# 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`

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I am a Software Developer - Java, Rust, SQL, TypeScript - with strong interest doing research in pure and applied Mathematics.