MATH41112
Three hours
THE UNIVERSITY OF MANCHESTER
ERGODIC THEORY
2nd June 2015
9:45am – 12:45pm
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.
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MATH41112
1.
(i) Let p(n) = αk nk + αk−1 nk−1 + · · · + α1 n + α0 , n ∈ N, be a degree k polynomial. State,
without proof, Weyl’s Theorem on Polynomials on the uniform distribution mod 1 of p(n).
[2 marks]
(ii) Let xn = (xn,1 , xn,2 ) ∈ R2 be a sequence of points in R2 . What does it mean to say that xn is
uniformly distributed mod 1?
State, without proof, Weyl’s Criterion for the sequence xn ∈ R2 to be uniformly distributed
mod 1.
[4 marks]
(iii) Let α1 , α2 ∈ R. Recall that α1 , α2 , 1 are said to be rationally independent if the only rationals
r1 , r2 , r ∈ Q such that r1 α1 + r2 α2 + r = 0 are r1 = r2 = r = 0.
Prove that the sequence xn = (nα1 , nα2 ) ∈ R2 is uniformly distributed mod 1 if, and only if,
α1 , α2 , 1 are rationally independent.
[8 marks]
(iv) Let p(n) = α1 n2 +β1 n+γ1 , q(n) = α2 n2 +β2 n+γ2 where α1 , α2 , β1 , β2 , γ1 , γ2 ∈ R and α1 , α2 6= 0.
Show that if at least one of α1 , α2 , 1 or β1 , β2 , 1 are rationally independent then the sequence
xn = (p(n), q(n)) ∈ R2 is uniformly distributed mod 1.
(You may use any results from the course without proof provided that you state them clearly.)
[8 marks]
(v) Let p(n) =
√
2n2 +
√
3n +
√
5.
Consider each of the polynomials q(n) below. In each case, decide whether xn = (p(n), q(n)) ∈
R2 is uniformly distributed mod 1. Justify your answers.
√
√
√
(a) q(n) = 3n2 + 3n + 5.
√
√
√
(b) q(n) = 2n2 + 3n + 3.
[8 marks]
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MATH41112
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 ?
What does it mean to say that µ is an ergodic measure for T ?
Suppose that T has a fixed point at x ∈ X. Prove from the definition that the Dirac
delta measure δx is an invariant measure for T . Prove that δx is an ergodic measure for
T.
[8 marks]
Define the map T : [0, 1] → [0, 1] by T (x) = (n + 1) − n(n + 1)x if x ∈ (1/(n + 1), 1/n] and T (0) = 0.
(This map is called the alternating Lüroth map.) Let B be the Borel σ-algebra on [0, 1] and let µ
denote Lebesgue measure on [0, 1].
(ii) Sketch the graph of T .
By expressing
1
n(n+1)
in terms of partial fractions, or otherwise, prove that
∞
X
n=1
1
= 1.
n(n + 1)
Hence prove that µ is an invariant measure for T .
[10 marks]
The following result was proved in the course: Let A0 be a collection of intervals and let A
denote the set of finite unions of intervals in A0 . Suppose that A is an algebra of subsets that
generates the Borel σ-algebra B. Suppose that B ∈ B is such that there exists K > 0 for which
µ(B)µ(I) ≤ Kµ(B ∩ I) for all sets I ∈ A0 . Then µ(B) = 0 or 1.
(iii) Prove that µ is an ergodic measure for the alternating Lüroth map T defined above.
(You may use without proof any criterion stated in the course for an algebra A to generate the
Borel σ-algebra.)
[12 marks]
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3. Let Σ = {x = (xj )∞
j=0 | xj ∈ {0, 1, 2}} denote the shift space on 3 symbols. For x, y ∈ Σ, x 6= y,
define n(x, y) by n(x, y) = n if xj = yj for 0 ≤ j ≤ n − 1 but xn 6= yn . Define
1
if x 6= y,
2n(x,y)
d(x, y) =
0
if x = y.
This is a metric on Σ (you do not need to prove this).
(i) Let σ : Σ → Σ denote the shift map defined by (σ(x))j = xj+1 . Equip the circle R/Z with the
standard metric. Define π : Σ → R/Z by
π(x) = π(x0 , x1 , . . .) =
x0 x1
xn
+ 2 + · · · + n+1 + · · · .
3
3
3
Show that π is continuous.
[4 marks]
(ii) Let T : R/Z → R/Z be the map T (x) = 3x mod 1. Show that π ◦ σ = T ◦ π.
[2 marks]
(iii) Let µ be a σ-invariant Borel probability measure on Σ. Define π∗ µ by π∗ µ(B) = µ(T −1B) for
a Borel subset B ⊂ R/Z. Show that π∗ µ is a T -invariant Borel probability measure. (You may
assume that π is a surjection.)
[2 marks]
(iv) Show that if µ is an ergodic measure for σ then π∗ µ is an ergodic measure for T .
Conclude that there are uncountably many different ergodic measures for T .
(You may use without proof the fact that Bernoulli measures are ergodic for the shift map.)
[8 marks]
(v) Let C = [i0 , . . . im−1 ] = {x = (xj )∞
j=0 ∈ Σ | xj = ij , 0 ≤ j ≤ m − 1} be a cylinder and let χC be
its characteristic function. Prove that χC is continuous.
Recall that x ∈ Σ is said to be periodic with period n if σ n (x) = x. How many periodic points
of period n are there in Σ?
Let Per(n) denote the set of all periodic points of period n. Define
µn =
1
3n
X
δx
x∈Per(n)
and let µ denote the Bernoulli (1/3, 1/3, 1/3)-measure.
R
R
Let C be any cylinder. Prove that limn→∞ χC dµn = χC dµ.
Use the Stone-Weierstrass Theorem and any standard results from the course to show that
µn ⇀ µ (here ⇀ denotes weak∗ convergence).
[14 marks]
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4. Let X be a compact metric space and let B denote the Borel σ-algebra of X. Let T : X → X be
a continuous transformation.
(i) Recall that M(X, T ) denotes the set of all T -invariant Borel probability measures.
What does it mean to say that µ ∈ M(X, T ) is an extreme point? State, without proof, the
relationship between extreme points of M(X, T ) and ergodic measures.
[4 marks]
(ii) Recall that T is said to be uniquely ergodic if there is a unique invariant Borel probability
measure µ.
Briefly explain why, in this case, µ must be ergodic.
[2 marks]
(iii) Recall Oxtoby’s Ergodic Theorem: Let T be a continuous transformation of a compact metric
space X. Then T is uniquely ergodic if, and only if, for each f ∈ C(X, R) there exists a
constant c(f ) such that
n−1
1X
lim
f (T j (x)) = c(f )
n→∞ n
j=0
uniformly for x ∈ X.
Let X = R/Z. Use Oxtoby’s Ergodic Theorem and the Stone-Weierstrass Theorem to prove
that T (x) = x + α mod 1 is uniquely ergodic (with Lebesgue measure as the unique invariant
measure) if, and only if, α 6∈ Q.
[12 marks]
(iv) Suppose that T : X → X is a homeomorphism and is uniquely ergodic. Show that every orbit
of T is dense if, and only if, µ(U) > 0 for all non-empty open sets U.
(You may use, without proof, the fact that a continuous transformation of a compact metric
space has at least one invariant probability measure.)
[10 marks]
(v) Suppose that T : X → X is a uniquely ergodic homeomorphism of a compact metric space and
that every orbit of T is dense. Does it follow that µ(C) > 0 for every non-empty closed subset
C of X?
[2 marks]
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5.
(i) Let (X, B, µ) be a probability space and let T : X → X be an ergodic measure-preserving
transformation. Let f ∈ L1 (X, B, µ).
State, without proof, Birkhoff’s Ergodic Theorem.
[2 marks]
(ii) Let (X, B, µ) and T be as in (i). Suppose that f : X → R, f ≥ 0, is measurable but
Deduce from Birkhoff’s Ergodic Theorem that for µ-a.e. x ∈ X we have
R
f dµ = ∞.
n−1
1X
lim
f (T j (x)) = ∞.
n→∞ n
j=0
(You may assume, without proof, the Monotone Convergence Theorem.)
[8 marks]
(iii) Let X = [0, 1], equipped with the Borel σ-algebra B and Lebesgue measure λ. Let T (x) =
10x mod 1.
Let x ∈ [0, 1] and write x as a decimal: x = ·x0 x1 x2 . . ., xj ∈ {0, 1, . . . , 9}.
Relate the decimal expansion of T (x) to the decimal expansion of x.
By applying Birkhoff’s Ergodic Theorem to a suitable function f , prove that for Lebesgue a.e.
x ∈ [0, 1] we have
n−1
1X
lim
xj = 4.5.
n→∞ n
j=0
(You may assume without proof that T is ergodic with respect to Lebesgue measure λ.)
[8 marks]
(iv) Let X = [0, 1], equipped with the Borel σ-algebra B and Gauss’ measure µ. Recall that µ is
defined by
Z
dx
1
, B ∈ B.
µ(B) =
log 2 B 1 + x
Prove that there exist constants c, C > 0 such that for all B ∈ B we have
cλ(B) ≤ µ(B) ≤ Cλ(B).
[2 marks]
(v) Let S(x) = 1/x mod 1 denote the Gauss map. Let x ∈ (0, 1] have continued fraction expansion
1
x=
1
x0 +
x1 +
, xj ∈ N.
1
x2 + · · ·
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Relate the continued fraction of S(x) to the continued fraction expansion of x.
By applying the result you proved in part (ii) of this question to a suitable function f , prove
that for Lebesgue a.e. x ∈ [0, 1] we have
n−1
1X
xj = ∞.
n→∞ n
j=0
lim
(∗)
(You may assume without proof that S is ergodic with respect to Gauss’ measure µ.)
[8 marks]
(vi) Suppose that x ∈ [0, 1] is rational. What can you say about the limit in (*) in this case?
[2 marks]
END OF EXAMINATION PAPER
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