About literature related to the topic of this question:

In the answer to the question f(f(x))=exp(x)-1 and other functions “just in the middle” between linear and exponential. one can find an interesting link with many references related to the problem:

• Lars Kindermann: Iterative Roots and Fractional Iteration

There is also Kindermanns PhD thesis about finding solutions to iterative functional equations using a neural network (in german only)

• Kindermann(2001): Neuronale Netze zur Berechnung Iterativer Wurzeln und Fraktionaler Iterationen

which might be helpful.

About literature related to the topic of this question:

In the answer to the question f(f(x))=exp(x)-1 and other functions “just in the middle” between linear and exponential. one can find an interesting link with many references related to the problem:

• Lars Kindermann: Iterative Roots and Fractional Iteration

There is also Kindermanns PhD thesis about finding solutions to iterative functional equations using a neural network (in german only)

• Kindermann(2001): Neuronale Netze zur Berechnung Iterativer Wurzeln und Fraktionaler Iterationen

which might be helpful.