On the Fixed Points of the Map x -> x^x Modulo a Prime
  In this paper, we show that for almost all primes p there is an integer solution x in [2,p-1] to the congruence x^x == x mod p. The solutions can be interpretated as fixed points of the map x -> x^x mod p, and we study numerically and discuss some unexpected properties of the dynamical system associated with this map. 
P\"ar Kurlberg, Florian Luca, Igor Shparlinski
math.NT math.DS