MA 771 Exercises
1.1. Let X = [0, 1]. Dene f : X X by f (x) = 1 x. Describe the suspension M of
f : X X and the dynamics of the suspended ow t : M M .
1.2. Let f be complex conjugation of the circle S 1 . In other words, if S 1 is thought of as
all comple
MA 771 Exercises
1.4. Give an example of a compact space X and a map f : X X such that
(f |) = (f ).
1.5. Suppose that X is a compact space and f : X X is a homeomorphism. If U is
a neighborhood of (f ) and x X , show that there exists an integer N such t
MA 771 Exercises
2.1. Suppose p and q = 0 are relatively prime integers and that k is an integer multiple
of q . Let
F (x) = x + sin(2kx),
where 0 < < 1/(2k ). The map F : R R induces a map f : S 1 S 1 using the
universal covering map u : R S 1 . Let g =
MA 771 Exercises
2.5. Given an orientation-preserving homeomorphism f : S 1 S 1 with no periodic points,
we dened a number sf in class using next nearest neighbors and continued fractions.
Show that sf = f in R/Z (f denotes the rotation number of f ).
Las
MA 771 Exercises
3.1. Robinson 4.1 (p. 129)
3.2. Use Jordan canonical form and equivalence of norms (see handout on web page) to
prove Theorem 9.1 in Chapter 4 of Robinson.
3.3. Using the theorem cited in the previous exercise, prove Theorem 6.1 in Chapte