Research Article | Open Access
Ergodicity Space for Measure-Preserving Transformations
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.
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 [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 be a probability space and let be a measure-preserving transformation. Define the relation “” on as follows: “” is clearly an equivalence relation on . Put ~. We again denote the members of by instead of for simplicity. Indeed, we identify the functions in whose difference is constant —almost everywhere.
Now we define the relation “” on as follows: where , for . “” is clearly a well-defined equivalence relation on . Now put Define the addition operation as and the scalar multiplication as on . Clearly “+” and “·” are well-defined and is a vector space on which is called the ergodicity space of . The following theorem shows that the vector space measures the ergodicity of .
Theorem 1. Let be a measure-preserving transformation on a probability space :(1)if , then ;(2) is ergodic if and only if .
Proof. (i) If , then for all ; equivalently for all and so for all and this is equivalent to .
(ii) Let be ergodic. For we have Therefore and so .
Now let . If , then and since by assumption for all , then and hence ; therefore is ergodic (, Theorem ).
In general , and will attain its largest size if and only if is ergodic, while it will be of its smallest size if . The middle states of will present the weight of ergodicity of .
Definition 2. Let be a probability space and let be a transformation on . The semistable set of is defined as follows:
The following theorem presents the relationship between the ergodicity spaces of two measure-preserving transformations and their semistable sets.
Theorem 3. Let and be two measure-preserving transformations on the probability space . Then
Proof. Let ; then . So, for , if and only if or equivalently if and only if . Now let . For , if and only if or equivalently if and only if and hence if and only if ; therefore
Conversely, suppose that ; then if and only if for all or equivalently if and only if for all . In the following, first we show that .
As observed in the last paragraph, if and only if for all characteristic functions .
Let , where ’s are distinct real numbers and ’s are nonempty disjoint measurable sets. Suppose and let be arbitrary but fixed. Choose . By assumption we have This will give us that , so . The opposite inclusion similarly follows and then we will have . Since , then and so . Similarly, if , then . Hence for simple functions we have if and only if .
By a standard measure theoretical argument we have if and only if for any complex function .
The above arguments show that if , then . Now consider . If , then and so ; hence . Therefore . Similarly and so . Since was arbitrary we have and the proof is complete.
Corollary 4. Let and be two injective measure-preserving transformations on a Lebesgue space such that . Then .
Proof. Since , then, by Theorem 3, . Now we have
Theorem 5. Let be an invertible measure-preserving transformation on the probability space . Then for all .
Proof. Note that if and only if .
3. Invariance of the Ergodicity Space
In this section we show that the ergodicity space 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 and be two measure-preserving transformations on probability spaces and , respectively. One says that and are similar if there exists an invertible measure-preserving transformation such that .
Theorem 7. If and are similar measure-preserving transformations on and , respectively, then (i.e., and are isomorphic vector spaces).
Proof. Let be an invertible measure-preserving transformation such that . Define by . Since it is easily seen that is well-defined. Moreover, one can easily check that is a linear transformation.
Now if , then or ; hence and so . Since is bijective we have and thus ; therefore is one-to-one. Finally, for , put ; then and so is onto. Therefore is isomorphic to .
Remark 8. If the measure-preserving transformation in Definition 6 is onto but not necessarily invertible, then is isomorphic to a subspace of . The map , defined by , is the injective isomorphism embedding into .
Definition 9. Let and be two measure-preserving transformations on probability spaces and , respectively. We say that is isomorphic to if there exist and with such that (i);(ii)there is an invertible measure-preserving transformation with for all .
The following is an example of isomorphism of transformations .
Example 10. Let be the transformation , where is equipped by the Borel -algebra and Lebesgue measure. Let, also, be the -sided -shift map. Define by Let , where is the set of points of which have eventually constant coordinates, and , where is the set of dyadic rational numbers in . Clearly both and have full measure and maps to bijectively. Also for , so is isomorphic to .
Definition 11. Let and be two measure-preserving transformations on probability spaces and , respectively. We say that is conjugate to if there is a measure algebra isomorphism such that .
Definition 12. Measure-preserving transformations on are spectrally isomorphic, if there is a linear operator such that (i) is invertible;(ii) for all ;(iii).
Definition 13. Let be a probability space such that is separable. An invertible measure-preserving transformation on is said to have countable Lebesgue spectrum if there is a sequence , with , of members of such that is an orthogonal basis of .
Theorem 14 (see ). Let and be two measure-preserving transformations on probability spaces and , respectively. Then (i)if is isomorphic to , then is conjugate to ;(ii)if and are conjugate, then and are spectrally isomorphic;(iii)any two invertible measure-preserving transformations with countable Lebesgue spectrum are spectrally isomorphic.
Now we are ready to prove the invariance of .
Theorem 15. Let and be two measure-preserving transformations on probability spaces and , respectively. If and are spectrally isomorphic, then .
Proof. Let be an isomorphism of Hilbert spaces such that . Define by . Since , then is well-defined. Obviously is a linear transformation. Moreover, if , then so and hence . Thus . Since is bijective, then and so ; therefore is injective. Finally, for , let . Then ; hence is onto and therefore an isomorphism.
Corollary 16. If and are conjugate, then .
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 on a probability space which is spectrally isomorphism invariant. Any basis of the ergodicity space of is denoted by .
In the following theorem we observe that the ergodicity space decreases in size as we compose with itself.
Theorem 18. Let be a measure-preserving transformation on a probability space . Then is isomorphic to a subspace of .
Proof. Let be a basis for . Then for we have for some and . Hence . So generates the linear space . Therefore and so is isomorphic to a subspace of .
Analogously we have the following corollary.
Corollary 19. Let be a measure-preserving transformation on a probability space and . If then is isomorphic to a subspace of .
Roughly speaking, Corollary 19 states that if , then is more ergodic than .
Theorem 20. If and are two invertible measure-preserving transformations on the probability space , then .
Proof. Let be a basis for . For we have for some and . In other words ; hence Thus or Therefore and this means thatHence generates and so is isomorphic to a subspace of . Similarly is isomorphic to a subspace of .
Corollary 21. If and are two invertible measure-preserving transformations on the probability space , then for all .
Proof. Since and are invertible then any composition of them is also invertible. For , applying Theorem 20 to and , we will have Finally, if is a negative integer, one may apply Theorem 5 to complete the proof.
Example 22. Let and . Define the set function by . Clearly is a probability measure on and therefore is a probability space. Note that So we may consider any as a vector in ; say , where . Let be the permutation , where . In other words, Then is a measure-preserving transformation on .
Now let . For , we have Applying the previous relation on we will have Therefore if and only if for . Hence Now define by where . Clearly is a well-defined linear transformation. Moreover, if , then for so and hence is one-to-one.
On the other hand let . If we put , where and (), then . So is onto. Therefore is an isomorphism between and and so .
Corollary 23. For any permutation , where and is equipped with normalized counting measure, one has .
Corollary 24. For any there is a measure-preserving transformation on a probability space such that .
Example 25. Let , let be the -algebra of Borel subsets of , and let be the Haar measure on . Let not be a root of unity and define by . Then is an ergodic measure-preserving transformation and so by Theorem 1. On the other hand, Therefore .
4. Concluding Remarks
In this paper we assigned a vector space to a measure-preserving map on a probability space . It is an invariant space under conjugacy of dynamical systems and is an algebraic weight for ergodicity of . The map is ergodic if and only if has its greatest possible size. The middle states give the weighted ergodicity. So, for any two measure-preserving maps and on a probability space , we say that is more ergodic than if and only if . We summarize the following results:(1)For any measure-preserving map , is more ergodic than . This was expected since, exactly as in the case of Theorem 18, if is ergodic, then is ergodic.(2)If , then is more ergodic than . As in the previous part, this was expected since if and are ergodic, then is ergodic.(3)If and are invertible measure-preserving maps on a probability space , then and have similar ergodicity treatment.(4)If is invertible, then and have similar ergodicity treatment.
The authors declare that they have no competing interests.
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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.