I was tinkering around logistic map today, I accidentally discovered a property of it. That is, if you look at histogram of the sequence generated by it, it will be same independent of the input. This immediately hit me that there may be an underlying probability density of this map. In fact, after Googled, I found out that Ulam and von Neumann studied this! So here we go.

A bit of logistic map: It is a deterministic recurrence equation. For example, consider the following map,
\begin{align}
x_{n+1} = a x_n (1 - x_n)
\end{align}
where $x_n \in (0,1)$. For $a = 4$, this system exhibits a chaotic behaviour: Its response to a small uncertainty in the input is unpredictable. In other words, if you generate two sequences with very small difference in the initial condition, the sequences differ vastly.

But as it turns out, it is not totally unpredictable. Take an initial condition $x_0 \in (0,1)$ (we assume we do not choose unstable points for now), and simulate $N$ steps. We have a sequence $(x_n)_{n=1}^N$. Treating these as samples, if you plot the histogram, you will get:

This is true for any $x_0 \in (0,1)$ except for three unstable points $0.25, 0.5, 0.75$. Ulam and von Neumann found out that, if we see this sequence as a sequence of iid samples, the underlying probability density would be: \begin{align} p(x) = \frac{1}{\pi \sqrt{x(1-x)}} \end{align} More interestingly, it is possible to obtain uniform density by applying the following transformation: \begin{align} y_n = (2/\pi) \sin^{-1}(\sqrt{x_n}) \end{align}See the histogram where I used same samples with above after applying this transformation:

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