Krieger s finite generator theorem for ergodic actions of countable groups

Transcription

1 Krieger s finite generator theorem for ergodic actions of countable groups Brandon Seward University of Michigan UNT RTG Conference Logic, Dynamics, and Their Interactions II June 6, 2014 Brandon Seward Krieger s theorem for countable groups June 6, / 13

2 Generating Partitions Let G be a countably infinite group. Let G (X, µ) be a probability-measure-preserving (p.m.p.) action. Brandon Seward Krieger s theorem for countable groups June 6, / 13

3 Generating Partitions Let G be a countably infinite group. Let G (X, µ) be a probability-measure-preserving (p.m.p.) action. Definition A countable partition α is generating if the smallest G-invariant sub-σ-algebra containing α {null sets} is the entire Borel σ-algebra. Generating partitions precisely correspond to representations of the action. Brandon Seward Krieger s theorem for countable groups June 6, / 13

4 Generating Partitions Let G be a countably infinite group. Let G (X, µ) be a probability-measure-preserving (p.m.p.) action. Definition A countable partition α is generating if the smallest G-invariant sub-σ-algebra containing α {null sets} is the entire Borel σ-algebra. Generating partitions precisely correspond to representations of the action. Folklore There exists a generating partition α with α = k N if and only if there is a G-equivariant isomorphism (X, µ) = (k G, ν) for some G-invariant probability measure ν. Brandon Seward Krieger s theorem for countable groups June 6, / 13

5 Shannon Entropy Definition The Shannon entropy of a countable partition α is H(α) = A α µ(a) log µ(a). This satisfies 0 H(α) log α. H(α) is a numerical measure of how fine α is. Brandon Seward Krieger s theorem for countable groups June 6, / 13

6 Shannon Entropy Definition The Shannon entropy of a countable partition α is H(α) = A α µ(a) log µ(a). This satisfies 0 H(α) log α. H(α) is a numerical measure of how fine α is. Definition A probability vector is a finite or countable sequence p = (p i ) with p i > 0 and p i = 1. The Shannon entropy of p is H( p) = i p i log(p i ). Brandon Seward Krieger s theorem for countable groups June 6, / 13

7 Generating Partitions and Entropy Theory Rokhlin s Generator Theorem Krieger s Finite Generator Theorem Brandon Seward Krieger s theorem for countable groups June 6, / 13

8 Generating Partitions and Entropy Theory Rokhlin s Generator Theorem If G is amenable and G (X, µ) is free and ergodic, then the entropy h G (X, µ) satisfies h G (X, µ) = inf{h(α) : α is a cntbl generating partition}. Krieger s Finite Generator Theorem Brandon Seward Krieger s theorem for countable groups June 6, / 13

9 Generating Partitions and Entropy Theory Rokhlin s Generator Theorem (G = Z Rokhlin 67; G amenable S. Tucker-Drob 14) If G is amenable and G (X, µ) is free and ergodic, then the entropy h G (X, µ) satisfies h G (X, µ) = inf{h(α) : α is a cntbl generating partition}. Krieger s Finite Generator Theorem Brandon Seward Krieger s theorem for countable groups June 6, / 13

10 Generating Partitions and Entropy Theory Rokhlin s Generator Theorem (G = Z Rokhlin 67; G amenable S. Tucker-Drob 14) If G is amenable and G (X, µ) is free and ergodic, then the entropy h G (X, µ) satisfies h G (X, µ) = inf{h(α) : α is a cntbl generating partition}. Krieger s Finite Generator Theorem If G is amenable, G (X, µ) is free and ergodic, and h G (X, µ) < log(k) then there is a generating partition α with α = k. Brandon Seward Krieger s theorem for countable groups June 6, / 13

11 Generating Partitions and Entropy Theory Rokhlin s Generator Theorem (G = Z Rokhlin 67; G amenable S. Tucker-Drob 14) If G is amenable and G (X, µ) is free and ergodic, then the entropy h G (X, µ) satisfies h G (X, µ) = inf{h(α) : α is a cntbl generating partition}. Krieger s Finite Generator Theorem (G = Z Krieger 70; If G is amenable, G (X, µ) is free and ergodic, and h G (X, µ) < log(k) then there is a generating partition α with α = k. Brandon Seward Krieger s theorem for countable groups June 6, / 13

12 Generating Partitions and Entropy Theory Rokhlin s Generator Theorem (G = Z Rokhlin 67; G amenable S. Tucker-Drob 14) If G is amenable and G (X, µ) is free and ergodic, then the entropy h G (X, µ) satisfies h G (X, µ) = inf{h(α) : α is a cntbl generating partition}. Krieger s Finite Generator Theorem (G = Z Krieger 70; G amenable Danilenko Park 02) If G is amenable, G (X, µ) is free and ergodic, and h G (X, µ) < log(k 1) < log(k) then there is a generating partition α with α = k. Brandon Seward Krieger s theorem for countable groups June 6, / 13

13 Some Simple Open Questions When G is non-amenable or the action G X does not preserve a probability measure, many simple questions remain open. Brandon Seward Krieger s theorem for countable groups June 6, / 13

14 Some Simple Open Questions When G is non-amenable or the action G X does not preserve a probability measure, many simple questions remain open. (1) If a direct product action G (X Y, µ ν) has a finite generator, does either G (X, µ) or G (Y, ν) have a finite generator? Brandon Seward Krieger s theorem for countable groups June 6, / 13

15 Some Simple Open Questions When G is non-amenable or the action G X does not preserve a probability measure, many simple questions remain open. (1) If a direct product action G (X Y, µ ν) has a finite generator, does either G (X, µ) or G (Y, ν) have a finite generator? (2) If every ergodic component of G (X, µ) has a finite generator, is there a finite generator for G (X, µ)? Brandon Seward Krieger s theorem for countable groups June 6, / 13

16 Some Simple Open Questions When G is non-amenable or the action G X does not preserve a probability measure, many simple questions remain open. (1) If a direct product action G (X Y, µ ν) has a finite generator, does either G (X, µ) or G (Y, ν) have a finite generator? (2) If every ergodic component of G (X, µ) has a finite generator, is there a finite generator for G (X, µ)? (3) Does there exist an infinite group such that every action admits a finite generator? Brandon Seward Krieger s theorem for countable groups June 6, / 13

17 Some Simple Open Questions When G is non-amenable or the action G X does not preserve a probability measure, many simple questions remain open. (1) If a direct product action G (X Y, µ ν) has a finite generator, does either G (X, µ) or G (Y, ν) have a finite generator? (2) If every ergodic component of G (X, µ) has a finite generator, is there a finite generator for G (X, µ)? (3) Does there exist an infinite group such that every action admits a finite generator? (4) If G X is a Borel action which does not admit any invariant Borel probability measure, does it have a finite generator? (when X is σ-compact and G acts continuously, Tserunyan obtained a positive answer). Brandon Seward Krieger s theorem for countable groups June 6, / 13

18 Rokhlin Entropy For any cntbl group G and any p.m.p. action G (X, µ), we can define the Rokhlin entropy as hg Rok (X, µ) = inf{h(α) : α is a cntbl generating partition}. When G is amenable and the action is free and ergodic, Rokhlin s theorem says that h Rok (X, µ) is equal to the classical entropy. G Brandon Seward Krieger s theorem for countable groups June 6, / 13

19 A Generalization of Krieger s Theorem Theorem (S. 2014) Brandon Seward Krieger s theorem for countable groups June 6, / 13

20 A Generalization of Krieger s Theorem Theorem (S. 2014) Suppose that G (X, µ) is ergodic (but not necessarily free) and (X, µ) is non-atomic. If p = (p i ) is any finite or cntbl probability vector with hg Rok(X, µ) < H( p) then there exists a generating partition α = {A i} with µ(a i ) = p i for every i. Brandon Seward Krieger s theorem for countable groups June 6, / 13

21 A Generalization of Krieger s Theorem Theorem (S. 2014) Suppose that G (X, µ) is ergodic (but not necessarily free) and (X, µ) is non-atomic. If p = (p i ) is any finite or cntbl probability vector with hg Rok(X, µ) < H( p) then there exists a generating partition α = {A i} with µ(a i ) = p i for every i. This theorem is false under the assumption h Rok (X, µ) H( p). G Brandon Seward Krieger s theorem for countable groups June 6, / 13

22 A Generalization of Krieger s Theorem Theorem (S. 2014) Suppose that G (X, µ) is ergodic (but not necessarily free) and (X, µ) is non-atomic. If p = (p i ) is any finite or cntbl probability vector with hg Rok(X, µ) < H( p) then there exists a generating partition α = {A i} with µ(a i ) = p i for every i. This theorem is false under the assumption h Rok (X, µ) H( p). Corollary G Suppose that G (X, µ) is ergodic (but not necessarily free) and (X, µ) is non-atomic. If hg Rok (X, µ) < log(k) then there exists a generating partition α with α = k. Brandon Seward Krieger s theorem for countable groups June 6, / 13

23 Dependence of Generators on the Measure Corollary Brandon Seward Krieger s theorem for countable groups June 6, / 13

24 Dependence of Generators on the Measure Corollary Let G X be a Borel action. Then the map taking ergodic invariant prob. measures µ to h Rok (X, µ) is Borel. G Brandon Seward Krieger s theorem for countable groups June 6, / 13

25 Dependence of Generators on the Measure Corollary Let G X be a Borel action. Then the map taking ergodic invariant prob. measures µ to h Rok (X, µ) is Borel. G Corollary Fix k N and consider the shift-action G k G. Then the map taking ergodic invariant prob. measures µ to hg Rok(kG, µ) is upper-semicontinuous in the weak -topology. Brandon Seward Krieger s theorem for countable groups June 6, / 13

26 The Full Group Definition For an action G (X, µ) define E X G = {(x, y) : g G g x = y}. The full group, denoted [EG X ], is the set of all Borel bijections θ : X X with x EG X θ(x) for a.e. x X. Brandon Seward Krieger s theorem for countable groups June 6, / 13

27 The Full Group Definition For an action G (X, µ) define E X G = {(x, y) : g G g x = y}. The full group, denoted [EG X ], is the set of all Borel bijections θ : X X with x EG X θ(x) for a.e. x X. For θ [EG X ], there is a partition {Z g : g G} of X with θ(x) = g x for a.e. x Z g. Definition If C is a collection of sets with each Z g σ-alg G (C) then we call θ C-expressible. Brandon Seward Krieger s theorem for countable groups June 6, / 13

28 The Rokhlin Lemma All previous proofs of Krieger s theorem required the Rokhlin Lemma and the Shannon McMillan Breimann theorem. However, these tools are only available for the class of amenable groups. The Rokhlin Lemma Brandon Seward Krieger s theorem for countable groups June 6, / 13

29 The Rokhlin Lemma All previous proofs of Krieger s theorem required the Rokhlin Lemma and the Shannon McMillan Breimann theorem. However, these tools are only available for the class of amenable groups. The Rokhlin Lemma Let T : (X, µ) (X, µ) be meas.-preserving, invertible, and aperiodic. If n N and ɛ > 0 then there exists S X such that 0 i j n T i (S) T j (S) = and µ(s T (S) T n (S)) > 1 ɛ. Brandon Seward Krieger s theorem for countable groups June 6, / 13

30 Analysis of Rokhlin s Lemma Brandon Seward Krieger s theorem for countable groups June 6, / 13

31 Analysis of Rokhlin s Lemma Let T, X, S be as in Rokhlin s Lemma. The set S naturally decomposes each orbit into intervals (most of length n). Let θ [ET X ] be the function which cyclically permutes the points in these intervals. The action of θ induces a finite subequivalence relation E θ. Brandon Seward Krieger s theorem for countable groups June 6, / 13

32 Analysis of Rokhlin s Lemma Let T, X, S be as in Rokhlin s Lemma. The set S naturally decomposes each orbit into intervals (most of length n). Let θ [ET X ] be the function which cyclically permutes the points in these intervals. The action of θ induces a finite subequivalence relation E θ. For applications such as Krieger s theorem, the important properties are: S is small since µ(s) 1/n θ is S-expressible S is a transversal for E θ [x] Eθ = n for most x Brandon Seward Krieger s theorem for countable groups June 6, / 13

33 The Replacment to Rokhlin s Lemma and the Shannon McMillan Breimann Theorem Proposition (S. 2014) Let G (X, µ) with (X, µ) non-atomic. Let β be a finite collection of sets, let N N, and let ɛ > 0. Brandon Seward Krieger s theorem for countable groups June 6, / 13

34 The Replacment to Rokhlin s Lemma and the Shannon McMillan Breimann Theorem Proposition (S. 2014) Let G (X, µ) with (X, µ) non-atomic. Let β be a finite collection of sets, let N N, and let ɛ > 0. Then there exist n N S X with µ(s) < ɛ θ [E X G ] which is S-expressible and such that E θ admits a S-expressible transversal such that for a.e. x we have [x] Eθ = n Brandon Seward Krieger s theorem for countable groups June 6, / 13

35 The Replacment to Rokhlin s Lemma and the Shannon McMillan Breimann Theorem Proposition (S. 2014) Let G (X, µ) with (X, µ) non-atomic. Let β be a finite collection of sets, let N N, and let ɛ > 0. Then there exist n N S X with µ(s) < ɛ θ [E X G ] which is S-expressible and such that E θ admits a S-expressible transversal such that for a.e. x we have [x] Eθ = n and B β µ(b) ɛ < 1 n B [x] E θ < µ(b) + ɛ. Brandon Seward Krieger s theorem for countable groups June 6, / 13

36 Thank you! Brandon Seward Krieger s theorem for countable groups June 6, / 13