Table of Contents
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 \(f^n(x_0) \stackrel{n\to \infty}{\to} \sqrt{a}\).
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 \(\varphi\colon X\to X\) admits a fixed-point.
3. Lipschitz Contraction
- A function \(f\colon M \to M\) such that \(\exists L < 1, d(f(x), f(y)) \le L d(x,y)\)
- 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 \(\Phi\colon X\to X\) 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.