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 [1], 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 [2]. 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 [3–7], 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 L2(μ) as follows: (1)“f~gifff-g=cte”.“~” is clearly an equivalence relation on L2(μ). Put Lc2μ≔L2(μ)/~. We again denote the members of Lc2(μ) by f instead of [f]~ for simplicity. Indeed, we identify the functions in L2(μ) whose difference is constant μ—almost everywhere.

Now we define the relation “~T” on Lc2(μ) as follows: (2)“f~TgiffUTf-g=f-g,”where UTf≔f∘T, for f∈Lc2(μ). “~T” is clearly a well-defined equivalence relation on Lc2(μ). Now put (3)ΛT≔Lc2μ~T=fT:f∈Lc2μ.Define the addition operation (4)+:ΛT×ΛT⟶ΛT as (5)fT,gT⟼fT+gT≔f+gT and the scalar multiplication (6)·:C×ΛT⟶ΛTas (7)λ,fT⟼λfT≔λfTon ΛT. Clearly “+” and “·” are well-defined and (ΛT,+,·) is a vector space on C which is called the ergodicity space of T. The following theorem shows that the vector space ΛT measures the ergodicity of T.

Theorem 1.

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

if T=idX, then ΛT={[0]T};

T is ergodic if and only if ΛT=Lc2(μ).

Proof.

(i) If T=idX, then UT(f)=f∘T=f for all f∈Lc2(μ); equivalently f~T0 for all f∈Lc2(μ) and so [f]T=[0]T for all f∈Lc2(μ) and this is equivalent to ΛT={[0]T}.

(ii) Let T be ergodic. For f0∈L2(μ) we have (8)f∈f0T⟺f-f0∘T=f-f0⟺f-f0=cte⟺f∈f0~. Therefore [f0]T=[f0]~ and so ΛT=Lc2(μ).

Now let ΛT=Lc2(μ). If UT(f)=f, then [f]T=[0]T and since by assumption [f]T=[f]~ for all f∈L2(μ), then [f]~=[0]~ and hence f=cte; therefore T is ergodic ([8], Theorem 1.6).

In general {[0]T}⊆ΛT⊆Lc2(μ), and ΛT will attain its largest size if and only if T is ergodic, while it will be of its smallest size if T=idX. The middle states of ΛT 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)ST≔A∈B:T-1A=A.

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

Theorem 3.

Let T1 and T2 be two measure-preserving transformations on the probability space (X,B,μ). Then (10)ΛT1=ΛT2⟺ST1=ST2⟺0T1=0T2.

Proof.

Let ΛT1=ΛT2; then [0]T1=[0]T2. So, for f∈L2(μ), f~T10 if and only if f~T20 or equivalently f∘T1=f if and only if f∘T2=f. Now let A∈B. For f=χA, χA∘T1=χA if and only if χA∘T2=χA or equivalently χT1-1(A)=χA if and only if χT2-1(A)=χA and hence T1-1(A)=A if and only if T2-1(A)=A; therefore S(T1)=S(T2).

Conversely, suppose that S(T1)=S(T2); then T1-1(A)=A if and only if T2-1(A)=A for all A∈B or equivalently χA∘T1=χA if and only if χA∘T2=χA for all A∈B. In the following, first we show that [0]T1=[0]T2.

As observed in the last paragraph, f∘T1=f if and only if f∘T2=f for all characteristic functions f=χA.

Let f=∑j=1ncjχAj, where cj’s are distinct real numbers and Aj’s are nonempty disjoint measurable sets. Suppose f∘T1=f and let 1≤k≤n be arbitrary but fixed. Choose x∈Ak. By assumption we have (11)∑j=1ncjχT1-1Ajx=∑j=1ncjχAjx=ck.This will give us that x∈T1-1(Ak), so Ak⊆T1-1(Ak). The opposite inclusion similarly follows and then we will have T1-1(Ak)=Ak. Since S(T1)=S(T2), then T2-1(Ak)=Ak and so f∘T2=f. Similarly, if f∘T2=f, then f∘T1=f. Hence for simple functions we have f∘T1=f if and only if f∘T2=f.

By a standard measure theoretical argument we have f∘T1=f if and only if f∘T2=f for any complex function f∈L2(μ).

The above arguments show that if S(T1)=S(T2), then [0]T1=[0]T2. Now consider f0∈L2(μ). If f∈[f0]T1, then f-f0∈[0]T1 and so f-f0∈[0]T2; hence f∈[f0]T1. Therefore [f0]T1⊆[f0]T2. Similarly [f0]T2⊆[f0]T1 and so [f0]T1=[f0]T2. Since f0 was arbitrary we have ΛT1=ΛT2 and the proof is complete.

Corollary 4.

Let T1 and T2 be two injective measure-preserving transformations on a Lebesgue space (X,B,μ) such that ΛT1=ΛT2. Then Fix(T1)=Fix(T2).

Proof.

Since ΛT1=ΛT2, then, by Theorem 3, S(T1)=S(T2). Now we have(12)x∈FixT1⟺T1-1x=x⟺x∈ST1=ST2⟺T2-1x=x⟺x∈FixT2.

Theorem 5.

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

Proof.

Note that f~Tf0 if and only if f~T-1f0.

3. Invariance of the Ergodicity Space

In this section we show that the ergodicity space ΛT 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 [8]).

Definition 6.

Let T1 and T2 be two measure-preserving transformations on probability spaces (X1,B1,μ1) and (X2,B2,μ2), respectively. One says that T1 and T2 are similar if there exists an invertible measure-preserving transformation φ:X1→X2 such that φ∘T1=T2∘φ.

Theorem 7.

If T1 and T2 are similar measure-preserving transformations on (X1,B1,μ1) and (X2,B2,μ2), respectively, then ΛT1≅ΛT2 (i.e., ΛT1 and ΛT2 are isomorphic vector spaces).

Proof.

Let φ:X1→X2 be an invertible measure-preserving transformation such that φ∘T1=T2∘φ. Define Ψ:ΛT1→ΛT2 by Ψ(f]T1≔[f∘φ-1]T2. Since φ∘T1=T2∘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]T1)=[0]T2, then [f∘φ-1]T2=[0]T2 or f∘φ-1~T20; hence f∘φ-1∘T2=f∘φ-1 and so f∘T1∘φ-1=f∘φ-1. Since φ is bijective we have f∘T1=f and thus [f]T1=[0]T1; therefore Ψ is one-to-one. Finally, for [g]T2∈ΛT2, put f≔g∘φ; then Ψ([f]T1)=[g]T2 and so Ψ is onto. Therefore ΛT1 is isomorphic to ΛT2.

Remark 8.

If the measure-preserving transformation φ:X1→X2 in Definition 6 is onto but not necessarily invertible, then ΛT2 is isomorphic to a subspace of ΛT1. The map Ψ:ΛT2→ΛT1, defined by Ψ(f]T2≔[f∘φ]T1, is the injective isomorphism embedding ΛT2 into ΛT1.

Definition 9.

Let T1 and T2 be two measure-preserving transformations on probability spaces (X1,B1,μ1) and (X2,B2,μ2), respectively. We say that T1 is isomorphic to T2 if there exist M1∈B1 and M2∈B2 with μ1(M1)=μ2(M2)=1 such that

Ti(Mi)⊆Mi(i=1,2);

there is an invertible measure-preserving transformation ϕ:M1→M2 with ϕ∘T1(x)=T2∘ϕ(x) for all x∈M1.

The following is an example of isomorphism of transformations [8].

Example 10.

Let T1:[0,1)→[0,1) be the transformation T1x=2xmod1, where [0,1) is equipped by the Borel σ-algebra and Lebesgue measure. Let, also, T2=σ:Σ2→Σ2 be the 1-sided (1/2,1/2)-shift map. Define ϕ:Σ2→[0,1) by (13)ϕa1,a2,a3,…≔∑i=1+∞ai2i.Let M1=Σ2∖D1, where D1 is the set of points of Σ2 which have eventually constant coordinates, and M2=[0,1)∖D2, where D2 is the set of dyadic rational numbers in [0,1). Clearly both M1 and M2 have full measure and ϕ maps M1 to M2 bijectively. Also ϕ∘T2(x)=T1∘ϕ(x) for x∈M1, so T1 is isomorphic to T2.

Definition 11.

Let T1 and T2 be two measure-preserving transformations on probability spaces (X1,B1,μ1) and (X2,B2,μ2), respectively. We say that T1 is conjugate to T2 if there is a measure algebra isomorphism Φ:(B2~,μ2~)→(B1~,μ1~) such that Φ∘T2~-1=T1~-1∘Φ.

Definition 12.

Measure-preserving transformations Ti on (Xi,Bi,μi)(i=1,2) are spectrally isomorphic, if there is a linear operator W:L2(μ2)→L2(μ1) such that

W is invertible;

〈Wf,Wg〉=〈f,g〉 for all f,g∈L2(μ2);

UT1W=WUT2.

Definition 13.

Let (X,B,μ) be a probability space such that L2(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 {fj}j=0∞, with f0≡1, of members of L2(μ) such that {f0}∪{UTnfj∣j≥1,n∈Z} is an orthogonal basis of L2(μ).

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

Let T1 and T2 be two measure-preserving transformations on probability spaces (X1,B1,μ1) and (X2,B2,μ2), respectively. Then

if T1 is isomorphic to T2, then T1 is conjugate to T2;

if T1 and T2 are conjugate, then T1 and T2 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 ΛT.

Theorem 15.

Let T1 and T2 be two measure-preserving transformations on probability spaces (X1,B1,μ1) and (X2,B2,μ2), respectively. If T1 and T2 are spectrally isomorphic, then ΛT1≅ΛT2.

Proof.

Let W:L2(μ2)→L2(μ1) be an isomorphism of Hilbert spaces such that UT1W=WUT2. Define Ψ:ΛT1→ΛT2 by Ψ([f]T1)≔[W-1f]T2. Since UT1W=WUT2, then Ψ is well-defined. Obviously Ψ is a linear transformation. Moreover, if Ψ([f]T1)=[0]T2, then [W-1f]T2=[0]T2 so W-1f~T20 and hence UT2W-1f=W-1f. Thus W-1UT1f=W-1f. Since W is bijective, then UT1f=f and so [f]T1=[0]T1; therefore Ψ is injective. Finally, for [g]T2∈ΛT2, let f≔Wg. Then Ψ([f]T1)=[g]T2; hence Ψ is onto and therefore an isomorphism.

Corollary 16.

If T1:(X1,B1,μ1)→(X1,B1,μ1) and T2:(X2,B2,μ2)→(X2,B2,μ2) are conjugate, then ΛT1≅ΛT2.

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 BT.

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 ΛT2 is isomorphic to a subspace of ΛT.

Proof.

Let BT={[fα]T}α∈J be a basis for ΛT. Then for f∈L2(μ) we have [f]T=∑i=1kλi[fαi]T for some k∈N and λi∈C. Hence [f]T2=∑i=1kλi[fαi]T2. So {[fα]T2}α∈J generates the linear space ΛT2. Therefore card(BT2)≤card(BT) and so ΛT2 is isomorphic to a subspace of ΛT.

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 ΛTn is isomorphic to a subspace of ΛTm.

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

Theorem 20.

If T1 and T2 are two invertible measure-preserving transformations on the probability space (X,B,μ), then ΛT1∘T2≅ΛT2∘T1.

Proof.

Let {[fα]T2∘T1}α∈J be a basis for ΛT2∘T1. For f∈L2(μ) we have (14)f∘T2-1T2∘T1=∑i=1kλifαiT2∘T1 for some λi∈C and αi∈J. In other words f∘T2-1~T2∘T1∑i=1kλifαi; hence (15)f∘T2-1-∑i=1kλifαi∘T2∘T1=f∘T2-1-∑i=1kλifαi.Thus (16)f-∑i=1kλifαi∘T2∘T1=f-∑i=1kλifαi∘T2∘T2-1or (17)f-∑i=1kλifαi∘T2∘T1∘T2=f-∑i=1kλifαi∘T2.Therefore f~T1∘T2∑i=1kλifαi∘T2 and this means that(18)fT1∘T2=∑i=1kλifαi∘T2T1∘T2.Hence {[fα∘T2]T1∘T2}α∈J generates ΛT1∘T2 and so ΛT1∘T2 is isomorphic to a subspace of ΛT2∘T1. Similarly ΛT2∘T1 is isomorphic to a subspace of ΛT1∘T2.

Corollary 21.

If T1 and T2 are two invertible measure-preserving transformations on the probability space (X,B,μ), then Λ(T1∘T2)n≅Λ(T2∘T1)n for all n∈Z.

Proof.

Since T1 and T2 are invertible then any composition of them is also invertible. For n∈N, applying Theorem 20 to T2 and T1∘(T2∘T1)n-1, we will have (19)ΛT2∘T1n=ΛT2∘T1∘T2∘T1n-1≅ΛT1∘T2∘T1n-1∘T2=ΛT1∘T2n.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 μA≔1/ncard(A). Clearly μ is a probability measure on B and therefore (X,B,μ) is a probability space. Note that (20)L2μ=f:X⟶C:∫Xf2dμ<∞=f:X→C:∑j=1nfj2<∞=CX. So we may consider any f∈L2(μ) as a vector in Cn; say (c1,c2,…,cn), where cj=f(j). Let T:X→X be the permutation T=σk=(12⋯k), where 1≤k≤n. In other words, (21)Tj=j+1forj=1,…,k-1,Tk=1,Tj=jforj>k.Then T is a measure-preserving transformation on (X,B,μ).

Now let f0=(c1,c2,…,cn)∈L2(μ). For f=(z1,z2,…,zn)∈L2(μ), we have (22)f~Tf0ifff-f0∘T=f-f0.Applying the previous relation on j=1,2,…,k-1 we will have (23)zj+1-zj=cj+1-cjforj=1,2,…,k-1.Therefore f~Tf0 if and only if zj+1-zj=cj+1-cj for j=1,2,…,k-1. Hence (24)f0T=z1,z2,…,zn∈Cn:zj+1-zj=cj+1-cjforj=1,2,…,k-1.

Now define Ψ:ΛT→Ck-1 by (25)ΨfT≔z2-z1,z3-z2,…,zk-zk-1,where f=(z1,z2,…,zn). Clearly Ψ is a well-defined linear transformation. Moreover, if Ψ([f]T)=(0,0,…,0), then zj+1-zj=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 (c1,c2,…,ck-1)∈Ck-1. If we put f=(d0,d1,…,dk-1,0,…,0), where d0=0 and dj=∑i=1jci (j=1,2,…,k-1), then Ψ([f]T)=(c1,c2,…,ck-1). So Ψ is onto. Therefore Ψ is an isomorphism between ΛT and Ck-1 and so ΛT≅Ck-1.

Corollary 23.

For any permutation σ∈Sn, where σ=(i1i2⋯ik) and X={1,2,…,n} is equipped with normalized counting measure, one has Λσ≅Ck-1.

Corollary 24.

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

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)=az. Then T is an ergodic measure-preserving transformation and so by Theorem 1ΛT=Lc2(K,m). On the other hand, (26)dimLc2K,m=dimL2K,m=dimL20,1,m=c.Therefore dimΛT=c.

4. Concluding Remarks

In this paper we assigned a vector space ΛT 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 ΛT has its greatest possible size. The middle states give the weighted ergodicity. So, for any two measure-preserving maps T1 and T2 on a probability space (X,B,μ), we say that T2 is more ergodic than T1 if and only if ΛT1⩽ΛT2. We summarize the following results:

For any measure-preserving map T:(X,B,μ)→(X,B,μ), T is more ergodic than T2. This was expected since, exactly as in the case of Theorem 18, if T2 is ergodic, then T is ergodic.

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

If T1 and T2 are invertible measure-preserving maps on a probability space (X,B,μ), then T1∘T2 and T2∘T1 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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