Nielsen periodic point theory for periodic maps on orientable surfaces

Let denote a periodic self map of minimal period m on the orientable surface of genus g with g>1. We study the calculation of the Nielsen periodic numbers NPn(f) and NΦn(f). Unlike the general situation of arbitrary maps on such surfaces, strong geometric results of Jiang and Guo allow for straightforward calculations when n≠m. However, determining NPm(f) involves some surprises. Because fm=idFg, fm has one Nielsen class Em. This class is essential because L(idFg)=χ(Fg)=2−2g≠0. If there exists k<m with L(fk)≠0 then Em reduces to the essential fixed points of fk. There are maps g (we call them minLef maps) for which L(gk)=0 for all k<m. We show that the period of any minLef map must always divide 2g−2. We prove that for such maps Em reduces algebraically iff it reduces geometrically. This result eliminates one of the most difficult problems in calculating the Nielsen periodic point numbers and gives a complete trichotomy (non-minLef, reducible minLef, and irreducible minLef) for periodic maps on Fg.We prove that reducible minLef maps must have even period. For each of the three types of periodic maps we exhibit an example f and calculate both NPn(f) and NΦn(f) for all n. The example of an irreducible minLef map is on F₄ and is of maximal period 6. The example of a non-minLef map is on F₂ and has maximal period 12 on F₂. It is defined geometrically by Wang, and we provide the induced homomorphism and analyze it. The example of an irreducible minLef map is a map of period 6 on F₄ defined by Yang. No algebraic analysis is necessary to prove that this last example is an irreducible minLef map. We explore the algebra involved because it is intriguing in its own right. The examples of reducible minLef maps are simple inversions, which can be applied to any Fg. Using these examples we disprove the conjecture from the conclusion of our previous paper.