How can one prove chaos in logistic map?

Question

Many introductions to chaos start with logistic map $$ x_{n+1}=\lambda x_n(1-x_n) $$ and claim it is chaotic at some values of $\lambda$. Unfortunately, all proofs of chaos I saw were numerical and not rigorous. How does one prove that such a map is chaotic at a particular point in general (I suspect it may be well known in the field, but I couldn't find an anwer easily)?

Answer 1

Here is the proof of typical dense orbits for $\lambda=4$:

Start with the doubling map $\quad \theta_{n+1}=2\theta_n \mod 1$ on $[0,1) \,$. By considering the binary expansion of $\theta_0$, the law of large numbers implies that almost all orbits of the doubling map are dense. Write $x_n=\sin^2(\pi \theta_n)$ and observe that it satisfies the given recursion with $\lambda=4$. It follows that this map has typically dense orbits in $[0,1]$ as well.