MATH497C Midterm
Solutions.
Give solutions to the following problems. Points are awarded for correctness and clarity.
You may use results from class or the homeworks, but refer to any results you use with as much
accuracy as possible.
Problem 1 (4 Points)
Name:
MATH497C Midterm
Give solutions to the following problems. Points are awarded for correctness and clarity.
You may use results from class or the homeworks, but refer to any results you use with as much
accuracy as possible.
Problem 1 (4 Points). Let
MASS 2012 - 497C
In-class practice problems
October 25, 2012
Problem 1. Find the rotation number of the following maps:
(i) R (x) = x + , R/Z
(ii) f ([x]) = [x2 ], x is identied with representative in [0, 1)
(iii) ft ([x]) = [x2 + t], x is identied with r
MASS 2012 - 497C
In-class practice problems
October 18, 2012
A metric space X is said to be connected if whenever X = U V and U , V are open and nonempty, then
U V = .
Problem 1. Show that the following are equivalent:
(i) X is connected
(ii) If X = C1 C2
MASS 2012 - 497C
In-class practice problems
September 27, 2012
Problem 1. Let X and Y be metric spaces, and let X Y be equipped with the metric d(x1 , y1 ), (x2 , y2 ) =
max cfw_dX (x1 , x2 ), dY (y1 , y2 ). Show that:
1. (xn , yn ) (x, y ) if and only if
MASS 2012 - 497C
In-class practice problems
September 20, 2012
Let X and Y be metric spaces, X be compact, and C (X, Y ) = cfw_f : X Y be the set of all continuous
functions from X to Y . Show that:
1. The function d(f, g ) = sup cfw_d(f (x), g (x) : x X
MASS 2012 - 497C
In-class practice problems
September 13, 2012
Problem 1. Let G1 and G2 be abelian groups with translation-invariant metrics. Show that:
(i) If : G1 G2 is a continuous homomorphism, then ker is a closed subgroup
(ii) Let H be a closed subg
MATH497C In-Class Problems
September 6, 2012
These problems will not be graded, and are here for you to practice.
Problem 1. Let fc : Rn Rn be dened by fc (x) = cx. Classify the orbit types of fc for
values of c R
Problem 2. Let G be a group. Dene Lg : G
MATH497C Assignment 11
Solutions.
Problem 1. Show that the group of continuous homomorhpisms of Td has a canonical isomorphism with Md (Z) = cfw_d d matrices with entries in Z, where the group operation on continuous homomorhpisms and matrices is both add
MATH497C Assignment 10
Solutions.
Problem 1. * Let p be an irreducible integral polynomial (ie, p cannot be written as p = p1 p2 ,
where p1 and p2 are polynomial of degree greater than 1 with coecients in Q). Show that p
must have distinct roots (ie, each
MATH497C Assignment 9
Solutions
Problem 1. Let f : I 2 R2 be the horseshoe map that appeared in the lecture and notes. See
the notes or lectures for the denitions of W (x, y ) and W (x, y ) for = s, u. Show that for
every (x, y ) f and suitably small :
W
MATH497C Assignment 8
Solutions
Problem 1. Suppose that f and g are two commuting homeomorphisms (ie, f g = g f ).
Show that (f g ) = (f ) + (g ).
Proof. Note that in this case, we can conlude by induction that (f g )n = f n g n . This is trivial
when n =
MATH497C Assignment 7
Solutions
Problem 1. Let an R be a sequence of real numbers, and suppose that there is some L > 0
such that am+n am + an + L. Then limn an converges to some r R cfw_. [Hint: First
n
show that the sequence is bounded above, then show
MATH497C Assignment 5
Due: October 5, 2012
Problem 1. Let X be a compact metric space. We say that x X is an isolated point if there
is some > 0 such that B (x) = cfw_x, i.e. cfw_x is an open set. Prove that if X is countable then
there is a point x X tha
MATH497C Assignment 4
Due: September 28, 2012
Problem 1.
(a) Prove that if f : R/Z R/Z has degree d then for any x R/Z, card(f 1 (x) |d|.
(b) Prove that if f, g : R/Z R/Z then deg(f g ) = deg(f ) deg(g ).
(c) Show that if h : R/Z R/Z is a homeomorphism, t
MATH497C Assignment 3
Due: September 21, 2012
Problem 1. Let G be an abelian group, and d be a metric on G such that d(gh1 , gh2 ) = d(h1 , h2 )
for every g, h1 , h2 G (such a metric is translation invariant). Show that if H is a closed
subgroup of G (equ
MATH497C Assignment 2
Solutions
Problem 1. Dene cfw_x to be the fractional part of x, i.e. 0 cfw_x < 1 and x = cfw_x + [x] where
[x] is an integer (the integer part of x).
(a) Prove that given any irrational there are innitely many solutions p, q Z, q > 0
MATH497C Assignment 1
Solutions
Problem 1. Write a careful proof of Exercise 2.6 from the notes (you may use results from
other exercises therein).
Solution of Problem 1. A simple, closed curve in R2 is a continuous map : [0, 1] R2 wich
is injective on [0