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1 Three hours MATH41112 THE UNIVERSITY OF MANCHESTER ERGODIC THEORY 31st May :00 17:00 Answer FOUR of the FIVE questions. If more than four questions are attempted, then credit will be given for the best four answers. Electronic calculators are permitted, provided they cannot store text. 1 of 6 P.T.O.

2 1. (i) Let x n R be a sequence of real numbers. What does it mean to say that x n is uniformly distributed mod 1? (ii) Recall that Weyl s criterion says that the following two statements are equivalent: (a) the sequence x n R is uniformly distributed mod 1; (b) for each l Z \ {0}, we have 1 n 1 lim e 2πilx j = 0. n n j=0 Prove that (b) (a). (You may use without proof the fact that trigonometric polynomials are uniformly dense in the space of real-valued continuous functions defined on R/Z.) [14 marks] (iii) Let α, β R. Define the sequence x n by x n = αn + β. Suppose that α Q. Use Weyl s Criterion to prove that x n is uniformly distributed mod 1. (iv) Let x n be a sequence. Let m 1 and define the sequence x (m) n of mth differences by x (m) n = x n+m x n. It was proved in the course that if, for every m 1, the sequence of mth differences is uniformly distributed mod 1 then x n is uniformly distributed mod 1. Using this result and (iii) above, prove that x n = αn 2 + βn + γ is uniformly distributed mod 1 if α Q. (v) Are either of the sequences n 2 + 2n + 1 n 2 + n + 2 [6 marks] uniformly distributed mod 1? Justify your answers. 2 of 6 P.T.O.

3 2. (i) Let T be a measurable transformation of the probability space (X, B, µ). What does it mean to say that µ is an invariant measure for T? (ii) Define T : [0, 1] [0, 1] by T (x) = 4x(1 x). Show that T does not preserve Lebesgue measure. Define a probability measure µ on [0, 1] by µ(b) = 1 π B dx x(1 x). Let [a, b] [0, 1]. Show that ([ 1 1 a µ, 1 ]) 1 b = 1 µ([a, b]) and hence show that µ is a T -invariant measure. [16 marks] (iii) Let X R 3 denote the sphere with centre (0, 0, 1) and radius 1, let S = (0, 0, 0) denote the south pole and let N = (0, 0, 2) denote the north pole. Define the stereographic projection map φ : X \ {N} R 2 {0} in the following way: for x X \ {N} draw the straight line through N and x and define φ(x) to be the unique point of intersection between this line and R 2 {0} R 3, as illustrated in Figure 1. Figure 1: See Question 2(iii). Define T : X X by T (x) = φ 1 ( 1 2 φ(x)) for x N and T (N) = N. Then T is a homeomorphism of X (you do not need to prove this). Determine all invariant Borel probability measures for T. [12 marks] 3 of 6 P.T.O.

4 3. (i) Let (X, B, µ) be a probability space and let T : X X be a measurable transformation. Suppose that µ is a T -invariant measure. What does it mean to say that T is ergodic with respect to µ? (ii) Let α, β R. Define T : R 2 /Z 2 R 2 /Z 2 : (x, y) + Z 2 (x + α, x + y + β) + Z 2. Show that T is ergodic with respect to Lebesgue measure µ if and only if α Q. (You may use any standard characterisations of ergodicity from the course, provided that you state them clearly. You may also assume without proof that µ is a T -invariant measure.) [16 marks] (iii) Let S = {1,..., k} and let Σ = {x = (x j ) j=0 x j S}. Let σ : Σ Σ denote the shift map: σ(x 0, x 1, x 2,...) = (x 1, x 2, x 3,...). For i j S, define cylinder sets by [i 0, i 1,..., i n 1 ] = {x = (x j ) j=0 x j = i j, 0 j n 1}. Let (p(1),..., p(k)) be a probability vector (so that p(j) > 0 and k j=1 p(j) = 1). Define a Bernoulli measure µ by defining it on cylinders by µ([i 0, i 1,..., i n 1 ]) = p(i 1 )p(i 2 ) p(i n 1 ) and then extending it to the Borel σ-algebra using the Hahn-Kolmogorov Extension Theorem. One can prove that µ is a σ-invariant measure (you do not need to do this). Let I, J be two cylinders. Prove that provided that n is sufficiently large. µ(i σ n J) = µ(i)µ(j) Hence prove that σ is ergodic with respect to µ. (You may use without proof the fact that if there exists K > 0 such that µ(b)µ(i) Kµ(B I) for all cylinders I then µ(b) = 0 or 1.) [12 marks] 4 of 6 P.T.O.

5 4. Let X be a compact metric space, let B denote the Borel σ-algebra, and let T : X X be continuous. Let M(X) denote the space of Borel probability measures on X. Let C(X, R) denote the space of all continuous real-valued functions defined on X. (i) What does it mean to say that a sequence of measures µ n M(X) weak* converges to µ M(X)? (ii) The measure T µ is defined, for µ M(X), by (T µ)(b) = µ(t 1 B). (You may assume without proof that T µ M(X).) Prove that integration with respect to T µ is given by the formula f d(t µ) = f T dµ for f L 1 (X, B, µ). [6 marks] (iii) Prove that the map T : M(X) M(X) is weak* continuous. Prove that the space M(X, T ) of all T -invariant Borel probability measures is a weak* closed subset of M(X). [6 marks] (iv) State, without proof, the Riesz Representation Theorem. Prove that the following two conditions are equivalent: (i) µ M(X) is T -invariant; (ii) for all f C(X, R) we have f T dµ = f dµ. [8 marks] (v) Suppose that T : [0, 1] [0, 1] is continuous. Prove that there exists some x [0, 1] such that δ x is T -invariant. Does this remain true if [0, 1] is replaced by the circle R/Z? [8 marks] 5 of 6 P.T.O.

6 5. (i) Let (X, B, µ) be a probability space and let T : X X be an ergodic measure-preserving transformation. Let f L 1 (X, B, µ). State, without proof, Birkhoff s Ergodic Theorem. (ii) Suppose that T is ergodic with respect to µ and let A B. Use Birkhoff s Ergodic Theorem to show that the frequency with which the orbit of µ-almost every point x X visits A is equal to µ(a). (iii) Let (X, B, µ) be a probability space and let T : X X be an ergodic measure-preserving transformation. Let f L 1 (X, B, µ). Let A, B B. Deduce from Birkhoff s Ergodic Theorem and the definition of the Lebesgue integral that (a) implies (b) in the following: (a) T is ergodic with respect to µ (b) for all f, g L 2 (X, B, µ) we have 1 n 1 lim n n j=0 f(t j x)g(x) dµ = f dµ g dµ. (1) (One can also prove that (b) implies (a) but you are not required to do this.) [12 marks] (iv) Suppose that for all f, g L 2 (X, B, µ) we have f(t n x)g(x) dµ = lim n Prove that T is ergodic with respect to µ. f dµ g dµ. (2) Let α R be irrational. Recall that the irrational circle rotation T : R/Z R/Z, T (x) = x + α mod 1 is ergodic with respect to Lebesgue measure (you do not need to prove this). Give an example of a pair of L 2 functions f, g to show, however, that (2) does not hold for T. (Hint: consider exponential functions.) [8 marks] (v) Suppose that (X, B, µ) is a probability space and T : X X is a measure-preserving, but not necessarily ergodic, transformation. Let A, B B be arbitrary sets and suppose that µ(a), µ(b) > 0. Is it necessarily true that for µ-almost every point x of A there are infinitely many n for which T n (x) A? Is it necessarily true that for µ-almost every point x of B there are infinitely many n for which T n (x) A? In each case justify your answer by either quoting a theorem from the course or by giving a counter-example. END OF EXAMINATION PAPER 6 of 6

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