Pseudorandomness and Dynamics of Fermat Quotients

We obtain some theoretic and experimental results concerning various properties (the number of fixed points, image distribution, cycle lengths) of the dynamical system naturally associated with Fermat quotients acting on the set $\{0,\mathrm{.}\mathrm{.}\mathrm{.},p-1\}$. We also consider pseudorandom properties of Fermat quotients such as joint distribution and linear complexity.