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Table of Contents

1. Attractor
- 1.1. Babylonian Method
2. Brouwer Fixed Point Theorem
- 2.1. Statement
3. Lipschitz Contraction
4. Banach Fixed Point Theorem
- 4.1. Statement
5. Theorem (DeBlasi, Myjak)
6. Theorem (Bargetz, Reich, Thimm)
7. See Also
8. Reference

1. Attractor

Multiple application of the same function $f$ will likely ends up on the fixed-point of $f$ .
$f$ is on the fixed-points.

1.1. Babylonian Method

Method of finding the square roots.
$f (x) = \frac{x + a / x}{2}$ admits a fixed point $\sqrt{a}$ .
Start with a guess $x_{0}$ , then $f^{n} (x_{0}) \overset{n \to \infty}{\to} \sqrt{a}$ .

2. Brouwer Fixed Point Theorem

The theorem is for finite dimensional spaces.

2.1. Statement

For a convex compact subset of $X$ , any continuous function $φ : X \to X$ admits a fixed-point.

3. Lipschitz Contraction

A function $f : M \to M$ such that $\exists L < 1, d (f (x), f (y)) \leq L d (x, y)$
Lipschitz continuous function with $K < 1$ .
- Intuitively, the function is under the line with the slope $K$ at every point.

4. Banach Fixed Point Theorem

4.1. Statement

For a complete metric space $X$ , the Lipschitz contraction $Φ : X \to X$ has a unique fixed-point $x^{*}$ .
This can be any finite or infinite dimensional space.

5. Theorem (DeBlasi, Myjak)

The set of non-expansive functions defined on a bounded set that admits fixed-points are co-porous with respect to the ((66478ab5-ed8d-4525-a7e1-a41671c644f9)).
That is, almost all such functions admits fixed-point.
There are almost no Lipschitz contraction, but most of them are Rakotch contractions which are the functions that are below an arbitrary curve.

6. Theorem (Bargetz, Reich, Thimm)

There are almost no Rakotch contraction on an unbounded set. But almost all of them admits fixed-point, since most of them are locally Rakotch contraction.

7. See Also

8. Reference

Created: 2025-05-06 Tue 23:34