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 differential 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 structural 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*) < ε for all k ≥ N.
We often write xk → x*. Such an x* is called a limit point, and in dynamical systems we often want to
know about such points. To this end, the notions of open and closed sets are fundamental; these are
most easily defined via open balls:
DEFINITION 3: If x∈X and ε > 0 then
B(x,ε)
{ y∈X : d(y,x) < ε }
=
is called the open ball or neighbourhood of radius ε centred on x (draw a picture). It also often denoted Bε (x) and often N(x,ε) is used instead (N for neighbourhood). We also sometimes need the
closed ball { y∈X : d(y,x) ≤ ε }. One can use open balls to make a rigorous definition of the limit of a
function:
DEFINITION 4 : A function f : X → R is said to converge to a value f* as x tends to x* if for every value
of ε > 0 there exists some positive number δ > 0 such that
f(x) - f*
ε
<
for all d(x,x*) < δ , or in other words
f(B(x*,δ))
⊂
B(f*,ε)
where we have adopted the notation f(A) = { f(x) : x∈A }. We write
lim f(x)
x→x *
=
f*
or f(x) → f*.
The idea behind this definition is that f(x) can be made arbitrarily close (the ε part) to f* by forcing x
to be sufficiently close (the δ part) to x*. Using this notion of convergence, we can make a precise
definition of continuity:
DEFINITION 5 A function f is said to be continuous at x* if
lim f(x)
x→x *
=
f(x*)
EXERCISE: Prove that f is continuous at x* if and only if f(xk) → f(x*) for every sequence xk such that
xk → x*.
To proceed further, we need some more definitions.
DEFINITION 6: A set U ⊂ X is called an open set if for every x∈X there exists an ε > 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 < x < 1 and ε x = min { x, 1-x } then B(x,εξ) ⊂
(0,1).
EXERCISE: If A and B are open sets, show that i) A ∩ B is open, ii) A ∩ B is open.
EXERCISE: If Ai is a countable collection of open sets show that i) the union of all the Ai is open but
ii) the intersection need not be open.
EXERCISE: [Harder] Show that a function f : X → R is continuous at every point x ∈ X if and only if
f-1(U) is open for every open set U ⊂ X. Recall that f-1 (U) = { x∈X : f(x)∈U}
DEFINITION 7: A set V ⊂ X is called a closed set if
X\V
=
{ x∈X : x∉V }
is an open set.
Closed sets include their boundaries, whereas open sets do not.
EXERCISE: Prove that V is a closed set if and only if for every sequence xk in V such that xk → x*, we
have x*∈V.
To see why the above exercise expresses a non-trivial concept, suppose that U = (0,1) and consider
1
the sequence xk = k . Then xk → 0 but 0∉(0,1), so (0,1) is not a closed set. Often, a set V will be defined
as the collection of states satisfying some dynamical condition, and we usually hope that such conditions are preserved under taking limits.
In case a set is not closed, we may still be interested in its closure.
_
DEFINITION 8: If U ⊂ X, then the closure U ⊂ X is the collection of limits of sequences in U.
_
EXERCISE: If V is closed, show that V = V.
_
DEFINITION 9: If U ⊂ V is such that U = V, then U is said to be dense in V.
Before completing discussion of the closure of sets, there is one important example which we will
use in the proof of the existence theorem for solutions to odes.
Finally, there is one more result from the theory of metric spaces that is used in many constructions
in dynamical systems (including the existence theorem for solutions to ordinary differential equations).
DEFINITION 10: A sequence x k is called Cauchy if given any ε > 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 simple 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 exercise 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
d(Pn(x),Pm(x))
λm
≤
1 − λn −m
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
∑
k =m
λk
=
λm
n −m −1
∑λ
k
k =0
=
λm
1 − λn −m
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 example which has no fixed points, and one in which the fixed point is not unique. Hint: see
Q3, example sheet 1.