Algorithms for Nielsen type periodic numbers of maps with remnant on surfaces with boundary and on bouquets of circles I

Abstract

Abstract. In this paper and its sequel we present a method that, under loose restrictions, is algorithmic for calculating the Nielsen type numbers N Φn(f ) and N Pn(f ) of self maps f of hyperbolic surfaces with boundary and also of bouquets of circles. Because self maps of these surfaces have the same homotopy type as maps on wedges of circles, and the Nielsen periodic numbers are homotopy type invariant, we need concentrate only on the latter spaces. Of course the results will then automatically apply to the former spaces as well. The algorithm requires only that f has minimal remnant, by which we mean that there is limited cancellation between the f * images of generators of the fundamental group. These methods often work even when the minimal remnant condition is not satisfied.Our methodology involves three separate techniques. Firstly, beginning with an endomorphism h on the fundamental group, we adapt an algorithm of Wagner to our setting, allowing us to distinguish non-empty Reidemeister classes for iterates of a special representative map for h, which we introduce. Secondly, using techniques reminiscent of symbolic dynamics, we assign key algebraic information to the actual periodic points of this special representative. Finally, we use word length arguments to prove that the remaining information required for the calculation of N Φn(f ) and N Pn(f ) can be found with a finite computer search. We include many illustrative examples.In this first paper we give the tools we need in order to present and give the algorithm for N Pn(f ). All the tools introduced here will be needed in the sequel where we develop the extra tools needed in order to compute N Φn(f ).

Remnant inequalities and doubly-twisted conjugacy in free groups

“…Extensions of Wagner's technique have been made in [8,10], and for Nielsen periodic point theory in [7].…”

“…Wagner, in [15], defined the remnant condition for free group endomorphisms which would become a key tool for several later techniques for computation of the Nielsen number in fixed point theory (the special case where ψ is the identity) for certain mappings on surfaces with boundary. Extensions of Wagner's technique have been made in [7] and [9], and for Nielsen periodic point theory in [6].…”

Combinatorial scheme of finding minimal number of periodic points for smooth self-maps of simply connected manifolds

“…Graff and J. Jezierski JFPTA introduced by Jiang [18] in 1983 as a lower bound for the number of r-periodic points in the homotopy class, and it was proved in 2006 that NF r (f ) is the best such lower bound, i.e., it is equal to the minimum in (1.1); see [16]. During the last decade, NF r (f ) was computed in many particular cases; see [13,14,15,19,20,21]. Recent investigations of the authors showed that the smooth and continuous theories do not coincide.…”

Minimal number of periodic points of smooth boundary-preserving self-maps of simply-connected manifolds

“…Finding minimal number of r -periodic points in the homotopy class (for a fixed r ) is an important challenge in modern homotopy periodic point theory, with an increasing number of valuable results obtained in the last decade in many particular cases [1][2][3][4][5][6][7].…”