MATH36206 - MATHM6206
Dynamical Systems and Ergodic Theory
Teaching Block 1, 2014/15
Lecturers:
Prof. Jens Marklof
Dr. Corinna Ulcigrai
PART II: LECTURES 8-15
course web site: www.maths.bris.ac.uk/majm/DSET/
Copyright c University of Bristol 2010 & 2014.
MATH36206 - MATHM6206
Dynamical Systems and Ergodic Theory 2014/15
Solutions for Homework 7
Feedback
Q1: (Level 3) Perfectly done by all the students.
(Level M) Well done. Some students forgot to mention the Riemann-Lebesgue Lemma in their
arguments to pr
MATH36206 - MATHM6206
Dynamical Systems and Ergodic Theory 2014/15
Solutions for Homework 7
Feedback
Q1: (Level 3) Many students had problems in part (b), seemingly due to some confusion with the
denition of ergodicity.
(Level M) Most students were correc
MATH36206 - MATHM6206
Dynamical Systems and Ergodic Theory 2014/15
Solutions for Homework 5
Feedback
Solutions
Solutions to Exercise 1
Part (a) Let us prove that the set Sn1 is (n, )separated. Let x = x0 , . . . , xn1 , 1, 1, . . . and
y = y0 , . . . , yn
MATH36206 - MATHM6206
Dynamical Systems and Ergodic Theory 2014/15
Solutions for Homework 4
Feedback
Q1: Although all the students got the correct idea, there were small mistakes in most of the scripts,
namely:
in part (a), the fact that F acts like the d
MATH36206 - MATHM6206
Dynamical Systems and Ergodic Theory 2014/15
Homework 10
Hand in on Thursday, 11 December 2014, during class.
Exercise 1. Consider the baker map F : [0, 1]2 [0, 1]2 and consider the Lebesgue measure
on [0, 1]2 . You can use that F p
MATH36206 - MATHM6206
Dynamical Systems and Ergodic Theory 2014/15
Solutions for Homework 3
Feedback
Q1: Well done in general. In part (b) some students draw the image of the unit square [0, 1)2 in R2
and forgot to project the result into T2 .
Q2: Part (a
MATH36206 - MATHM6206
Dynamical Systems and Ergodic Theory 2014/15
Homework 7
Hand in on Thursday, 20 November 2014, during class.
Exercise 1. Let X = T2 and TA : T2 T2 be the toral automorphism
TA (x, y) = (x mod 1, x + y mod 1).
Let B be algebra of Bore
MATH36206 - MATHM6206
Dynamical Systems and Ergodic Theory 2014/15
Solutions for Homework 1
Feedback
1) Overall well done, although a number of careless mistakes that could have been avoided, especially in part a).
Note: to show a period is minimal it is
MATH36206 - MATHM6206
Dynamical Systems and Ergodic Theory 2014/15
Homework 8
Hand in on Thursday, 27 November 2014, during class.
Exercise 1. Let X = R/Z, B the Borel algebra, the 1dimensional Lebesgue measure.
Let Tm (x) = mx mod 1 (where m > 1 is an in
MATH36206 - MATHM6206
Dynamical Systems and Ergodic Theory 2014/15
Solutions for Homework 9
Feedback
Q1: (Level 3) Everyone used the correct strategy; although, many mistakes were present when comparing the Fourier coecients, e.g. more than one student wr
MATH36206 - MATHM6206
Dynamical Systems and Ergodic Theory 2014/15
Homework 1
Hand in on Thursday, 9 October 2014, during class.
Exercise 1.1. [Level 3 only]
Let R : S 1 S 1 where = p/q and p and q are coprime, i.e., gcd(p, q) = 1.
(a) Draw an orbit of R
MATH36206 - MATHM6206
Dynamical Systems and Ergodic Theory
Teaching Block 1, 2014/15
Lecturers:
Prof. Jens Marklof
Dr. Corinna Ulcigrai
PART III: LECTURES 16-30
course web site: www.maths.bris.ac.uk/maxcu/DynSysErgTh/
Copyright c University of Bristol 201
MATH36206 - MATHM6206
Dynamical Systems and Ergodic Theory
Teaching Block 1, 2014/15
Lecturers:
Prof. Jens Marklof
Dr. Corinna Ulcigrai
PART I: LECTURES 1-7
course web site: www.maths.bris.ac.uk/majm/DSET/
Copyright c University of Bristol 2010 & 2014. Th
MATH36206 - MATHM6206
Dynamical Systems and Ergodic Theory 2014/15
Homework 5
Hand in on Thursday, 6 November 2014, during class.
Exercise 1. Let + : + + be the full shift on N = cfw_1, . . . , N N symbols with the
N
N
distance the distance
+
|xi yi |
,
MATH36206 - MATHM6206
Dynamical Systems and Ergodic Theory 2014/15
Homework 4
Hand in on Thursday, 30 October 2014, during class.
Exercise 1. Let F : [0, 1]2 [0, 1]2 be the baker map.
(a) Show that the baker map has sensitive dependence on initial conditi
MATH36206 - MATHM6206
Dynamical Systems and Ergodic Theory 2014/15
Homework 6
Hand in on Thursday, 13 November 2014, during class.
Exercise 1. Let d be the metric on + , where > N and let + + be the subshift
N
A
N
space associated to an N N irreducible tr
MATH36206 - MATHM6206
Dynamical Systems and Ergodic Theory 2014/15
Homework 2
Hand in on Thursday, 16 October 2014, during class.
Exercise 1.1. Let f (x) = 2x mod 1 be the doubling map on X = [0, 1). Code x [0, 1)
with its itinerary with respect to P0 = [
MATH36206 - MATHM6206
Dynamical Systems and Ergodic Theory 2014/15
Solutions for Homework 10
Feedback
Q1: (Level 3) Although the strategy was generally correct, many mistakes appeared in most of the
scripts. Important: ergodicity does not imply mixing, th
MATH36206 - MATHM6206
Dynamical Systems and Ergodic Theory 2014/15
Solutions for Homework 2
Feedback
Q1: Done very well in general. Many students struggled in parts (b) and (c) to present their inductive
ideas formally - most had the right idea however. S
MATH36206 - MATHM6206
Dynamical Systems and Ergodic Theory 2014/15
Homework 9
Hand in on Thursday, 4 December 2014, during class.
Exercise 1. Let X = T2 be the two dimensional torus with the two dimensional Lebesgue
measure . Let R\Q be an irrational numb
7
Modes of Convergence
As usual we let (X, X, ) be any measure space.
Remark. We are already familiar with uniform convergence, pointwise convergence, convergence -a.e. and convergence in Lp (1 p ). Now we
will introduce several new modes of convergence,
Dynamical Systems and Ergodic Theory
Solutions Homework 9
Solutions for Problem Set 9
Feedback
Again most of the questions were done well. Remember when you use a result from the notes
to quote the result you are using. In particular remember to state whe
Dynamical Systems and Ergodic Theory
Solutions Homework 9
Solutions for Problem Set 9
Feedback
Again most of the questions were done well. Remember when you use a result from the notes
to quote the result you are using. A lot of solutions to questions 1 a
Dynamical Systems and Ergodic Theory
Solutions Homework 8
Solutions for Problem Set 8
Feedback
On the whole most of the questions were done well. However only one of the M level homeworks had a correction solution of question 8. For questions 1 and 3 near
Dynamical Systems and Ergodic Theory
Solutions Homework 6
Solutions for Problem Set 7
Feedback
On the whole most of the questions were done well. However a common mistake was to
confuse the inverse function of T with the preimage of a set A under the map
Dynamical Systems and Ergodic Theory Solutions and Feedback Homework 4
Solutions for Problem Set 4
Feedback
This problem set went generally well and several people achieved quite high marks. Most
mistakes were in Exercise 5.6. In the computations of entro
Dynamical Systems and Ergodic Theory Solutions and Feedback Homework 2
Solutions and feedback to set problems of Problem Set 2
Feedback
Also this Homework went quite well. Level 3 did a little worst on average then last time. A
few level M did not hand in