Whipping-in-mainstream-movies-

fractal geometry and fixed point theorem Banach

Introduction.

Let (E, d) a metric space complet.Pbf (E) denotes


all parties bounded and closed EK (E) denotes all

the compact subsets of E. We define on Pbf (E) Hausdorff distance

:

For A, B Pbf (E), D (A, B) = max (d '(B, A), (A, B))

or (B , A) = sup {d (x, A), x B} and (A, B) = sup {d (x, B), x in A};

d (x, A) (Resp . d (x, B)) denotes the distance from x to A (resp. B)

We prove that (Pbf (E), D) and (K (E), D) are complete metric spaces

.

Exercise.

(E, d) denotes a metric space complet.Soit system

iterated function (or ISF) on E, ie of a set of applications

{f_ {i}} (1

application k_ {i} strictly-contracting of E in E.


It defines a mapping F on K (E):

F (A) = U f_ {i} (A).
1

i) Check that K (E) is stable under F.

ii) Prove that F is strictly k-contracting or

k = max (k_ {i}, i = 1,2, ... n) (K (E) D) (K (E), D)

Deduce that there exists a C unique K (E) (called Attractor ISF)

as: C = U

f_ {i} (C).


1 C is a fractal.

iii) Show that for any x in C, F ^ {p} converges to x in C (K (E), D). <=i<=n),telle que chaque f_{i} est

iv) Examples.

a) in the real line.

Let f_ {1} {2} f_ and two applications of IR into IR defined by:

f_ {1} (x) = (1 / 3) x, f_ {2} (x) = (1 / 3 ) x +2 / 3. <=i<=n

is taken as initial value C_ {0} = [0,1]; check

that the fixed point of F, the iterations F ^ {n} (C_ {0}) is

the Cantor set.

b) in the plane. Write

the previous example in the plan, and draw

the first three iterations C_ {0}, C_ {1} and C_ {2}.
<=i<=n