I have a map which I have to show is a conjugate to the Logistic Map ( $x_{n+1} = rx_n(1-x_n)$ ). The map in question is as follows.
$x_n = \sin^2(\pi\theta_n)$
$\theta_{n+1} = N^n\theta_0$ mod $1$
$\theta_0 = \pi^{-1}\arcsin(\sqrt{x_0})$
My idea for proving this is to plot this map and show the symbolic dynamics rather than finding some crazy transform. The problem is I'm having trouble deciphering the map. What is $N$? And how do I know what $x_0$ is?
Thanks
$N$ is a given constant, presumably an integer $> 1$. $x_0$ is the initial point. There's a misprint in your equations: you want $\theta_n = N^n \theta_0 \mod 1$, not $\theta_{n+1}$. The map here takes $x_0$ to $x_1$.
As for your idea, a plot is not a proof.
The Logistic Map, by the way, is a two-to-one function (i.e. for almost every $y$ in the range, there are two values of $x$ with $f(x) = y$. If you want this map to be conjugate to the Logistic Map, that will restrict the possibilities for $N$ rather drastically.