Ergo decomposition

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 . ${\ displaystyle (X, \ omega)}$ ${\ displaystyle G}$

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. ${\ displaystyle M ^ {G} (X)}$ ${\ displaystyle G}$ ${\ displaystyle E ^ {G} (X) \ subset M ^ {G} (X)}$

A decomposition map is a measurable map

{\ displaystyle \ beta \ colon X \ to E ^ {G} (X)}

{\ displaystyle x \ mapsto \ beta _ {x}}

with the following properties:

for all is

{\ displaystyle g \ in G, x \ in X}

{\ displaystyle \ beta _ {gx} = \ beta _ {x}}

for everyone is measurable and ${\ displaystyle \ eta \ in E ^ {G} (X)}$ ${\ displaystyle X _ {\ eta}: = \ beta ^ {- 1} (\ eta)}$ ${\ displaystyle \ eta (X _ {\ eta}) = 1}$

for all and all measurable subsets applies

{\ displaystyle \ mu \ in M ​​^ {G} (X)}

{\ displaystyle A \ subset X}

{\ displaystyle \ mu (A) = \ int _ {X} \ beta _ {x} (A) d \ mu (x)}

.

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. ${\ displaystyle G}$ ${\ displaystyle (X, \ omega)}$ ${\ displaystyle G}$ ${\ displaystyle M ^ {G} (X) \ not = \ emptyset}$ ${\ displaystyle E ^ {G} (X) \ not = \ emptyset}$ ${\ displaystyle \ beta \ colon X \ to E ^ {G} (X)}$

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 . ${\ displaystyle \ beta, \ beta ^ {\ prime} \ colon X \ to E ^ {G} (X)}$ ${\ displaystyle \ beta _ {x} = \ beta _ {x} ^ {\ prime}}$ ${\ displaystyle x \ in X \ setminus N}$ ${\ displaystyle N}$ ${\ displaystyle \ mu (N) = 0}$ ${\ displaystyle \ mu \ in M ^ {G} (X)}$

Examples

For consider the effect of on through for . Then it's for everyone ${\ displaystyle \ alpha = e ^ {2 \ pi i / n}, n \ in \ mathbb {N}}$ ${\ displaystyle G = \ mathbb {Z}}$ ${\ displaystyle X = S ^ {1} \ subset \ mathbb {C}}$ ${\ displaystyle (m, z) \ to \ alpha ^ {m} z}$ ${\ displaystyle m \ in \ mathbb {Z}, z \ in S ^ {1}}$ ${\ displaystyle z \ in S ^ {1}}$

{\ displaystyle X _ {\ beta _ {z}} = \ left \ {z, \ alpha z, \ alpha ^ {2} z, \ ldots, \ alpha ^ {n-1} z \ right \}}

and is the uniform distribution on the finite set .

{\ displaystyle \ beta _ {z}}

{\ displaystyle X _ {\ beta _ {z}}}

Be no root of unity and the effect of on given by for . Then it's for everyone ${\ displaystyle \ alpha \ in S ^ {1} \ subset \ mathbb {C}}$ ${\ displaystyle G = \ mathbb {Z}}$ ${\ displaystyle X = S ^ {1} \ times \ left [0,1 \ right]}$ ${\ displaystyle (m, (z, t)) \ to (\ alpha ^ {m} z, t)}$ ${\ displaystyle m \ in \ mathbb {Z}, (z, t) \ in S ^ {1} \ times \ left [0,1 \ right]}$ ${\ displaystyle (z, t) \ in S ^ {1} \ times \ left [0,1 \ right]}$