Connected components of the graph generated by power maps in prime finite fields

Consider the power pseudorandom-number generator in a finite field ${\mathbb F}_q$\mathrm{.}Thatis,forsomeinteger$e\ge2$,oneconsidersthesequence$u,u^e,u^{e^2},\dots$in${\mathbb F}_q$foragivenseed$u\in {\mathbb F}_q^\times$. This sequence is eventually periodic. One can consider the number of cycles that exist as the seed $u$ varies over ${\mathbb{F}}_q^{\times }$. This is the same as the number of cycles in the functional graph of the map $x\mapsto x^e$ in ${\mathbb{F}}_q^{\times }$. We prove some estimates for the maximal and average number of cycles in the case of prime finite fields.