Math 134: Notes
1. Notes
Definition 1.1 (Metric space). A metric space is a pair (X, d) where X is a set
and d : X × X → R≥0 such that
(1) For any x, y ∈ X, d(x, y) = 0 if and only if x = y.
(2) For any x, y ∈ X, d(x, y) = d(y, x).
(3) For any x, y, z ∈ X, d(x, y) ≤ d(x, z) + d(z, y).
Example 1.2. (R, d) where d(x, y) = |y − x| is a metric space. In this case, d is
called the “standard metric on R”.
Example 1.3. Let X be any set. Then we can turn X into a metric space by
considering it with the “discrete metric”. More explicitly, (X, d) is a metric space
where
(
1 if x 6= y,
d(x, y) =
0 if x = y.
p
Example 1.4. (Rn , d) where d((x1 , ..., xn ), (y1 , ..., yn )) = (x1 − y1 )2 + ... + (xn − yn )2
is a metric space. In this case, d is called the “standard metric on Rn ”.
Definition 1.5. A discrete dynamical metric system consists of a metric space and
a self-map. More explicitly, it is a triple (X, d, T ) where (X, d) is a metric space
and T : X → X.
Definition 1.6. Let (X, d, T ) be a discrete dynamical metric system. Then a
trajectory of a point x ∈ X is the sequence (T n (x))n∈N . Note that
(T n (x))n∈N = (x, T (x), T (T (x)), T (T (T (x))), ...).
Definition 1.7. Let (X, d, T ) be a discrete dynamical metric system and x ∈ X.
(1) We say that x is a fixed point if T (x) = x.
(2) We say that x is periodic if there exists some n ≥ 1 such that T n (x) = x.
(3) We say that x is recurrent if for every > 0 there exists some N ∈ N such
that d(T N (x), x) < .
(4) We say that x is (weakly) attracting if there exists δ > 0 such that for any
y ∈ X such that d(x, y) < δ, we have that limn→∞ T n (y) = x.
2
Example
1.8. Consider (R , d) with the standard metric and the map T =
0 1
. In this discrete dynamical system,
1 0
(1) Every point is periodic, i.e. for any (a, b) ∈ R2 , we have that T 2 ((a, b)) =
(a, b).
(2) For each point on the line x = y is a fixed point.
Example 1.9. Consider (R, d) with the standard metric and the map T : R → R
such that T (x) = x2 . We claim that the point 0 is both a fixed point and a (weakly)
attracting point.
Proof. Notice that T (0) = 02 = 0. Hence, 0 is fixed. Now we show that 0 is
attracting. Choose δ = 1 (in this case, we can choose δ equal to anything). Now
we need to show that if d(y, 0) < 1, then limn→∞ T n (y) = 0. Notice that
y
1
lim T n (y) = lim n = y lim n = y · 0 = 0.
n→∞
n→∞ 2
n→∞ 2
1
2
Hence 0 is weakly attracting.
Theorem 1.10. Suppose that (X, d, T ) is a DDMS. Suppose that T : (X, d) →
(X, d) is continuous. If x ∈ X is attracting, then x is a fixed point.
Proof. Since x is attracting, there exists some d ∈ X such that limn→∞ T n (d) = x.
Now notice that
T (x) = T ( lim T n (d)) = lim T n+1 (d) = lim T n (d) = x.
n→∞
n→∞
n→∞
Notice that the 2nd equality sign follows from the fact that the map T is continuous.
Example 1.11. Consider the interval [0, 1) with the standard metric. Fix α ∈
[0, 1). We define the map Tα : [0, 1) → [0, 1) where
(
x+α
if x + α ≤ 1,
Tα (x) =
x + α − 1 if x + α > 1.
(1) If α is rational, then every point in [0, 1) is periodic.
(2) If α is irrational, then no point is periodic. Every point is recurrent.