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 give a summary of
conjectures and results which follow from these heuristics, building again
on the aforementioned work. We also make some new conjectures and prove
some average versions of the results.