ Z such that x = X m ∈ Z x m ϕ ( − m ) = X m ∈ Z ( τ ⌊ δ − δ ⌋ y ) m ϕ ( − m + θ )= ( τ ⌊ δ − δ ⌋ y ) m ϕ ( θ − m ) . Taking into account that x , ( τ ⌊ δ − δ ⌋ y ) m ∈ {− , } , we must have ϕ ( θ − m ) = 1. Thus,by the conditions on the function ϕ , it implies that m = θ . Thus, θ ∈ [0 , ∩ Z which givesthat θ = 0 and the claim is proved. From the claim, we have that δ − ⌊ δ ⌋ = θ = δ − ⌊ δ ⌋ .Moreover, Λ x ( t + δ ) = X m ∈ Z x m ϕ ( t + δ − m ) = X m ∈ Z n x m ϕ ( t + θ + ⌊ δ ⌋ − m )= X m ∈ Z ( τ ⌊ δ ⌋ x ) m ϕ ( t + θ − m ) , for all t ∈ R , (2.1) JEAN SILVA and the same holds if we change x by y and δ by δ . Taking into account that Λ x ( t + δ ) =Λ y ( t + δ ) for all t ∈ R , we get: X m ∈ Z ( τ ⌊ δ ⌋ x ) m ϕ ( t + θ − m ) = X m ∈ Z ( τ ⌊ δ ⌋ x ) m ϕ ( t + θ − m ) , for all t ∈ R . Therefore, changing t by t − θ in the above equality, we have( τ ⌊ δ ⌋ x ) m = ( τ ⌊ δ ⌋ y ) m , for all m ∈ Z , which establishes the remark.2. Define H : Ω → Ω × [0 ,
1) by H (Λ x ( · + δ )) = (cid:0) τ ⌊ δ ⌋ x , δ − ⌊ δ ⌋ (cid:1) . By the step 1, the function H is well defined and is onto.3. We claim that the function H is one-to-one. In order to show this, let x , y ∈ Ω and δ , δ ∈ R be such that H (Λ x ( · + δ )) = H (Λ y ( · + δ )) . By the definition of H , we have: • τ ⌊ δ ⌋ x = τ ⌊ δ ⌋ y . • δ − ⌊ δ ⌋ = δ − ⌊ δ ⌋ = θ .Now, using these relations in definition of Λ x and the relation (2.1), we have the followingequality: Λ x ( t + δ ) = X m ∈ Z ( τ ⌊ δ ⌋ x ) m ϕ ( t + θ − m )= X m ∈ Z ( τ ⌊ δ ⌋ y ) m ϕ ( t + θ − m ) = Λ y ( t + δ ) , for all t ∈ R . (cid:3) By Lemma 2.2, the set Ω inherits the probabilistic and the topologicals features of thespace Ω × [0 ,
1) in a natural way. Let µ = µ q be the probability measure on Ω associatedto the measure on Ω × [0 ,
1) defined as the product of measure ν on Ω and the Lebesguemeasure on [0 , the space Ω × [0 , Lemma 2.3.
Let (cid:16) Ω , µ (cid:17) be the probability space constructed above and T : R × Ω → Ω bethe shift operator defined on Ω , that is, T ( z, ω ) := ω ( · + z ) . Then, we have the followingsproperties: (i) T ( z ) = H − ◦ S ( z ) ◦ H, where S ( z ) : Ω → Ω is given by (2.2) S ( z )( x , θ ) := (cid:16) τ ⌊ z + θ ⌋ x , z + θ − ⌊ z + θ ⌋ (cid:17) ( z ∈ R ) . RGODIC FUNCTIONS 7 (ii)
The function S ( z ) : Ω → Ω defined by (2.2) is an ergodic dynamical system withrespect to the product of measure ν on Ω and the Lebesgue measure on [0 , . (iii) The mapping T : R × Ω → Ω is an ergodic dynamical system.Proof.
1. Writing ω ( · ) = Λ x ( · + δ ) for some x ∈ Ω and some δ ∈ R , we have by definitionof H in the proof of the lemma 2.2(see step 2): H ( T ( z ) ω ) = H ( ω ( · + z )) = H (Λ x ( · + δ + z )) = (cid:16) τ ⌊ z + δ ⌋ x , z + δ − ⌊ z + δ ⌋ (cid:17) . On the other hand, S ( z ) ( H ( ω )) = S ( z ) (cid:0) τ ⌊ δ ⌋ x , δ − ⌊ δ ⌋ (cid:1) = (cid:16) τ ⌊ z + δ −⌊ δ ⌋⌋ + ⌊ δ ⌋ x , z + δ − ⌊ δ ⌋ − ⌊ z + δ − ⌊ δ ⌋⌋ (cid:17) = (cid:16) τ ⌊ z + δ ⌋ x , z + δ − ⌊ z + δ ⌋ (cid:17) = H ( T ( z ) ω ) , where we have used the property ⌊ t + k ⌋ = ⌊ t ⌋ + k for all t ∈ R and k ∈ Z . This proves theitem ( i ).2. By the item ( i ), we have that S ( z ) = H ◦ T ( z ) ◦ H − . Since T ( z ) is a shift operator,then it is clear that T (0) = I Ω and T ( z + z ) = T ( z ) ◦ T ( z )(group property). Hence, thesame happens with the function S ( z ), that is, S ( z + z ) = S ( z ) ◦ S ( z ).3. If E ⊆ Ω is a Borel set, then same occurs with S ( z )( E ) for any z ∈ R . This canbe seen by noting that the collection of all sets having this property forms a σ − algebra.Moreover, taking into account that the measurable sets of Ω are sent to measurable sets bythe transformation τ k for any k ∈ Z and the same happens with the space [0 ,
1) with respectto transformation · + z − ⌊· + z ⌋ for any z ∈ R , we can deduce that the rectangles are alsosent to measurable sets of Ω by the mapping S ( z ). Therefore, S ( z )( E ) is a Borel set if E isa Borel set for any z ∈ R . Let P be the measure defined as the product of the measure ν onΩ with the Lebesgue measure dθ on [0 , P ( S ( z )( E )) = P ( E ) forany Borel set E ⊆ Ω and any z ∈ R . Take a Borel set E ⊆ Ω and z ∈ R . Let 1 E ( · ) be thecharacteristic function of the set E . Since 1 E (cid:16) S ( − z )( x , θ ) (cid:17) = 1 S ( z )( E ) ( x , θ ) and S ( − z )( E ) isa Borel set, then the function ( x , θ ) E (cid:16) S ( z )( x , θ ) (cid:17) is measurable. Hence, by the Fubini JEAN SILVA
Theorem P (cid:16) S ( z )( E ) (cid:17) = Z Ω × [0 , S ( z )( E ) ( x , θ ) d P ( x , θ ) = Z Ω × [0 , E (cid:16) S ( − z )( x , θ ) (cid:17) d P ( x , θ )= Z Ω × [0 , E (cid:16) τ ⌊ θ − z ⌋ x , θ − z − ⌊ θ − z ⌋ (cid:17) d P ( x , θ )= Z [0 , ( Z Ω E (cid:16) τ ⌊ θ − z ⌋ x , θ − z − ⌊ θ − z ⌋ (cid:17) dν ( x ) ) dθ = Z [0 , ( Z Ω E (cid:16) x , θ − z − ⌊ θ − z ⌋ (cid:17) dν ( x ) ) dθ = Z Ω × [0 , E ( x , θ ) dν ( x ) dθ = P ( E ) .
4. We claim that the function ( z, x , θ ) f ( S ( z )( x , θ )) defined on the cartesian product R × Ω is measurable for any measurable function f : Ω → R . We can prove this claimreasoning as follows: Using an approximation argument, it is enough to consider f ( · ) = 1 E ( · ),where E is a measurable set of Ω and 1 E ( · ) is the characteristic function of the set E .Then, we observe that the class of all measurable sets E ⊆ Ω such that the function( z, x , θ ) E ( S ( z )( x , θ )) is measurable on R × Ω is a σ − algebra. Finally, we will finishthe proof of the claim by showing that the rectangles of Ω belongs to this σ − algebra. Forthis, write E = E × E , where E ⊆ Ω and E ⊆ [0 ,
1) are measurable sets. Note that1 E ( S ( z )( x , θ )) = 1 E (cid:0) τ ⌊ θ + z ⌋ x (cid:1) E ( z + θ − ⌊ z + θ ⌋ ) = f ( z, θ, x ) f ( z, θ, x ) , with obvious notations for f , f .Since the values of the function f are in the set { , } , it is sufficient to verify that f − (1) is a measurable set in R × [0 , × Ω . Note that ( z, θ, x ) ∈ f − (1) if and only if x ∈ τ −⌊ θ + z ⌋ ( E ). Furthermore, the value of ⌊ θ + z ⌋ can be ⌊ z ⌋ or ⌊ z ⌋ + 1 depending on thecase if the sum of the fractional part of z with θ is smaller or grater than 1. Due to this, wepartitioned the set R × [0 ,
1) as follows: R × [0 ,
1) = (cid:16) [ k ∈ Z V + ( k, (cid:17) [ (cid:16) [ k ∈ Z V c + ( k, (cid:17) , where V := { ( z, θ ) ∈ [0 , ; z + θ < } and V c := { ( z, θ ) ∈ [0 , ; z + θ ≥ } . Thus, takinginto account this decomposition, it is easily seen that f − (1) = (cid:16) [ k ∈ Z { V + ( k, } × τ − k ( E ) (cid:17) [ (cid:16) [ k ∈ Z { V c + ( k, } × τ − k − ( E ) (cid:17) , which provides the measurability of the function f . Similar procedures can be made for thefunction f . This completes the proof of the claim.5. In this step, we shall show that the dynamical system { S ( z ) } z ∈ R is ergodic. Let f : Ω → R be an invariant function, that is, f ( S ( z )( x , θ )) = f ( x , θ ) for all z ∈ R and for RGODIC FUNCTIONS 9 P− almost everywhere ( x , θ ) ∈ Ω . By definition of S ( z ), we have f (cid:0) τ ⌊ z + θ ⌋ x , z + θ − ⌊ z + θ ⌋ (cid:1) = f ( x , θ ) , P− almost everywhere ( x , θ ) and all z ∈ R . Taking z = k , we obtain f ( τ k x , θ ) = f ( x , θ ) , P− almost everywhere ( x , θ ) and all k ∈ Z . Since the mapping { τ } k ∈ Z is ergodic, we deduce that the function f does not depend of thefirst variable. Using this in the second equation above, we have f ( z + θ − ⌊ z + θ ⌋ ) = f ( θ ) , for all z ∈ R and almost everywhere θ ∈ [0 , . As the mapping ( z, θ ) z + θ − ⌊ z + θ ⌋ ∈ [0 ,
1) is ergodic, we see that the function f alsois independent of the second variable. Thus, f is equivalent to a constant. The item ( iii ) isa direct consequence of the itens ( i ) and ( ii ). Hence, we finish the proof of the lemma. (cid:3) To sum up, the elements of the space Ω are uniformly continuous functions ω defined in R and the shift operator T ( z, ω ) := ω ( · + z ) is an ergodic dynamical system by the Lemma2.3. Therefore, by the arguments in the proof of the Theorem 3 . f : Ω → R , for a.a. ω ∈ Ω, f ( T ( · ) ω ) belongs to an ergodic algebra.Defining the function f : Ω → R by f ( ω ) = ω (0), it is easy to see that f is continous on Ω andits realization by the dynamical system T satisfies f ( T ( · ) ω ) = ω ( · ). Hence, we conclude thatalmost all elements of Ω are ergodic functions. Therefore, at this point, a natural questionarise: What is the amount of the functions in the probability space Ω that are periodic,almost-periodic, weakly almost periodic or more generally almost periodic plus L − meanzero? The aim of the next lemma is to reckon the amount of the periodic functions in Ω. Lemma 2.4.
The µ -measure of the set (cid:26) ω ∈ Ω; ω ( · ) is a periodic function (cid:27) is null.Proof.
1. Let ω ∈ Ω be a periodic function. Thus, there exists a sequence x ∈ Ω , δ ∈ R and a number p ∈ R + (we can assume that p >
1) such that ω ( · ) = Λ x ( · + δ ) andΛ x ( · + k p + δ ) = Λ x ( · + δ ) , on R and for all k ∈ Z . Due to the identification given by lemma 2.2, we must have: • τ ⌊ p + δ ⌋ x = τ ⌊ δ ⌋ x . • p + δ − ⌊ p + δ ⌋ = δ − ⌊ δ ⌋ .Hence, from the second relation, we deduce that p ∈ Z and from the first we get τ p (cid:0) τ ⌊ δ ⌋ x (cid:1) = τ ⌊ δ ⌋ x , that is, the sequence τ ⌊ δ ⌋ x is also periodic with period [0 , p ) ∩ Z . Thus, (cid:26) ω ∈ Ω; ω ( · ) is a periodic function (cid:27) = Ω Per0 × [0 , , where Ω Per0 := { x ∈ Ω ; x is a periodic sequence } . Then, it is enough to prove that the setΩ Per0 has null measure.2. We claim that the set Ω
Per0 is at most countable. Indeed, observe thatΩ
Per0 = ∪ p ∈ Z + (cid:8) x ∈ Ω ; τ p x = x (cid:9) . Moreover, given p ∈ Z + define C := {− , } Z ∩ [0 ,p ) , be the set of finite sequence α = ( α m ) with m ∈ Z ∩ [0 , p ) which assumes its values in theset {− , } . Hence, we can note that (cid:8) x ∈ Ω ; τ p x = x , (cid:9) = ∪ α ∈C (cid:8) x α (cid:9) , where x α ∈ Ω is such that x αm = α m p . Since ♯ C = 2 p , the claim is verified.3. The proof of the lemma is completed by noting that the measure ν attributes valuezero to any point in Ω , which implies that any countable set in Ω has ν -measure zero. (cid:3) In the next Lemma, we analyze the amount of elements in the set Ω that are almost-periodic functions.
Lemma 2.5.
The µ -measure of the set (cid:26) ω ∈ Ω; ω ( · ) is an almost-periodic function (cid:27) is null.Proof. First, remember that a function ω ∈ C ( R ) is said to be an almost-periodic functionif only if the set (cid:8) ω ( · + T ) (cid:9) T ∈ R is strongly pre-compact in C ( R ). Let ω ∈ Ω be an almost-periodic function. By definition of Ω, there exist a sequence x ∈ Ω and δ ∈ R such that ω ( · ) = Λ x ( · + δ ) = X m ∈ Z x m ϕ ( · + δ − m ) . Since ω is an almost-periodic function, the set (cid:8) ω ( · + k ) (cid:9) k ∈ Z is strongly pre-compact in C ( R ).Thus, there exists a subsequence { k j } j ≥ such that the sequence o functions { ω ( · + k j ) } j ≥ converges uniformly in R as j → ∞ . Since ω ( · + k j ) = X m ∈ Z x m + k j ϕ ( · + δ − m ) , we have { τ k j x } j ≥ = { ω ( · − δ + k j ) } j ≥ converges uniformly in Z . Thus, the sequence x is an almost-periodic sequence. By the Lemma 2.1, it follows that it must be a periodicsequence. Taking into account the ν -negligibleness of the set of periodic sequence in Ω (seethe previous lemma), we conclude the proof of the lemma. (cid:3) In [9], the following theorem was established as the crucial point in the proof of the maintheorem therein.
RGODIC FUNCTIONS 11
Theorem 2.1 (see [9]) . Let (cid:16) E , P (cid:17) be a probability space and T : R n × E → E an ergodicdynamical system. Let γ : E → R be a measurable function and define F ( · , ω ) := γ ( T ( · ) ω ) .Suppose that we have the following: (F0) The family n F ( · , ω ); ω ∈ E o is equicontinuous. (F1) For P − a.a. ω ∈ E , F ( · , ω ) ∈ W ∗ AP( R n ) . Then, for P − a.a. ω ∈ E , F ( · , ω ) ∈ AP( R n ) .Sketch of the proof. The first and crucial step of the proof is to find a suitable way toextract the almost periodic component from F ( · , ω ), for each ω ∈ E , that is, since F ( · , ω ) = F ap ( · , ω ) + F N ( · , ω ), with F ap ( · , ω ) ∈ AP( R n ) and F N ( · , ω ) ∈ N ( R n ) for a.a. ω ∈ E , we needto devise a way to obtain F ap ( · , ω ) from F ( · , ω ) such that F ap : R n × E → R is a stationaryergodic process with F ap ( y, ω ) = ˜ γ ( T ( y ) ω ), where ˜ γ ( ω ) = F ap (0 , ω ). The way to process thisextraction of that almost periodic component is by using the approximation of the identity φ α which is known to exist, both from classical Bochner-Fej´er polynomials(see, e.g.,[4]) andfrom the fact that the Bohr compact is a topological group(see, e.g., [10]) for which theexistence of such approximation is known (see [10]) and it is a generalized sequence, or net,in AP( R n ). Using the approximation of the identity we have F ( · , ω ) ∗ φ α = F ap ( · , ω ) ∗ φ α ,for a.a. ω ∈ E , where ∗ is the convolution in the L − mean, and the equation follows fromdefinition of N ( R n ) which implies that F N ( · , ω ) ∗ φ α = 0, for a.a. ω ∈ E . Now, the fact that φ α is a net is an inconvenience to be overcome by reducing φ α to a sequence { φ j } j ≥ whichserves as well as an approximation of the unity for the family { F ap ( · , ω ); ω ∈ E ∗ } , where E ∗ ⊆ E and P ( E ∗ ) = 1. The way to achieve this reduction of φ α to a sequence φ j in [9] was tointroduce a topology in E as a dense subset of a separable compact space so that the family { F ( x, · ); x ∈ Q n } generates the topology given to E . This allows us to take a countable densesubset D ⊆ E and then we consider the separable closed subalgebra A ∗ of AP( R n ) generatedby the unit and { F ap ( · , ω ); ω ∈ D } , for which we may obtain from φ α from φ α a sequence φ j which is an approximation of the identity for the whole family { F ap ( · , ω ); ω ∈ E ∗ } . Indeed,one can consider the compact space associated with A ∗ by stone’s theorem(see, e.g., [6]),which is then, after passing to a quotient space if necessary, a group for which there is anapproximate identity sequence φ n that may be seen as a subsequence of φ α . The final stepis to prove that F ( y, ω ) = F ap ( y, ω ) for a.a. ω ∈ E and all y ∈ R n , which follows from thedecomposition of F ( · , ω ) = F ap ( · , ω ) + F N ( · , ω ) and Birkhoff theorem, namely, that for each y ∈ R n , we have, denoting M ( g ) the mean value of g , when g : R n → R possesses meanvalue, Z E | γ ( T ( y ) ω ) − ˜ γ ( T ( y ) ω ) | d P ( ω ) = Z E | γ ( ω ) − ˜ γ ( ω ) | d P ( ω )= M (cid:16) | γ ( T ( · ) ω ∗ ) − ˜ γ ( T ( · ) ω ∗ ) | (cid:17) = 0 , for P − a.a. ω ∗ ∈ E , The author thanks H. Frid for providing him the sketch of the proof presented here. by the invariance of P with respect to T ( y ), Birkhoff’s relation and by the fact that F ( · , ω ) ∈W ∗ AP( R n ). (cid:3) Lemma 2.6.
Define F : R × Ω → R by F ( x, ω ) := f ( T ( x ) ω ) , where the function f ( ω ) = ω (0) and the dynamical system is such that T ( x ) ω = ω ( · + x ) . Then, the following property holds: lim s → sup | x − z |
1. Let ϕ ∈ C c ( R ) be the function used in definition of the set Ω. Remember that bydefinition of the set Ω, given ω ∈ Ω there exists a sequence { x m } m ∈ Z ⊆ {− , } and δ ∈ R such that F ( · , ω ) = f ( T ( · ) ω ) = ω ( · ) = X m ∈ Z x m ϕ ( · + δ − m ) . Since ϕ is uniformly continuous, given ǫ > t > | x − z | < t ⇒ | ϕ ( x ) − ϕ ( z ) | < ǫ. Now, define t := 1 / ϕ, ∂ ( − / , / s := min { t , t } . Let x, z ∈ R besuch that | x − z | < s . Since R = ∪ m ∈ Z ( − / , /
2] + m , we have two possibilities: • There exists m ∈ Z such that x + δ, z + δ ∈ ( − / , /
2] + m . In this case, F ( x, ω ) = x m ϕ ( x + δ − m ) and F ( z, ω ) = x m ϕ ( z + δ − m ), which implies | F ( x, ω ) − F ( z, ω ) | < ǫ. • There exists m ∈ Z and l ∈ Z such that x + δ ∈ ( − / , /
2] + m , z + δ ∈ ( − / , /
2] + l and x + δ − m , z + δ − l / ∈ supp ϕ . Therefore, F ( x, ω ) = 0 = F ( z, ω ) = 0.In any case, | x − z | < s ⇒ | F ( x, ω ) − F ( z, ω ) | < ǫ for all ω ∈ Ω . (cid:3) Now, we are ready to prove the main result of this paper.
Theorem 2.2.
Let (cid:16) Ω , µ (cid:17) ⊆ BUC( R ) be the probability space introduced in the Lemma 2.3.Then, for µ − a.a. ω ∈ Ω , ω ( · ) / ∈ W ∗ AP( R ) . Proof.
1. Let (cid:16) Ω , µ (cid:17) be the probability space and T : R × Ω → Ω be the dynamical systemsconsidered in the Lemma 2.3. Define F := n ω ∈ Ω; ω ( · ) ∈ W ∗ AP( R ) o . Our aim is to showthat µ ( F ) = 0. But, the lack of measurability of the set F is a problem. We overcome thislack of measurability of the set F by showing the existence of an invariant and measurableset F + such that F ⊆ F + . Then, we use the Theorem 2.1 to conclude that µ ( F + ) = 0 andso finish the proof of the theorem. RGODIC FUNCTIONS 13
2. Let E ⊆ Ω(not necessarily measurable) be such that T ( x ) E = E for all x ∈ R . Due tothis invariance and the identification given by the Lemma 2.3 item ( i ), we have that H ( E )is invariant by the dynamical system S : R × Ω → Ω . This implies that we can find aset C ⊆ Ω (not necessarily measurable) such that H ( E ) = C × [0 , C ⊆ τ k ( C ) for all k ∈ Z , where { τ k } k ∈ Z is the shift operator acting in Ω . The claimcan be proved by reasoning in the following way: a sequence x ∈ C . Due to the invarianceof the set C × [0 ,
1) by the dynamical system S , we must have ( x , ∈ S ( k ) (cid:16) C × [0 , (cid:17) .Thus, there exists a sequence y ∈ C and δ ∈ [0 ,
1) such that( x ,
0) = S ( k )( y , δ ) = (cid:16) τ ⌊ k + δ ⌋ y , k + δ − ⌊ k + δ ⌋ (cid:17) = (cid:16) τ k y , δ (cid:17) , which gives that x = τ k y ∈ τ k ( C ). This proves the claim. Taking into account the arbi-trariness of k ∈ Z in the claim, we have C ⊆ τ − k ( C ). As a consequence, τ k ( C ) = C forall integer k .3. Now, observe that there exists a measurable set C ⊆ Ω such that C ⊆ C and ν ( C ) = inf n ν ( C ); C ⊆ Ω is measurable and C ⊆ C o . Since the set C is τ − invariant, we have that C = τ k ( C ) ⊆ τ k ( C ) for all integer k .Hence, C ⊆ ∩ k ∈ Z τ k ( C ) =: C +0 . Moreover, we can see that C +0 is an τ − invariant set and ν ( C +0 ) = ν ( C ). Therefore, the set (cid:16) C +0 , [0 , (cid:17) is invariant by the dynamical system S , thatis, S ( z ) (cid:16) C +0 , [0 , (cid:17) = (cid:16) C +0 , [0 , (cid:17) , for all z ∈ R . Consequently, we have the existence of a measurable set E + ⊆ Ω such that(P1) E ⊆ E + := H − (cid:16) C +0 , [0 , (cid:17) .(P2) T ( z )( E + ) = E + for all z ∈ R .(P3) µ ( E + ) = inf n µ ( C ); C ⊆ Ω is measurable and E ⊆ C o . Considering the ergodicity of the dynamical system T , we must have µ ( E + ) ∈ { , } .4. Let F be as in the step 1. Since the set F is invariant by the dynamical system T , wecan apply the step 3 for E = F and obtain the existence of a measurable set F + ⊆ Ω havingthe properties P1,P2 and P3 above. Furthermore, we have that µ ( F + ) ∈ { , } . We claimthat µ ( F + ) = 0. Suppose that the opposite happens, that is, µ ( F + ) = 1. In this case, wecan endow the set F with the following probability structure: Consider the σ − algebra A := n E ⊆ F ; E = A ∩ F for some measurable set A ⊆ Ω o and define the probability measure µ + : A → [0 ,
1] as µ + ( E ) := µ ( A ∩ F + ). Also, define themapping T : R × F → F by T ( x ) ω := T ( x ) ω . It is clear that T (0) = Id and T ( x + y ) = T ( x ) ◦ T ( y ). If E ∈ A , then, E = A ∩ F for some measurable set A ⊆ Ω. Hence, due to theinvariance of F by T , we have T ( x ) E = T ( x ) A ∩ F . Since T is a dynamical system on Ω, we have T ( x ) A is a measurable set. Consequently, T ( x )( E ) ∈ A and due to the invarianceof the set F + by T , we get µ + (cid:16) T ( E ) (cid:17) := µ (cid:16) T ( x ) A ∩ F + (cid:17) = µ (cid:16) T ( x ) (cid:0) A ∩ F + (cid:1) (cid:17) = µ (cid:16) A ∩ F + (cid:17) = µ + ( E ) . Therefore, T is a dynamical system acting on the probability space (cid:16) F , A , µ + (cid:17) . It remainsto show the ergodicity of T . For this, let E ∈ A be such that T ( x )( E ) = E for all x ∈ R .By definition of T , this means T ( x ) E = E for all x ∈ R . Using the step 3, we can find ameasurable set E + ⊆ Ω satisfying the properties P1, P2 and P3. Moreover, µ ( E + ) ∈ { , } .Suppose that µ ( E + ) = 1 and write E = A ∩ F . By the property P3, we have1 = µ ( E + ) = inf n µ ( C ); C ⊆ Ω is measurable and E ⊆ C o ≤ µ (cid:16) A ∩ F + (cid:17) ≤ , for E = A ∩ F ⊆ A ∩ F + . Therefore, µ + ( E ) := µ (cid:16) A ∩ F + (cid:17) = 1.It is clear that if µ ( E + ) = 0 then µ + ( E ) = 0. Thus, T is ergodic.Define f : Ω → R by f ( ω ) = ω (0). By the Lemma 2.6, the function F ( x, ω ) = f ( T ( x ) ω )satisfies the hypotheses of the Theorem 2.1. Consequently, µ + (cid:16) { ω ∈ F ; ω ( · ) ∈ AP( R ) } (cid:17) =1, which is a contradiction with the Lemma 2.5. Thus, we must have µ ( F + ) = 0 and thetheorem is proved. (cid:3) Acknowledgements
The author acknowledges GOD for the inspiration. He also thanks the many helpfulsuggestions of H. Frid during the preparation of the paper and the support from CNPq,through grant proc. 302331/2017-4.
References [1] L. Ambrosio, H. Frid.
Multiscale Young measures in almost periodic homogenization and applications .Archive for Rational Mechanics and Analysis (2009), 37–85.[2] L. Ambrosio, H. Frid and J.C. Silva.
Multiscale Young Measures in Homogenization of ContinuousStationary Processes in Compact Spaces and Applications . Journal of Functional Analysis (2009),1962–1997.[3] H. Frid and Jean Silva.
Homogenization of Nonlinear PDE’s in the Fourier-Stieltjes Algebras .SIAMJournal on Mathematical Analysis, Vol. 41, 1589-1620, 2009.[4] A.S. Besicovitch. Almost Periodic Functions. Cambridge University Press, 1932.[5] L. Caffarelli, P.E. Souganidis and C. Wang.
Homogenization of nonlinear, uniformly elliptic and par-abolic partial differential equations in stationary ergodic media . Comm. Pure Appl. Math. (2005),no. 3, 319–361.[6] N. Dunford and J.T. Schwartz. Linear Operators. Parts I and II. Interscience Publishers, Inc., NewYork, 1958, 1963.[7] W.F. Eberlein. Abstract ergodic theorems and weak almost periodic functions.
Trans. Am. Soc., 67:217-240, 1949.[8] W.F. Eberlein.
The point spectrum of weakly almost periodic functions.
Michigan Math. J., 3:137-139,1955-56.
RGODIC FUNCTIONS 15 [9] Hermano Frid.
A Note on the Stochastic Weakly* Almost Periodic Homogenization of Fully Non-LinearElliptic Equations.
Portugal. Math. Vol. 72, Fasc. 2-3, 2015, 207-227.[10] E. Hewitt and A. Ross. ”Abstract Harmonic Analysis”, Vol. I, Springer-Verlag, New York, 1963.[11] H. Ishii.
Almost periodic homogenization of Hamilton-Jacobi equations . International Conference onDifferential Equations, Vol. 1, 2 (Berlin, 1999), 600–605, World Sci. Publishing, River Edge, NJ, 2000.[12] V.V. Jikov, S.M. Kozlov & O.A. Oleinik. Homogenization of Differential Operators and Integral Func-tionals. Springer-Verlag, Berlin Heidelberg, 1994.[13] B.M. Levitan and V.V. Zhikov. “Almost Periodic Functions and Differential Equations”. CambidgeUniversity Press, New York, 1982.[14] Ricardo Ma˜n´e. ”Ergodic Theory and Differentiable Dynamics”. Springer Verlag.[15] W. Rudin.
Weak almost periodic functions and Fourier-Stieltjes transforms.
Duke Math. J. (1959),215-220.[16] Jean Silva. On the Almost Periodic Homogenization of Non-Linear Scalar Conservation Laws.
Calc.Var.(2015)54:3623-3641.[17] P.E. Souganidis.
Stochastic homogenization of Hamilton-Jacobi equations and some application . As-ymptotic Analysis (1999), 141–178.[18] V.V. Zhikov, E.V. Krivenko. Homogenization of singularly perturbed elliptic operators . Matem. Zametki (1983), 571-582. (English transl.: Math. Notes (1983), 294-300). Departamento de Matem´atica, Universidade Federal de Minas Gerais
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