CJM Chinese Journal of Mathematics 2314-8071 Hindawi Publishing Corporation 10.1155/2016/6274839 6274839 Research Article Ergodicity Space for Measure-Preserving Transformations http://orcid.org/0000-0003-0856-4559 Rahimi M. 1,2 http://orcid.org/0000-0002-9721-5678 Assari A. 1,2 Terracina Andrea 1 Department of Mathematics Faculty of Science University of Qom Qom 37161-46611 Iran qom.ac.ir 2 Department of Basic Science Jundi-Shapur University of Technology Dezful 64616-18674 Iran jsu.ac.ir 2016 21 8 2016 2016 22 04 2016 27 07 2016 2016 Copyright © 2016 M. Rahimi and A. Assari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce the concept of ergodicity space of a measure-preserving transformation and will present some of its properties as an algebraic weight for measuring the size of the ergodicity of a measure-preserving transformation. We will also prove the invariance of the ergodicity space under conjugacy of dynamical systems.

1. Introduction

In statistical mechanics , the ergodic hypothesis says that, over long periods of time, the time spent by a particle in some region of the phase space of microstates with the same energy is proportional to the volume of this region; that is, all accessible microstates are equiprobable over a long period of time.

The ergodic hypothesis is often assumed in statistical analysis. The analyst would assume that the average of a process parameter over time and the average over the statistical ensemble are the same. Most of the physical systems are assumed to be ergodic. But, in general, the systems are not necessarily ergodic . Therefore, the more one system is near to being ergodic, the more it could be considered as a model for physical systems. So one may ask the following question: how near is a dynamical system to ergodicity? In this paper, we assign an algebraic structure to any measure-preserving map on a probability space which is invariant under conjugacy of dynamical systems. This algebraic structure is indeed a vector space which will be bigger in size as the system shows more ergodic treatments. Beside the numerical invariants such as entropy , it is an algebraic invariant in ergodic theory. It is also an algebraic weight which takes its maximum size when the systems are ergodic. The middle size represents the size of ergodicity of the system. Briefly, we apply a linear structure to demonstrate the ergodicity weight of a nonlinear system.

2. Ergodicity Space

Let ( X , B , μ ) be a probability space and let T : ( X , B , μ ) ( X , B , μ ) be a measure-preserving transformation. Define the relation “ ~ ” on L 2 ( μ ) as follows: (1) f ~ g iff f - g = c t e . ~ ” is clearly an equivalence relation on L 2 ( μ ) . Put L c 2 μ L 2 ( μ ) / ~. We again denote the members of L c 2 ( μ ) by f instead of [ f ] ~ for simplicity. Indeed, we identify the functions in L 2 ( μ ) whose difference is constant μ —almost everywhere.

Now we define the relation “ ~ T ” on L c 2 ( μ ) as follows: (2) f ~ T g i f f U T f - g = f - g , where U T f f T , for f L c 2 ( μ ) . “ ~ T ” is clearly a well-defined equivalence relation on L c 2 ( μ ) . Now put (3) Λ T L c 2 μ ~ T = f T : f L c 2 μ . Define the addition operation (4) + : Λ T × Λ T Λ T as (5) f T , g T f T + g T f + g T and the scalar multiplication (6) · : C × Λ T Λ T as (7) λ , f T λ f T λ f T on Λ T . Clearly “+” and “·” are well-defined and ( Λ T , + , · ) is a vector space on C which is called the ergodicity space of T . The following theorem shows that the vector space Λ T measures the ergodicity of T .

Theorem 1.

Let T be a measure-preserving transformation on a probability space ( X , B , μ ) :

if T = i d X , then Λ T = { [ 0 ] T } ;

T is ergodic if and only if Λ T = L c 2 ( μ ) .

Proof.

(i) If T = i d X , then U T ( f ) = f T = f for all f L c 2 ( μ ) ; equivalently f ~ T 0 for all f L c 2 ( μ ) and so [ f ] T = [ 0 ] T for all f L c 2 ( μ ) and this is equivalent to Λ T = { [ 0 ] T } .

(ii) Let T be ergodic. For f 0 L 2 ( μ ) we have (8) f f 0 T f - f 0 T = f - f 0 f - f 0 = c t e f f 0 ~ . Therefore [ f 0 ] T = [ f 0 ] ~ and so Λ T = L c 2 ( μ ) .

Now let Λ T = L c 2 ( μ ) . If U T ( f ) = f , then [ f ] T = [ 0 ] T and since by assumption [ f ] T = [ f ] ~ for all f L 2 ( μ ) , then [ f ] ~ = [ 0 ] ~ and hence f = c t e ; therefore T is ergodic (, Theorem 1.6 ).

In general { [ 0 ] T } Λ T L c 2 ( μ ) , and Λ T will attain its largest size if and only if T is ergodic, while it will be of its smallest size if T = i d X . The middle states of Λ T will present the weight of ergodicity of T .

Definition 2.

Let ( X , B , μ ) be a probability space and let T be a transformation on X . The semistable set of T is defined as follows: (9) S T A B : T - 1 A = A .

The following theorem presents the relationship between the ergodicity spaces of two measure-preserving transformations and their semistable sets.

Theorem 3.

Let T 1 and T 2 be two measure-preserving transformations on the probability space ( X , B , μ ) . Then (10) Λ T 1 = Λ T 2 S T 1 = S T 2 0 T 1 = 0 T 2 .

Proof.

Let Λ T 1 = Λ T 2 ; then [ 0 ] T 1 = [ 0 ] T 2 . So, for f L 2 ( μ ) , f ~ T 1 0 if and only if f ~ T 2 0 or equivalently f T 1 = f if and only if f T 2 = f . Now let A B . For f = χ A , χ A T 1 = χ A if and only if χ A T 2 = χ A or equivalently χ T 1 - 1 ( A ) = χ A if and only if χ T 2 - 1 ( A ) = χ A and hence T 1 - 1 ( A ) = A if and only if T 2 - 1 ( A ) = A ; therefore S ( T 1 ) = S ( T 2 ) .

Conversely, suppose that S ( T 1 ) = S ( T 2 ) ; then T 1 - 1 ( A ) = A if and only if T 2 - 1 ( A ) = A for all A B or equivalently χ A T 1 = χ A if and only if χ A T 2 = χ A for all A B . In the following, first we show that [ 0 ] T 1 = [ 0 ] T 2 .

As observed in the last paragraph, f T 1 = f if and only if f T 2 = f for all characteristic functions f = χ A .

Let f = j = 1 n c j χ A j , where c j ’s are distinct real numbers and A j ’s are nonempty disjoint measurable sets. Suppose f T 1 = f and let 1 k n be arbitrary but fixed. Choose x A k . By assumption we have (11) j = 1 n c j χ T 1 - 1 A j x = j = 1 n c j χ A j x = c k . This will give us that x T 1 - 1 ( A k ) , so A k T 1 - 1 ( A k ) . The opposite inclusion similarly follows and then we will have T 1 - 1 ( A k ) = A k . Since S ( T 1 ) = S ( T 2 ) , then T 2 - 1 ( A k ) = A k and so f T 2 = f . Similarly, if f T 2 = f , then f T 1 = f . Hence for simple functions we have f T 1 = f if and only if f T 2 = f .

By a standard measure theoretical argument we have f T 1 = f if and only if f T 2 = f for any complex function f L 2 ( μ ) .

The above arguments show that if S ( T 1 ) = S ( T 2 ) , then [ 0 ] T 1 = [ 0 ] T 2 . Now consider f 0 L 2 ( μ ) . If f [ f 0 ] T 1 , then f - f 0 [ 0 ] T 1 and so f - f 0 [ 0 ] T 2 ; hence f [ f 0 ] T 1 . Therefore [ f 0 ] T 1 [ f 0 ] T 2 . Similarly [ f 0 ] T 2 [ f 0 ] T 1 and so [ f 0 ] T 1 = [ f 0 ] T 2 . Since f 0 was arbitrary we have Λ T 1 = Λ T 2 and the proof is complete.

Corollary 4.

Let T 1 and T 2 be two injective measure-preserving transformations on a Lebesgue space ( X , B , μ ) such that Λ T 1 = Λ T 2 . Then F i x ( T 1 ) = F i x ( T 2 ) .

Proof.

Since Λ T 1 = Λ T 2 , then, by Theorem 3, S ( T 1 ) = S ( T 2 ) . Now we have (12) x F i x T 1 T 1 - 1 x = x x S T 1 = S T 2 T 2 - 1 x = x x F i x T 2 .

Theorem 5.

Let T be an invertible measure-preserving transformation on the probability space ( X , B , μ ) . Then Λ T n = Λ T - n for all n 1 .

Proof.

Note that f ~ T f 0 if and only if f ~ T - 1 f 0 .

3. Invariance of the Ergodicity Space

In this section we show that the ergodicity space Λ T is an algebraic spectral isomorphism invariant and therefore is a conjugacy and isomorphism invariant. Entropy and other invariants introduced so far have mainly been numerical invariants while the ergodicity space, introduced in Section 2, is an algebraic invariant. Before stating the main theorem we recall the concept of conjugacy, isomorphism, and spectral isomorphism (see also ).

Definition 6.

Let T 1 and T 2 be two measure-preserving transformations on probability spaces ( X 1 , B 1 , μ 1 ) and ( X 2 , B 2 , μ 2 ) , respectively. One says that T 1 and T 2 are similar if there exists an invertible measure-preserving transformation φ : X 1 X 2 such that φ T 1 = T 2 φ .

Theorem 7.

If T 1 and T 2 are similar measure-preserving transformations on ( X 1 , B 1 , μ 1 ) and ( X 2 , B 2 , μ 2 ) , respectively, then Λ T 1 Λ T 2 (i.e., Λ T 1 and Λ T 2 are isomorphic vector spaces).

Proof.

Let φ : X 1 X 2 be an invertible measure-preserving transformation such that φ T 1 = T 2 φ . Define Ψ : Λ T 1 Λ T 2 by Ψ ( f ] T 1 [ f φ - 1 ] T 2 . Since φ T 1 = T 2 φ it is easily seen that Ψ is well-defined. Moreover, one can easily check that Ψ is a linear transformation.

Now if Ψ ( [ f ] T 1 ) = [ 0 ] T 2 , then [ f φ - 1 ] T 2 = [ 0 ] T 2 or f φ - 1 ~ T 2 0 ; hence f φ - 1 T 2 = f φ - 1 and so f T 1 φ - 1 = f φ - 1 . Since φ is bijective we have f T 1 = f and thus [ f ] T 1 = [ 0 ] T 1 ; therefore Ψ is one-to-one. Finally, for [ g ] T 2 Λ T 2 , put f g φ ; then Ψ ( [ f ] T 1 ) = [ g ] T 2 and so Ψ is onto. Therefore Λ T 1 is isomorphic to Λ T 2 .

Remark 8.

If the measure-preserving transformation φ : X 1 X 2 in Definition 6 is onto but not necessarily invertible, then Λ T 2 is isomorphic to a subspace of Λ T 1 . The map Ψ : Λ T 2 Λ T 1 , defined by Ψ ( f ] T 2 [ f φ ] T 1 , is the injective isomorphism embedding Λ T 2 into Λ T 1 .

Definition 9.

Let T 1 and T 2 be two measure-preserving transformations on probability spaces ( X 1 , B 1 , μ 1 ) and ( X 2 , B 2 , μ 2 ) , respectively. We say that T 1 is isomorphic to T 2 if there exist M 1 B 1 and M 2 B 2 with μ 1 ( M 1 ) = μ 2 ( M 2 ) = 1 such that

T i ( M i ) M i ( i = 1,2 ) ;

there is an invertible measure-preserving transformation ϕ : M 1 M 2 with ϕ T 1 ( x ) = T 2 ϕ ( x ) for all x M 1 .

The following is an example of isomorphism of transformations .

Example 10.

Let T 1 : [ 0,1 ) [ 0,1 ) be the transformation T 1 x = 2 x mod 1 , where [ 0,1 ) is equipped by the Borel σ -algebra and Lebesgue measure. Let, also, T 2 = σ : Σ 2 Σ 2 be the 1 -sided ( 1 / 2 , 1 / 2 ) -shift map. Define ϕ : Σ 2 [ 0,1 ) by (13) ϕ a 1 , a 2 , a 3 , i = 1 + a i 2 i . Let M 1 = Σ 2 D 1 , where D 1 is the set of points of Σ 2 which have eventually constant coordinates, and M 2 = [ 0,1 ) D 2 , where D 2 is the set of dyadic rational numbers in [ 0,1 ) . Clearly both M 1 and M 2 have full measure and ϕ maps M 1 to M 2 bijectively. Also ϕ T 2 ( x ) = T 1 ϕ ( x ) for x M 1 , so T 1 is isomorphic to T 2 .

Definition 11.

Let T 1 and T 2 be two measure-preserving transformations on probability spaces ( X 1 , B 1 , μ 1 ) and ( X 2 , B 2 , μ 2 ) , respectively. We say that T 1 is conjugate to T 2 if there is a measure algebra isomorphism Φ : ( B 2 ~ , μ 2 ~ ) ( B 1 ~ , μ 1 ~ ) such that Φ T 2 ~ - 1 = T 1 ~ - 1 Φ .

Definition 12.

Measure-preserving transformations T i on ( X i , B i , μ i ) ( i = 1,2 ) are spectrally isomorphic, if there is a linear operator W : L 2 ( μ 2 ) L 2 ( μ 1 ) such that

W is invertible;

W f , W g = f , g for all f , g L 2 ( μ 2 ) ;

U T 1 W = W U T 2 .

Definition 13.

Let ( X , B , μ ) be a probability space such that L 2 ( μ ) is separable. An invertible measure-preserving transformation T on ( X , B , μ ) is said to have countable Lebesgue spectrum if there is a sequence { f j } j = 0 , with f 0 1 , of members of L 2 ( μ ) such that { f 0 } { U T n f j j 1 , n Z } is an orthogonal basis of L 2 ( μ ) .

Theorem 14 (see [<xref ref-type="bibr" rid="B8">8</xref>]).

Let T 1 and T 2 be two measure-preserving transformations on probability spaces ( X 1 , B 1 , μ 1 ) and ( X 2 , B 2 , μ 2 ) , respectively. Then

if T 1 is isomorphic to T 2 , then T 1 is conjugate to T 2 ;

if T 1 and T 2 are conjugate, then T 1 and T 2 are spectrally isomorphic;

any two invertible measure-preserving transformations with countable Lebesgue spectrum are spectrally isomorphic.

Now we are ready to prove the invariance of Λ T .

Theorem 15.

Let T 1 and T 2 be two measure-preserving transformations on probability spaces ( X 1 , B 1 , μ 1 ) and ( X 2 , B 2 , μ 2 ) , respectively. If T 1 and T 2 are spectrally isomorphic, then Λ T 1 Λ T 2 .

Proof.

Let W : L 2 ( μ 2 ) L 2 ( μ 1 ) be an isomorphism of Hilbert spaces such that U T 1 W = W U T 2 . Define Ψ : Λ T 1 Λ T 2 by Ψ ( [ f ] T 1 ) [ W - 1 f ] T 2 . Since U T 1 W = W U T 2 , then Ψ is well-defined. Obviously Ψ is a linear transformation. Moreover, if Ψ ( [ f ] T 1 ) = [ 0 ] T 2 , then [ W - 1 f ] T 2 = [ 0 ] T 2 so W - 1 f ~ T 2 0 and hence U T 2 W - 1 f = W - 1 f . Thus W - 1 U T 1 f = W - 1 f . Since W is bijective, then U T 1 f = f and so [ f ] T 1 = [ 0 ] T 1 ; therefore Ψ is injective. Finally, for [ g ] T 2 Λ T 2 , let f W g . Then Ψ ( [ f ] T 1 ) = [ g ] T 2 ; hence Ψ is onto and therefore an isomorphism.

Corollary 16.

If T 1 : ( X 1 , B 1 , μ 1 ) ( X 1 , B 1 , μ 1 ) and T 2 : ( X 2 , B 2 , μ 2 ) ( X 2 , B 2 , μ 2 ) are conjugate, then Λ T 1 Λ T 2 .

Corollary 17.

Any two invertible measure-preserving transformations with countable Lebesgue spectrum have isomorphic ergodicity spaces.

So far, we have assigned a linear space to any given measure-preserving transformation T on a probability space ( X , B , μ ) which is spectrally isomorphism invariant. Any basis of the ergodicity space of T is denoted by B T .

In the following theorem we observe that the ergodicity space decreases in size as we compose T with itself.

Theorem 18.

Let T be a measure-preserving transformation on a probability space ( X , B , μ ) . Then Λ T 2 is isomorphic to a subspace of Λ T .

Proof.

Let B T = { [ f α ] T } α J be a basis for Λ T . Then for f L 2 ( μ ) we have [ f ] T = i = 1 k λ i [ f α i ] T for some k N and λ i C . Hence [ f ] T 2 = i = 1 k λ i [ f α i ] T 2 . So { [ f α ] T 2 } α J generates the linear space Λ T 2 . Therefore card ( B T 2 ) card ( B T ) and so Λ T 2 is isomorphic to a subspace of Λ T .

Analogously we have the following corollary.

Corollary 19.

Let T be a measure-preserving transformation on a probability space ( X , B , μ ) and m , n N . If m n then Λ T n is isomorphic to a subspace of Λ T m .

Roughly speaking, Corollary 19 states that if m n , then T m is more ergodic than T n .

Theorem 20.

If T 1 and T 2 are two invertible measure-preserving transformations on the probability space ( X , B , μ ) , then Λ T 1 T 2 Λ T 2 T 1 .

Proof.

Let { [ f α ] T 2 T 1 } α J be a basis for Λ T 2 T 1 . For f L 2 ( μ ) we have (14) f T 2 - 1 T 2 T 1 = i = 1 k λ i f α i T 2 T 1 for some λ i C and α i J . In other words f T 2 - 1 ~ T 2 T 1 i = 1 k λ i f α i ; hence (15) f T 2 - 1 - i = 1 k λ i f α i T 2 T 1 = f T 2 - 1 - i = 1 k λ i f α i . Thus (16) f - i = 1 k λ i f α i T 2 T 1 = f - i = 1 k λ i f α i T 2 T 2 - 1 or (17) f - i = 1 k λ i f α i T 2 T 1 T 2 = f - i = 1 k λ i f α i T 2 . Therefore f ~ T 1 T 2 i = 1 k λ i f α i T 2 and this means that (18) f T 1 T 2 = i = 1 k λ i f α i T 2 T 1 T 2 . Hence { [ f α T 2 ] T 1 T 2 } α J generates Λ T 1 T 2 and so Λ T 1 T 2 is isomorphic to a subspace of Λ T 2 T 1 . Similarly Λ T 2 T 1 is isomorphic to a subspace of Λ T 1 T 2 .

Corollary 21.

If T 1 and T 2 are two invertible measure-preserving transformations on the probability space ( X , B , μ ) , then Λ ( T 1 T 2 ) n Λ ( T 2 T 1 ) n for all n Z .

Proof.

Since T 1 and T 2 are invertible then any composition of them is also invertible. For n N , applying Theorem 20 to T 2 and T 1 ( T 2 T 1 ) n - 1 , we will have (19) Λ T 2 T 1 n = Λ T 2 T 1 T 2 T 1 n - 1 Λ T 1 T 2 T 1 n - 1 T 2 = Λ T 1 T 2 n . Finally, if n is a negative integer, one may apply Theorem 5 to complete the proof.

Example 22.

Let X = { 1,2 , , n } and B = P ( X ) . Define the set function μ : B [ 0,1 ] by μ A 1 / n card ( A ) . Clearly μ is a probability measure on B and therefore ( X , B , μ ) is a probability space. Note that (20) L 2 μ = f : X C : X f 2 d μ < = f : X C : j = 1 n f j 2 < = C X . So we may consider any f L 2 ( μ ) as a vector in C n ; say ( c 1 , c 2 , , c n ) , where c j = f ( j ) . Let T : X X be the permutation T = σ k = ( 12 k ) , where 1 k n . In other words, (21) T j = j + 1 f o r j = 1 , , k - 1 , T k = 1 , T j = j f o r j > k . Then T is a measure-preserving transformation on ( X , B , μ ) .

Now let f 0 = ( c 1 , c 2 , , c n ) L 2 ( μ ) . For f = ( z 1 , z 2 , , z n ) L 2 ( μ ) , we have (22) f ~ T f 0 iff f - f 0 T = f - f 0 . Applying the previous relation on j = 1,2 , , k - 1 we will have (23) z j + 1 - z j = c j + 1 - c j f o r j = 1,2 , , k - 1 . Therefore f ~ T f 0 if and only if z j + 1 - z j = c j + 1 - c j for j = 1,2 , , k - 1 . Hence (24) f 0 T = z 1 , z 2 , , z n C n : z j + 1 - z j = c j + 1 - c j f o r j = 1,2 , , k - 1 .

Now define Ψ : Λ T C k - 1 by (25) Ψ f T z 2 - z 1 , z 3 - z 2 , , z k - z k - 1 , where f = ( z 1 , z 2 , , z n ) . Clearly Ψ is a well-defined linear transformation. Moreover, if Ψ ( [ f ] T ) = ( 0,0 , , 0 ) , then z j + 1 - z j = 0 for j = 1,2 , , k - 1 so [ f ] T = ( 0,0 , , 0 ) and hence Ψ is one-to-one.

On the other hand let ( c 1 , c 2 , , c k - 1 ) C k - 1 . If we put f = ( d 0 , d 1 , , d k - 1 , 0 , , 0 ) , where d 0 = 0 and d j = i = 1 j c i ( j = 1,2 , , k - 1 ), then Ψ ( [ f ] T ) = ( c 1 , c 2 , , c k - 1 ) . So Ψ is onto. Therefore Ψ is an isomorphism between Λ T and C k - 1 and so Λ T C k - 1 .

Corollary 23.

For any permutation σ S n , where σ = ( i 1 i 2 i k ) and X = { 1,2 , , n } is equipped with normalized counting measure, one has Λ σ C k - 1 .

Corollary 24.

For any n N there is a measure-preserving transformation T on a probability space ( X , B , μ ) such that Λ T C n .

Example 25.

Let K = { z C : | z | = 1 } , let B be the σ -algebra of Borel subsets of K , and let m be the Haar measure on K . Let a K not be a root of unity and define T : K K by T ( z ) = a z . Then T is an ergodic measure-preserving transformation and so by Theorem 1 Λ T = L c 2 ( K , m ) . On the other hand, (26) dim L c 2 K , m = dim L 2 K , m = dim L 2 0,1 , m = c . Therefore dim Λ T = c .

4. Concluding Remarks

In this paper we assigned a vector space Λ T to a measure-preserving map T on a probability space ( X , B , μ ) . It is an invariant space under conjugacy of dynamical systems and is an algebraic weight for ergodicity of T . The map T is ergodic if and only if Λ T has its greatest possible size. The middle states give the weighted ergodicity. So, for any two measure-preserving maps T 1 and T 2 on a probability space ( X , B , μ ) , we say that T 2 is more ergodic than T 1 if and only if Λ T 1 Λ T 2 . We summarize the following results:

For any measure-preserving map T : ( X , B , μ ) ( X , B , μ ) , T is more ergodic than T 2 . This was expected since, exactly as in the case of Theorem 18, if T 2 is ergodic, then T is ergodic.

If m n , then T m is more ergodic than T n . As in the previous part, this was expected since if m n and T n are ergodic, then T m is ergodic.

If T 1 and T 2 are invertible measure-preserving maps on a probability space ( X , B , μ ) , then T 1 T 2 and T 2 T 1 have similar ergodicity treatment.

If T is invertible, then T and T - 1 have similar ergodicity treatment.

Competing Interests

The authors declare that they have no competing interests.

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