TOPOLOGY PROBLEMS (1) State the Arzela-Ascoli ... - Neil Strickland

TOPOLOGY PROBLEMS 7
(30) Given a complex number c, define q c (z) = z 2 + c and
f n (c) = q c (q c (q c . . . (0) . . .))
where q c is applied n times. In other words:
f 0 (c) = 0
f n+1 (c) = q c (f n (c)) = f n (c) 2 + c
(a) Give formulae for f 1 (c), f 2 (c) and f 3 (c).
(b) Prove that f n : C −→ C is continuous. A little care is needed here to avoid talking
nonsense.
(c) The Mandelbrot set M is defined as
M = {c ∈ C | |f n (c)| ≤ 2 for all n}
Prove that M is bounded and closed.
Later on, we shall prove some other topological properties of M, including the fact that
it has no holes. It is also known that M is connected, but this is a deep result, relying
on some substantial work in complex analysis. It is also known that M is the closure
of its interior. One of the most important questions about M, which I believe is still
unanswered, is whether it locally connected. It is known that many interesting things
would follow from this.
(31) Let X be a compact Hausdorff space, and C(X) the set of continuous functions u: X −→ R.
Define two maps b and t (“bottom” and “top”) from C(X) to R by
b(u) = min{u(x) | x ∈ X}
t(u) = max{u(x) | x ∈ X}
(why are these finite ?). Prove that t and b are continuous.
A measure on X is defined to be a map µ: C(X) −→ R such that
(a) (linearity) µ(au + bv) = aµ(u) + bµ(v) ( where a, b ∈ R and u, v ∈ C(X)).
(b) (positivity) if u ≥ 0 (i.e. u(x) ≥ 0 for all x) then µ(u) ≥ 0.
(c) (normalisation) µ(1) = 1 (where the first 1 is the constant function).
Here are some examples with X = [0, 1]:
(a) δ(u) = u(0) (the Dirac measure).
(b) λ(u) = ∫ 1
u(x)dx (Lebesgue measure).
0
(c) ρ n (u) = 1 ∑ n
n+1 k=0 u(k/n)
We shall consider the following infinite product space:
Z =
∏
[b(u), t(u)] ⊂
∏
R
u∈C(X)
u∈C(X)
(a) Describe the elements of Z. This is not supposed to be a deep question; just transcribe
the general definition of an infinite product and perhaps tidy up the result a bit.
(b) Give two examples of elements of Z.
(c) Show that if µ is a measure then b(u) ≤ µ(u) ≤ t(u).
(d) Show that every measure µ is a continuous map C(X) −→ R. (This is not actually
relevant to what follows).
(e) Define an injective function j : M(X) −→ Z (there is only one reasonable one).
(f) Given u, v ∈ C(X) consider the set
A u,v = {α ∈ Z | π u (α) + π v (α) = π u+v (α)}
Show that A u,v is closed in Z and contains j(M(X)). By elaborating on this idea,
show that j(M(X)) is closed in Z and thus compact. We identify M(X) with
j(M(X)) and give it the subspace topology.
(g) Show that µ n −→ µ in M(X) iff µ n (u) −→ µ(u) for every u ∈ C(X). In particular, you
can show that ρ n −→ λ in M([0, 1]).
(0) Which of the following pairs (X, d) is a metric space ?
(a) X = R n , d(x, y) = ∑ n
k=1 k|x k − y k |