JOININGS, FACTORS, AND BAIRE CATEGORY

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1 JOININGS, FACTORS, AND BAIRE CATEGORY Abstract. We discuss the Burton-Rothstein approach to Ornstein theory. 1. Weak convergence Let (X, B) be a metric space and B be the Borel sigma-algebra generated by the open sets of X. Recall that in probability theory, we say that for probability measures on (X, B) that µ n µ weakly if µ n (f) µ(f) for all continuous and bounded functions f : X R, where µ(f) = f(x)dµ(x). In fact, this is would be called weak convergence in functional analysis. Exercise 1. Consider the case X = [0, 1] with the usual metric. Let µ n := 1 n 1 δ k/n, n where δ x is the usual point mass. Show that µ converges weakly. What is the limit? Exercise 2. Sometimes weak convergence is also defined equivalently by saying µ n µ weakly if lim inf µ n (G) µ(g) for all open sets G. Using Exercise 1, find an open set G such that lim inf µ n (G) > µ(g). Let [N] = {0, 1..., N 1}. Assign [N] the discrete metric. We are mainly interested in the case where X = [N] Z or [N] Z [N] Z. We give [N] Z the usual product metric d(x, y) := i Z k=0 2 i +1 1[x i y i ]. The following explicit metric between measures on [N] = {0, 1..., N 1} is compatible with the notition of weak convergence. For a measure µ on [N] Z, let µ i be the restriction of µ to [ i, i]. d (µ, ν) := 2 (i+1) d T V (µ i, ν i ). i=0 We similarly define the weak -metric for probability measures on [N] Z [N] Z. 1

2 2 JOININGS, FACTORS, AND BAIRE CATEGORY Exercise 3. Recall that if two processes are d close, then are also close in entropy. Show that this is not the case for the metric d. 2. Baire category Recall that a metric space (X, d) is said to be a Baire space if a countable intersection of open dense sets is still dense, and in particular non-empty. Theorem 4 (Baire category theorem). A complete metric space is a Baire space. Exercise 5. Show that the intersection of two open dense sets is again an open dense set. What if the sets are not assumed to be open? Exercise 6. Prove the Baire category theorem. Let ε > 0. Let z X. Let G i be open dense sets. In order to find x i N G i, with d(x, z) < ε, inductively apply the previous exercise to obtain a Cauchy sequence, which converges by completeness. Exercise 7. Where did you use a version of the axiom of choice in Exercise 6? Exercise 8. Use the Baire category theorem to show that R is uncountable. 3. Joinings Let µ and ν be shift-invariant measures on [N] Z. Recall that a joining of µ and ν is a shift-invariant coupling of µ and ν. Let J(µ, ν) denote the set of joinings of µ and ν. Exercise 9. Let T be the left-shift. Show that ξ J(µ, ν) may not be ergodic with repsect to T T even if µ and ν are. The set J(µ, ν) is a Baire space with the weak -metric. In fact as a consequence Helly s selection theorem the space is compact; tightness is automatic since the underlying metric space is compact. Another approach is to invoke the Banach-Alaoglu theorem. We will use the following variation of this fact. Theorem 10. Let µ and ν be ergodic invariant measures on [N] Z. Then the set J e (µ, ν) of ergodic joinings of µ and ν is a Baire space.

3 JOININGS, FACTORS, AND BAIRE CATEGORY 3 4. Factors Let p and q be probability measures on [N]. Suppose that φ : [N] Z [N] Z is a factor from B(p) to B(q). Let µ = p Z and ν = q Z. Let F be usual product sigma-algebra for [N] Z and let T = {, [N] Z} denote the trivial sigma-algebra. Note that the measure defined by ξ(a B) := µ ( A φ 1 (B) ) is a joining of µ and ν. For subsigma-algebras G, H F F, we write G H mod ξ if for every G G there exists H H such that ξ(g H) = 0. It is not hard to show that ξ has the property that T F F T mod ξ. (1) The converse is also true; that is, if ξ is joining such that (1) holds, then ξ is supported on a graph of a factor φ and ξ(a B) = µ(a φ 1 (B)). Consider the following idea. Let y [N] Z. Let B n := { z [N] Z : z i = y i, for all n i n }. Clearly, n 1 B n = {y}. Suppose ξ is such a joining, so that there exists A n such that ξ ( (A n [N] Z ) (B n [N] Z ) ) = 0. Consider A = n 1 A n. For each x A, set φ(x) = y. Thus if (1) holds for a joining ξ, then we can identify it with a factor φ. Similarly, if ξ is joining such that (1) holds and then in fact ξ is an isomorphism. F T T F mod ξ, (2) Theorem 11 (Residual Sinai theorem). If H(p) H(q), then the set of factors from B(p) to B(q) is an intersection of dense open sets in J e (µ, ν) and is nonempty. Exercise 12. Prove that if H(p) = H(q), then there exists an isomorphism of B(p) and B(q). Exercise 13. Show that if the Sinai factor theorem holds in the case of equal entropies H(p) = H(q), then this implies the version were H(p) > H(q). Hint: perturb q so that it has equal entropy to p: suppose q(0) > 0, and split the state 0 into two separate states a, a. Define q to agree with q except at the state 0.

4 4 JOININGS, FACTORS, AND BAIRE CATEGORY 5. Almost factors Let ε > 0. We write G ε H mod ξ, if for every G G there exists H H such that ξ(g H) < ε. Let P = {P 0,..., P N 1 } where P i = { x [N] Z : x 0 = i }. We say that ξ J e (µ, ν) is an ε-almost factor if T σ(p) ε F T mod ξ. Let V ε be the set of ε-factors. It is not hard to verify that V ε is an open set. Clearly, the intersection of V ε gives the set of factor from B(p) to B(q). Thus to prove Theorem 11, it suffices by the Baire category theorem, to prove that the V ε are dense. So we will be handed a joining ξ J(µ, ν) and we will need to make a pertubation of it to obtain a ε- almost factor. The following combinatorial theorem will help us make this perturbation. Theorem 14 (Hall s marriage theorem). Let B and G be finite sets. Let R B G. For each b B, let R(b, ) := {g G : (b, g) R} and set R(B, ) := b B R(b, ). Suppose that for every B B we have that B R(B, ). Then there exists an injection φ : B G such that φ R. Remark 15. The assumption in Theorem 14 is sometimes called the marriage condition. It is clearly a necessary condition for the conclusion of the theorem, but it turns out it is also sufficient. Exercise 16. Suppose there exists K such that for every b B, we have R(b, ) K, and for every g G, we have R(, g) K. Show that there exists a injection φ : B G such that φ R. Exercise 17 (Solitaire). Consider any arrangment of a standard deck of cards into a 4 13 array. Show that you can select you can select one card from each column so that you obtain a pile of 13 cards one from each rank (A, 2, 3,4,5,6,7,8,9,10,J,Q,K). Proof of Theorem 14. We will do a proof by induction of the size of B. The case B = 1 is trivial. Suppose B = n + 1. The proof consists of two cases. Suppose we are so lucky that for every B B with B B, we have B R(B, ) 1. Then fix (b, g) R and consider S := R \ {(b, g)}, M := B \{b}, and F := G\{g}. Clearly, the marriage condition is still satisfied with the triple (M, F, S), and M = n, so that the induction hypothesis applies. On the other hand, suppose that there exists B B with B B such that B = R(B, ). We have no choice but to map the elements

5 JOININGS, FACTORS, AND BAIRE CATEGORY 5 of B to R(B, ), which we can by the induction hypothesis. Let B be given by the disjoint union B = B M and G = R(B, ) F. Let S = R (M F ). So it suffices to verify the marriage condition for (M, F, S); this follows from the fact that B was chosen to obtain equality in the marriage conditon; if H M such that H < S(H, ), then we have that the set H B would contradict the marriage condition for (B, G, R). References [1] R. Burton, M. Keane, and J. Serafin. Residuality of dynamical morphisms. Colloq. Math., 85: , [2] R. Burton and A. Rothstein. Isomorphism theorems in ergodic theory. Technical report, Oregon State University, [3] T. Downarowicz and J. Serafin. A short proof of the Ornstein theorem. Ergodic Theory and Dynamical Systems, 32: , [4] R. Dudley. Real analysis and probability. Wadsworth, Pacific Grove, CA, [5] R. Durrett. Probability: theory and examples. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, fourth edition, [6] P. Hall. On representatives of subsets. J. London Math. Soc.(1), 10:26 30, [7] D. B. Leep and G. Myerson. Marriage, magic, and solitaire. Amer. Math. Monthly, 106(5): , [8] T. de la Rue. An introduction to joinings in ergodic theory. Discrete Contin. Dyn. Syst., 15: , 2006.