Dynamical systems of non-algebraic origins: fixed points, orbit lengths and distribution

We give a short survey of several theoretic and heuristic results, about the fixed points and orbit lengths of several dynamical system associated with iterations of functions of number theoretic nature. These include (in the historical order of study):

• discrete logarithm in finite fields $x \mapsto \log_g x$ for which we establish some of the conjectures of J. Holden and P. Moree (joint work with J. Bourgain and S. Konyagin);
• Fermat quotients $x \mapsto (x^{p-1} -1)/p \pmod p$ (joint work with A. Ostafe);
• self-exponential map $x \mapsto x^x \pmod p$ (joint work with P. Kurlberg and F. Luca);.
• fixed base exponentiation $x \mapsto g^x \pmod p$ (joint work with L. Glebsky and also with J. Kaszian and P. Moree).

Unfortunately theoretic results are rather scarce and mostly concern fixed points of these maps. Studying longer cycles is much more difficult and in some cases even a right heuristic model is not clear.

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