  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.
