Nielsen periodic point theory for periodic maps on orientable surfaces

摘要

Let f : F_g → F_g 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 NP_n(f) and NΦ_n(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 NP_m(f) involves some surprises. Because f~m = id_(F_g), f~m has one Nielsen class E_m. This class is essential because L(id_(F_g)) = χ(F_g) = 2 - 2g ≠ 0. If there exists k < m with L(f~k) ≠ 0 then E_m reduces to the essential fixed points of f~k. There are maps g (we call them minLef maps) for which L(g~k) = 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 E_m 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 F_g. 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 NP_n(f) and NΦ_n(f) for all n. The example of an irreducible minLef map is on F_4 and is of maximal period 6. The example of a non-minLef map is on F_2 and has maximal period 12 on F_2. 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 F_4 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 F_g. Using these examples we disprove the conjecture from the conclusion of our previous paper.