3 hours UNIVERSITY OF MANCHESTER. 22nd May and. Electronic calculators may be used, provided that they cannot store text.

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1 3 hours MATH40512 UNIVERSITY OF MANCHESTER DYNAMICAL SYSTEMS AND ERGODIC THEORY 22nd May Answer ALL four questions in SECTION A (40 marks in total) and THREE of the four questions in SECTION B (20 marks each) Electronic calculators may be used, provided that they cannot store text. 1 of 7 P.T.O.

2 SECTION A MATH40512 Answer ALL four questions A1. (i) Let T be an ergodic measure-preserving transformation of the probability space (X, B, µ). State (without proof) Birkhoff s Ergodic Theorem. (ii) Let r 2 be an integer. What does it mean to say that a real number x (0, 1) is normal in base r? Prove that Lebesgue a.e. real number in (0, 1) is normal in base r. (You may assume that the map T(x) = rx mod 1 is ergodic with respect to Lebesgue measure.) (iii) Hence prove that Lebesgue a.e. real number x (0, 1) is simultaneously normal in every base r 2. A2. (i) Let Σ + 2 = {(x j) j=0 x j {0, 1}} denote the full one-sided two-shift. Let x = (x j ) j=0, y = (y j ) j=0 Σ + 2. Define { 1/2 n(x,y) if x y, d(x, y) = 0 otherwise, where n(x, y) = sup{n x j = y j for j = 0, 1,..., n 1}. Prove that d satisfies the triangle inequality: d(x, z) d(x, y) + d(y, z) for all x, y, z Σ (ii) Recall that the shift map σ : Σ + 2 Σ+ 2 is defined by (σx) j = x j+1. Prove that the periodic points for σ are dense in Σ + 2. A3. (i) Let T be a measurable transformation of a probability space (X, B, µ). What does it mean to say that µ is a T-invariant measure? 2 of 7 P.T.O.

3 1 1/γ 1/γ 2/γ 1 Figure 1: The graph of T. See Question A3(ii). (ii) Let γ > 1 satisfy γ 2 2γ 1 = 0. Define T : [0, 1] [0, 1] by T(x) = γx mod 1. See Figure 1. Define the probability measure µ on [0, 1] by µ(b) = k(x) dx where B k(x) = γ 2 γ 2(γ 1) if x [0, 1/γ) if x [1/γ, 1]. Show, using the Kolmogorov Extension Theorem, that µ is a T-invariant measure. A4. (i) Recall that if γ is a sub-σ-algebra of B and f L 1 (X, B, µ) then E(f γ) is the unique function such that (a) E(f γ) is γ-measurable, and (b) E(f γ) dµ = f dµ for all C γ. C C Recall that if B B then the conditional probability of B given γ is defined by µ(b γ) = E(χ B γ). Using the usual convention that we identify a partition with the σ-algebra it generates, show that if γ is a countable partition of X then µ(b γ)(x) = C γ χ C (x) µ(b C). µ(c) 3 of 7 P.T.O.

4 (ii) Recall that if β and γ are countable partitions then the conditional information of β given γ is defined to be I(β γ) = B β χ B (x) log µ(b γ)(x). Also recall that we write β γ if every element of the partition β is a union of sets in the partition γ. Show that if β γ then I(β γ) = 0. (iii) Recall that if α = {A i } and β = {B j } are countable partitions then their join is defined to be the partition α β = {A i B j A i α, B j β}. Prove that if α, β, γ are countable partitions of X then I(α β γ) = I(α γ) + I(β α γ). 4 of 7 P.T.O.

5 SECTION B MATH40512 Answer THREE of the four questions B5. (i) Let X be a compact metric space and let B denote the Borel σ-algebra. Let M(X) denote the space of Borel probability measures on X. What does it mean to say that a sequence µ n M(X) weak converges to µ M(X)? (ii) Let T : X X be a continuous transformation. Let x n X be a sequence of points in X. Define the sequence of measures µ n = 1 n 1 δ n T k (x n) where δ y denotes the Dirac delta measure at y. Show that the limit of any weak convergent subsequence of µ n is T-invariant. k=0 (You may use any characterisation of T-invariant measures without proof, provided that you state this clearly.) (iii) Recall that a continuous transformation of a compact metric space X is said to be uniquely ergodic if there is just one invariant probability measure. Suppose that T : X X is uniquely ergodic. Prove that for each continuous real-valued function f there exists a constant c(f) R such that 1 n 1 f(t k x) c(f) (1) n k=0 uniformly in x as n. (You may assume the Riesz Representation Theorem and the fact that M(X) is weak compact.) (iv) Let α R be irrational. Define T : R 2 /Z 2 R 2 /Z 2 by T(x, y) = (x + α, x + y). It can be shown that T is uniquely ergodic and that 2-dimensional Lebesgue measure is the unique invariant measure. By considering the function f(x, y) = e 2πiy g(x) in (1), prove that for any continuous g : R/Z R we have uniformly in x as n. (Hint: First calculate T k (x, y).) 1 n 1 e 2πikx e πik(k 1)α g(x + kα) 0 n k=0 5 of 7 P.T.O.

6 B6. (i) Let x n = (x 1 n, x2 n ) R2 be a sequence of points in R 2. What does it mean to say that x n is uniformly distributed mod 1? (ii) State, without proof, Weyl s criterion for the sequence x n R 2 to be uniformly distributed mod 1. (iii) Recall that α 1, α 2 R are said to be rationally independent if the only integers r 1, r 2, r such that r 1 α 1 + r 2 α 2 + r = 0 are r 1 = r 2 = r = 0. Prove that the sequence x n = (nα 1, nα 2 ) is uniformly distributed mod 1 if and only if α 1, α 2 are rationally independent. (iv) Let p 1 (n) = α 1 n 2 + β 1 n + γ 1, p 2 (n) = α 2 n 2 + β 2 n + γ 2, α 1, α 2 0. Determine a condition on the coefficients α 1, α 2, β 1, β 2, γ 1, γ 2 that ensures that x n = (p 1 (n), p 2 (n)) R 2 is uniformly distributed. (You may assume any results that were proved in the course, provided that you state them clearly.) B7. (i) Let T be a measure-preserving transformation of the probability space (X, B, µ). What does it mean to say that T is ergodic? (ii) Prove that the following are equivalent: (a) T is ergodic with respect to µ, (b) if f L 1 (X, B, µ) is such that f T = f µ-a.e. then we have that f is constant µ-a.e. (iii) Let α, β R. Define the map T : R 2 /Z 2 R 2 /Z 2 by T(x, y) = (x + α mod 1, x + y + β mod 1). [10 marks] By using Fourier series, show that if α is irrational then T is ergodic with respect to Lebesgue measure. 6 of 7 P.T.O.

7 B8. (i) Recall that if T is an ergodic measure-preserving transformation of (X, B, µ) and α is a countable partition of X such that H(α) < then we define ( n 1 ) 1 h(t, α) = lim n n H T j α. j=0 Show that ( ) h(t, α) = H α T j α. j=1 (You may assume the Increasing Martingale Theorem, any identities and inequalities regarding entropy that were proved in the course, and the fact that if a n R is a decreasing sequence such that (a n + + a 1 )/n l then a n l, without proof.) (ii) What does it mean to say that a countable partition α is a strong generator? State without proof Sinai s theorem on the use of strong generators to calculate entropy. (iii) Define T : [0, 1] [0, 1] by T(x) = 4x(1 x). Define the measure µ by µ(b) = 1 1 dx. π x(1 x) One can show that µ is an ergodic T-invariant probability measure. B By using Sinai s theorem, show that h(t) = log 2. (You may assume that the partition α = {[0, 1/2], [1/2, 1]} is a strong generator and any other identities from the course provided that you state them clearly.) [10 marks] END OF EXAMINATION PAPER 7 of 7