# Chaos from the Logistic Map. Part 1: “Periodic Points.”

The aim with this story is to present one of the most famous theorems in the research of chaotic maps. Before getting shrug-off too much, let me note early, this theorem is mathematically not more involved from what one learns in a first course of university level mathematics.

Throughout this account we will assume any considered map `f`

to be a continuous real-valued function defined on some compact interval `I`

. Moreover, we will assume `f(I) \subset I`

. Therefore, we are able to build iterates like `f^n(x)`

, that denote `n`

-times application of `f`

onto the point `x`

.

My intend is to make the text as readable as possible, but due to this environment’s lack of support for latex symbols, please apologize for making the following abbreviations:

f^k(x): denotes k-times composition of f, that is: f^3(x) f(f(f(x)))A \intersect B: means the intersection of the sets A and BA \subset B: means A is a subset of BA \superset B: means A is a superset of B