Nielsen periodic point theory for periodic maps on orientable surfaces

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 − 2 g ≠ 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 2 g − 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 . ve 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.