Is the logistic map $x_{n+1}=r x_n (1-x_n)$ exactly solvable for any $r$ other than $-2,2,4$?

Question

It is known 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.

Answer 1

Explicit solutions for arbitrary $r$ exist in various forms:

• Logistic map: an analytical solution (1995) represents the solution as a power of a transfer matrix.

• An explicit solution for the logistic map (1999) gives a functional integral solution.

• A note on exact solutions of the logistic map (2020, paywall) gives the solution in terms of a power series.