Here is a picture of Brian's map.
It has three main areas. In the top left, green region, all orbits go to a stable attractor at (1,1). In the red area, bottom right, all orbits go to infinity.
If p and h are chosen from the central black region, the resulting fractal will have an invariant cycle. This is the wiggly line generated by the Volterra program if you use the default settings.
If p and h are chosen from one of the coloured flags in the central black area, then the fractal has a periodic attractor instead of the invariant cycle.
Finally there are some areas where several different attractors co-exist. The default settings in the volterra program illustrate one such fractal. Here we have the invariant cycle (wiggly line) and also a cycle of period 9.