  The map x -> x^x modulo p is related to a variation of the digital signature
scheme in a similar way to the discrete exponentiation map, but it has received
much less study. We explore the number of fixed points of this map by a
statistical analysis of experimental data. In particular, the number of fixed
points can in many cases be modeled by a binomial distribution. We discuss the
many cases where this has been successful, and also the cases where a good
model may not yet have been found.
