Elliptic Superfunctions

[bo198214]

From the double-angle formulas for the Jacobi elliptic functions we can get superfunctions to other rational functions.

Actually this topic was already considered in:
Schröder, E. (1871). Ueber iterirte Functionen. (On iterated functions.). Clebsch Ann., 3, 296–322.

Personally interesting for me would be the iterates/superfunctions of rational functions that dont have a real fixed point. Are there some amongst this class obtained from elliptic addition theorems?

In the case of several real fixed points there is still always the question at which fixpoint the obtained elementary iteration/superfunction is the regular iteration/superfunction.