Brizolis asked the question: does every prime p have a pair (g,h)
such that h is a fixed point for the discrete logarithm with base
g?  The first author previously extended this question to ask about
not only fixed points but also two-cycles, and gave heuristics
(building on work of Zhang, Cobeli, Zaharescu, Campbell, and
Pomerance) for estimating the number of such pairs given certain
conditions on g and h.  In this paper we extend these heuristics
and prove results for some of them, building again on the
aforementioned work.  We also make some new conjectures and prove some
average versions of the results.