Families of fixed-point cellular rotations

A cellular rotation is a pseudofree cellular automorphism, with no non-fixed pseudofixed points, of a graph embedded in an orientable surface. A family of cellular rotations is a collection of cellular rotations having one embedding of each genus above some fixed minimum genus, all sharing the same quotient embedding and, in an appropriate sense, the same voltage-assignment data. We provide a complete catalog of all families of cellular rotations having at least one fixed point, and provide preliminary results regarding families of cellular rotations having no fixed points.