C2 FUNDAMENTAL THEORY OF DYNAMICAL SYSTEMS
HANDOUT 2
METRIC SPACES
The theory of abstract metric spaces is less concrete than many of the topics we will cover in lectures.
This handout introduces and explains some concepts that you will need to become familiar with. The
material is not explicitly examinable, but you should make some attempt to understand it, as many of
the themes appear again and again in dynamical systems. Don’t be put off by the structure of this
handout. Mathematical analysis is often presented in this way. Part of the purpose of this handout is
to remind you that dynamical systems is also a branch of pure mathematics (although this is not our
emphasis in C2)!
Metric spaces can be thought of as collections of states on which there is a well-defined notion of
distance (the metric). In C2, we have two basic reasons for studying metric spaces: (i) the phase
space of a dynamical system is usually a metric space. The metric induces a topological structure,
and many of the questions one asks in dynamical systems have a topological character; (ii) we need
some results from metric spaces to prove the local existence theorem for solutions to ordinary differ-
ential equations in the next handout.
DEFINITION 1: Let X be a collection of states, and let d: X × X → R be a function with the following
properties:
i) d(x,y) = 0 if and only if x = y.
ii) d(y,x) = d(x,y) ≥ 0 for all x,y∈R.
iii) d(x,y) ≤ d(x,z) + d(z,y) for all x,y,z∈R.
Such a d (d for distance) is called a metric (on X), and (X,d) is called an (abstract) metric space. Here
are a couple of examples of metric spaces:
EXAMPLES:
1) X = R, d(x,y) = |x - y|
2) X = R2 and
d(x,y) = (x 1 − y 1 )2 + (x 2 − y 2 )2
3) X = {collection of all continuous functions on [0,1] } and
d(f,g) = max f(x) - g(x)
[
x∈ 0 ,1]
The third example above can be generalised to the situation where the functions in X are r times
differentiable, and the rth derivative is continuous. Such functions are denoted by C r(X), and the
metric
d(f,g) = f-g + f(1) - g(1) + … + f(r) - g(r)
Metric Spaces 2
where the rth derivatives of f and g have been denoted by f(r) and g(r). Such spaces are important for
the existence theory of ordinary differential equations, and will re-appear when we discuss struc-
tural stability towards the end of the course.
Metric spaces are important because they have a natural topological structure. We will often need
topological concepts in dynamical systems because we want to be rigorous about issues such as
convergence.
DEFINITION 2: A sequence xk of points in X is said to converge to a point x*∈X if d(xk,x*) → 0 as k →
∞. Recall that this means that given any ε > 0 there exists an N∈Ν such that d(xk,x*) 0 then
B(x,ε) = { y∈X : d(y,x) 0 there exists some positive number δ > 0 such that
f(x) - f* 0 such that B(x,ε)
⊂ U.
Metric Spaces 3
The number ε depends on x, can be arbitrarily small, but must be positive. The intuition behind an
open set is that every point in an open set is an interior point (i.e. is a non-zero distance from the
boundary). Note that since an empty set contains no points x∈X it trivially satisfies the definition.
EXAMPLE: The interval (0,1) is an open set because if 0 0 there exists an N∈Ν such that
d(xn,xm) < ε for all n,m ≥ N. A metric space is called complete if every Cauchy sequence converges to
some limit x*∈X.
It is a standard result in elementary analysis that Rn is complete, and from the definitions it is sim-
ple to see that any closed subset of a complete metric space is also complete. Hence typically all the
state spaces we encounter in dynamics are complete. We then have the following classic theorem,
whose proof can be found in any analysis textbook, but is sufficiently straightforward to be an exer-
cise for the interested reader.
Metric Spaces 4
CONTRACTION MAPPING THEOREM: Let (X,d) be a complete metric space, and let P: X → X be a
transformation. If there exists a constant 0 ≤ λ < 1 such that
d(P(x),P(y)) ≤ λ d(x,y)
for all x,y∈X, then P is called a contraction mapping (with contraction rate λ). In such a case there
exists a unique x*∈X such that P(x*) =x*. Moreover, Pk(x) → x* as k → ∞ for every initial x∈X.
EXERCISE: (Proof of Contraction Mapping Theorem)
i) Show that for any m < n, and x∈X
1 − λn −m
d(Pn(x),Pm(x)) ≤ λm d(P(x),x)
1− λ
Hint: recall that Pn - Pm = Pm(P - Id) + P(Pm(P - Id + … + Pn-m-1(Pm(P - Id and that
)) ))
n −1 n −m −1
1 − λn −m
∑
k =m
λk = λm ∑λ
k =0
k
= λm
1− λ
ii) Hence deduce that the sequence { Pk(x) : k ≥ 0 } is Cauchy and hence has a limit point x*∈X.
iii) Show that P(x*) = x*. Hint: show that P is continuous.
iv) Prove that if P(y*) = y*, then x* = y*. This shows uniqueness. Hint: compute d(P(x*),P(y*)).
EXERCISE: This illustrates the role that the hypotheses of the theorem play .
i) Can you think of a contraction mapping on a non-complete metric space that does not have
a fixed point (in that space)? Show that if it does have a fixed point, it must be unique.
ii) Give an example to show that if λ = 1, the result may fail. See if you can think of both an ex-
ample which has no fixed points, and one in which the fixed point is not unique. Hint: see
Q3, example sheet 1.