In the theory of dynamic systems , especially the theory of dimensionally preserving maps , the ergodic decomposition is an important tool in order to be able to trace the investigation of general dynamic systems back to the investigation of ergodic systems .
In general, invariant measures cannot simply be broken down as a sum or linear combination of ergodic measures , but more complicated decomposition maps are required, in which the ergodic measures must be integrated over the space .
Disassembly illustration
It is a standard Borel room with a measurable impact of a group .
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
We use the space of the -invariant probability measures as a subset of the ( locally convex ) topological vector space of the signed Radon measures with the weak - * - topology and Borel's σ-algebra . Furthermore, let the ( compact and convex ) subspace of the ergodic probability measures be.
![M ^ G (X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc966a585d5579177a05c96012eb844ff213cc8e)
![E ^ G (X) \ subset M ^ G (X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2808cdad5b5e75008f1df7d671bf7ca1647bc6c)
A decomposition map is a measurable map
![\ beta \ colon X \ to E ^ G (X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/73728ea038d67a636c4dce45f3d5523b5cde4e00)
![x \ mapsto \ beta_x](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7443d19f828dd89b91960b73e0b2deeda935b2d)
with the following properties:
- for all is
![g \ in G, x \ in X](https://wikimedia.org/api/rest_v1/media/math/render/svg/0207e50819089c645e02d1567e27ebf6c8efc1a5)
- for everyone is measurable and
![\ eta \ in E ^ G (X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4299cf9103f6fd3a29175c4ccca62f27a484e89)
- for all and all measurable subsets applies
![\ mu \ in M ^ G (X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad05a96bdf22ea5fb87946960d5e0624fd37fd50)
![A \ subset X](https://wikimedia.org/api/rest_v1/media/math/render/svg/826569be03f873b81cdc6f12637ef5520c369d21)
-
.
Ergo decomposition
Let it be a countable group and a standard Borel space with a measurable effect of the group . If , then is and there is a decomposition map with the above properties.
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![M ^ G (X) \ not = \ emptyset](https://wikimedia.org/api/rest_v1/media/math/render/svg/51bbff28499f50c86c7a5c00ff0b20b043efb0ed)
![E ^ G (X) \ not = \ emptyset](https://wikimedia.org/api/rest_v1/media/math/render/svg/b74bedaa6909af3df869ae353ce3a4a9656a3338)
![\ beta \ colon X \ to E ^ G (X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/73728ea038d67a636c4dce45f3d5523b5cde4e00)
Uniqueness
The decomposition of ergodes is clear in the following sense:
- If there are two mappings with the above properties, then holds for all with a set that satisfies for all .
![\ beta, \ beta ^ \ prime \ colon X \ to E ^ G (X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/3658a1b2d609685f5eaed0573b1b8bacdbcb00a6)
![\ beta_x = \ beta ^ \ prime_x](https://wikimedia.org/api/rest_v1/media/math/render/svg/634d5b1402fe432fbdf31279eba33e8be9e5a624)
![{\ displaystyle x \ in X \ setminus N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ac5f4c46b357f0ef96554380a66218bc8606658)
![N](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3)
![\ mu (N) = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b976c47b5d229e78e08df8c63c14919a9c774e4)
![{\ displaystyle \ mu \ in M ^ {G} (X)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad05a96bdf22ea5fb87946960d5e0624fd37fd50)
Examples
- For consider the effect of on through for . Then it's for everyone
![\ alpha = e ^ {2 \ pi i / n}, n \ in \ N](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc0e3a87fb5355ec801242f3d484ce2917c4153)
![G = \ mathbb {Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcd1288d63a9a1bb85f421f4e164ce7feda9e478)
![{\ displaystyle X = S ^ {1} \ subset \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d94c6481cc30fa3f49ac6aabebb6c19bb12724c)
![(m, z) \ to \ alpha ^ mz](https://wikimedia.org/api/rest_v1/media/math/render/svg/c010dea1c6790a511a7d8194970092a5c2235c52)
![m \ in \ Z, z \ in S ^ 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/becd8880208933f2fa6fe5f2a28e79e78ad82312)
![X _ {\ beta_z} = \ left \ {z, \ alpha z, \ alpha ^ 2 z, \ ldots, \ alpha ^ {n-1} z \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b961b9b75bdba92ec984138c09097b1bd7a57841)
- and is the uniform distribution on the finite set .
![\ beta_z](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7c42a4b65d52ebec696fd3d18d9560fa60b0760)
![X _ {\ beta_z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbba239dc16682a8af53b9690f1e8443cb348ef6)
- Be no root of unity and the effect of on given by for . Then it's for everyone
![G = \ mathbb {Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcd1288d63a9a1bb85f421f4e164ce7feda9e478)
![X = S ^ 1 \ times \ left [0.1 \ right]](https://wikimedia.org/api/rest_v1/media/math/render/svg/468bef13b24ff7c9b4c9e711e070c091677f0c89)
![(m, (z, t)) \ to (\ alpha ^ mz, t)](https://wikimedia.org/api/rest_v1/media/math/render/svg/38465682eb6f23d48f9bebb8d39851d967b34ac9)
![m \ in \ Z, (z, t) \ in S ^ 1 \ times \ left [0,1 \ right]](https://wikimedia.org/api/rest_v1/media/math/render/svg/a93b8f0abaa253439dda33398750292ea424ff96)
![X _ {\ beta _ {(z, t)}} = S ^ 1 \ times \ left \ {t \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bad6ee749b00749b1f490ee8508cfdc40dbc39e4)
- and is the uniform distribution (the normalized Lebesgue measure ) .
![\ beta _ {(z, t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcfcbd14abf10e06b9c6159c21b60efa0d937da6)
![S ^ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60796c8d0c03cf575637d3202463b214d9635880)
literature
-
VS Varadarajan : Groups of automorphisms of Borel spaces. Trans. Amer. Math. Soc. 1963, 109: 191-220. pdf
Web links