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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

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  • $\begingroup$ What do you mean by "conjugate"? I understand what is a topological conjugacy between two MAPS but here the second thing is only a sequence. In particular there is not (at least it is not shown that there is) any map $f$ such that $x_{n+1} = f(x_n)$. $\endgroup$ – demitau May 29 '15 at 7:51
  • $\begingroup$ However if you find this map $f$ I mentioned above, I think the best approach would be to construct a conjugacy by hands, because it is probably difficult to say something about symbolic dynamics of such a thing with nonlinear functions. $\endgroup$ – demitau May 29 '15 at 7:58
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$N$ 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.

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