It is [known](https://mathworld.wolfram.com/LogisticMap.html) that for $r=-2,2,4$ the logistic map $x\_{n+1}=r x\_n (1-x\_n)$ has exact solutions of the form
$$
x\_n=\frac12 \left\{ 1- f\left(r^n f^{-1}(1-2x\_0)\right)\right\} \qquad \qquad{(\*)}
$$
for suitable functions $f$. The same source further claims, with reference to a private communication, that $r=-2,2,4$ are the only values of $r$ for which the logistic map has exact solutions of the form $(\*)$.


However, one wonders whether



> 
> are there other values of $r$ for which the logistic map is exactly solvable?
> 
> 
> 


Of course, for these other values of $r$, if any, the exact solutions would likely be of the form other than (\*); alas so far I found in the literature nothing that would answer the above question.



Is the logistic map $x_{n+1}=r x_n (1-x_n)$ exactly solvable for any $r$ other than $-2,2,4$?
Explicit solutions for arbitrary $r$ exist in various forms:


* [Logistic map: an
analytical solution](http://havlin.biu.ac.il/PS/rbbsh260.pdf) (1995) represents the solution as a power of
a transfer matrix.
* [An explicit solution
for the logistic map](http://havlin.biu.ac.il/PS/rmh365.pdf) (1999) gives a functional integral solution.
* [A note on
exact solutions of the logistic map](https://aip.scitation.org/doi/10.1063/1.5125097) (2020, paywall) gives the
solution in terms of a power series.


