  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.
