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EXERCISE SHEET #2 Exercise 1 (Rigidity of the Dyadic odometer): Let X = {0, 1}N be the dyadic odometer and for 0 < p < 1, µp is the product measure on X with µp [xj = 0] = p = 1 − µp [xj = 1] . In this exercise you will prove the rigidity property along 2n for the dyadic odometer on X. That is for any f : X → R measurable and > 0, n µp f ◦ τ 2 − f > −−−→ 0. n→∞ Steps: (a) Show that if f = g (x1 , .., xn ) depends only on the first n coordinates then for all k ≥ n, f ◦ τ 2k ≡ f. Hint: Odometer property. ψ(x) dµp ◦τ −1 1−p (b) Let ψ(x) := min {n ∈ N : xn = 1} − 2 so that dµ (x) = . Show that if q > 1 p p q < 1 then, so that p1 1−p p 1 − p ψ q M := < ∞. p Lq(X,µp ) (c) Show that for all n ∈ N, the chain rule to show that 2−n dµp ◦τ dµp dµp ◦τ 2 dµp = −n = 1−p p 1−p p ψ◦S n P2n −1 k=0 with S : X → X the shift map. Hint: Use ψ◦τ −k . (d) Use steps (b) and (c) to show that for any A ∈ BX with q > 1 as in step (b), −n ˆ dµ ◦ τ 2−n q−1 p µp τ 2 A = dµp ≤ M µp (A) q . dµp A Hints: S preserves µp and Hoelder’s inequality. (e) Finish the proof. Hint: Since functions which depend only on finitely many coordinates are dense, it follows that for general f : X → R measurable and > 0 there exists g = g (x1 , .., xn ) s.t. µp |f − g| > 2 < . It then follows that n n n µp f ◦ τ 2 − f > ≤ µp f ◦ τ 2 − g ◦ τ 2 > /2 + µp (|f − g| > /2) . Exercise #2 (Ergodic Maharam extension for the dyadic odometer): Let (X, B, µp , τ ) be the dyadic odometer and τ̃ (x, n) := (τ (x), y + φ(x)) : X × Z → X × Z, with φ(x) := min {n ∈ N : xn = 0} − 2. Let mp be the measure on X × Z defined by n p mp (A × {n}) = µp (A) 1−p . (a) Show that τ̃ preserves mp (mp ◦ τ̃ = mp ). Hint: For any z ∈ Z and A ∈ B, A × {z} := ∪∞ l=−1 (A ∩ [φ = l]) × {z}. 1 EXERCISE SHEET #2 2 (b) Prove ergodicity of (X × Z, B × BZ , mp , τ̃ ). P n −1 φ ◦ τ k (x) =: φ2n (x) = φ ◦ S n (x) and thus for any Steps: (b.1) Recall that 2k=0 n n F ∈ L∞ (X × Z, mp ) and n ∈ N, F ◦ τ̃ 2 (x, y) = F τ 2 x, n + φ ◦ S n (x) (b.2) Show that by the rigidity proposition (exercise 1) for all F ∈ L∞ (X × Z, mp ) there exists nk → ∞ s.t. n F τ 2 k x, z −−−→ F (x, z), ∀z ∈ Z and a.e. x ∈ X. k→∞ (b.3) Show that for any nk → ∞ there exists a subseqence nkl → ∞ s.t. for a.e. x ∈ X φ ◦ S nkl (x) = −1, ∀l ≥ 1. Hint: the events An = [φ ◦ S n = −1] are independent with µp (An ) = p, Borel Cantelli Lemma. (b.4) Assume now that F ∈ L∞ (X, mp ) is τ̃ invariant. Show that there exists a susequence nkl → ∞ s.t for a.e. x ∈ X and for all z ∈ Z, nk nk F (x, z) = F ◦ τ̃ 2 l (x, z) = F τ 2 l x, z − 1 −−−→ F (x, z − 1) l→∞ (b.5) Show that if F ∈ L∞ (X, mp ) is τ̃ invariant, then ∃f : X → R such that f (x) = F (x, z). Conclude that F is a.e. constant by ergodicity of (X, B, µp , τ ). Exercise #3 (Maharm’s Theorem): In this exercise you will prove Maharam’s Theorem which says that if (X, B, m, T ) is an invertible conservative non singular transformation then it’s Maharam extension T̃ : X × R is a conservative measure preserving transformation of (X × R, B × BR , µ) with dµ◦T y dµ(x, y) := dm(x)e dy and T̃ (x, y) := T x, y − log dµ (x) . (a) Show that T̃ preserves µ. (b) Show that if T is conservative then for any f ∈ L1 (X, m), if T̂ f ≤ f then T̂ f = f a.e.. The second property is called “dual incompressibility”. Here T̂ : L1 (X, m) is the dual operator ´ ´ defined by X T̂ f gdm = X f g ◦ T dm for all g ∈ L∞ (X, m). Hint: Hopf’s reccurence theorem (see exercise sheet #1). n (c) Writing (T n )0 (x) := dµ◦T and F (x) = supn≥0 (T n )0 (x), show that F ◦ T ≤ F/T 0 hence dµ since T is dual incompressible (by part (b)) F ◦ T = F/T 0 . (d) For a ∈ R, define Xa = X × (−∞, a). Show that µ (Xa ) = ea < ∞ and the set −n X satisfies T̃ −1 K = K . Ka := ∪∞ a a a n=0 T̃ (e) Deduce that T̃ |Ka is conservative hence X × R = ∪a∈R Ka ⊂ C T̃ where C T̃ is the conservative part of T̃ . Exercise 4 (Lin’s Theorem): n Prove that a non singular transformation (X, B, m, T ) is exact if and only if T̂ f −−−→ 0 1 n→∞ ´ for every f ∈ L1 (X, m) with X f dm = 0. ´ (a) Show that if T is not exact then ∃f ∈ L (X, m) with 1 X f dm = 0 and α > 0 such that n −n B choose f = 1 − m(A) 1 c . Show T̂ f ≥ α for all n. Hint: for non trivial A ∈ ∩∞ A n=1 T m(Ac ) A ´1 n ´ that A T̂ f dm = A f dm > 0. EXERCISE SHEET #2 (b) From now on assume T is exact. (b.1) Show that ∃gn ∈ L∞ (X, m) such that 3 ´ n dm = T̂ n f . f g ◦ T n X 1 (b.2) Show that by exactness of T , every weak ∗ limit in L∞ of gn ◦ T n is constant. (b.3) Show that for every subsequence nj → ∞, there exists a subsubsequence njk → ∞ such that njk T̂ f −−−→ 0 1 k→∞ and deduce the final part of Lin’s Theorem. Exercise 5: Let X = [0, 1] and T x = 2x mod 1. Calculate Tˆ :L1 (X, m) and show that ˆ n T̂ f −−−→ f dm uniformly on X. n→∞ X (b) Deduce that T is exact. Exercise 6: An invertible measure preserving transformation (X, B, m, T ) is a K-system (K for Kolmogorov) if there exists F ⊂ B such that T −1 F ⊂ F (F- is a factor), ∩n∈N T −n F (exactness n of F) and ∨∞ n=0 T F = B. Show that if (X, B, m, T ) is a conservative K- system then T is ergodic. Hint: Similar to Parry’s Theorem.