My title - Departamento de Matemática da Universidade do Minho

15 DIMENSIONS, FRACTALS AND ENTROPY 98
Brownian trajectories. Other continuous curves which are not differentiable are “typical”
paths of a Brownian motion.
Trajectories of a random walk on the plane and a Wiener process in 3-dimensional space
(from http://en.wikipedia.org/wiki/Brownian_motion)
15.3 Self-similarity and iterated function systems
Iterated Function Systems.
An iterated function system (IFS) is a finite collection
f 1 : R n → R n , f 2 : R n → R n , . . . , f m : R n → R n
of contractions of the Euclidean space R n . An invariant set for the IFS is a compact subset
K ⊂ R n such that
K = f 1 (K) ∪ f 2 (K) ∪ · · · ∪ f m (K)
Let Comp(R n ) be the space of non-empty compact subsets X ⊂ R n , equipped with the Hausdorff
metric
d H (X, Y ) = max{sup
inf
x∈X y∈Y
d(x, y) , sup
y∈Y
inf
x∈X
d(x, y)}
= inf{ε > 0 s.t. X ⊂ Y ε and Y ⊂ X ε }
(above, C ε = ∪ c∈C {x ∈ R n s.t. d(x, c) < ε} denotes the ε-neighborhood of a subset C ⊂ R n ). One
can prove that (Comp(R n ), d H ) is a complete space. The Hutchinson operator 32 H : Comp(R n ) →
Comp(R n ), defined as
H(X) = ∪ m k=1f k (X) ,
is a contraction of Comp(R n ). There follows from the Banach fixed point theorem that
Theorem. There exists a unique non-empty compact subset K ⊂ R n such that H(K) = K.
Moreover, K = lim n→∞ H(C) for any non-empty compact set C ⊂ R n .
The “chaos game” to plot the attractor. To plot the attractor, one can start with a (small)
set of points, and apply randomly the contractions f k ’s.
Exercícios.
• Show that the middle-third Cantor set is the invariant set for the contractions
x ↦→ 1 3 x x ↦→ 1 3 x + 2 3
on the real line.
Sierpinski gasket.
32 J.E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30, no. 5 (1981), 713-747.