Reidemeister orbit sets

Abstract

Abstract. The Reidemeister orbit set plays a crucial role in the Nielsen type theory of periodic orbits, much as the Reidemeister set does in Nielsen fixed point theory. Extending Ferrario's work on Reidemeister sets, we obtain algebraic results such as addition formulae for Reidemeister orbit sets. Similar formulae for Nielsen type essential orbit numbers are also proved for fibre preserving maps. Introduction. Nielsen fixed point theory has been extended to aNielsen type theory of periodic orbits [6, Section III.3]. In fixed point theory, the computation of the Nielsen number often relies on the knowledge of the Reidemeister set, that is, the set of Reidemeister conjugacy classes in the fundamental group. Ferrario [2] made an algebraic study of the Reidemeister set in relation to an invariant normal subgroup. He obtained addition formulae for Reidemeister numbers, and applied them to the Nielsen number of fibre preserving maps. Our aim in this paper is similar, but in the more complicated setting of periodic orbits: to study the Reidemeister orbit set of a group endomorphism in relation to an invariant normal subgroup, to obtain addition formulae for Reidemeister orbit numbers, and as application, to find addition formulae for Nielsen type essential orbit numbers of fibre preserving maps.Given a group endomorphism f : G → G, the Reidemeister set of f , denoted by R(f ), is the set of orbits of the left action of G on G via γ g → 2000 Mathematics Subject Classification: Primary 55M20.

A Note on the Set of Root Classes

“…Example 3.7. As in [1, Example 6.1], let S 1 = {z : | z |= 1} be the unit circle in the complex plane and let f : S 1 → S 1 be defined by f (z) = z 3 . If we choose a = 1, then ρ(f, a) = 1.…”

“…(3) k of the root class z 3 k is reducible to 1 if and only if k − 2 ≡ 0 (mod 3 2 ). Thus we have d(z (3) 11 ) = 1 and (z (3) 11 ) = 2, so z (3) 11 does not have the full depth property.…”

“…f ) reduces is its depth. A Reidemeister orbit is said to have the full depth property if its depth equals its length, i.e., d = (see [3]).…”

A Nielsen Type Number of Fibre Preserving Maps

“…Our aim in this paper is similar. We define a Nielsen type number of fibre preserving maps which are essentially norbit fibre uniform, using the Reducing Lemma in [8]. We show that a relation between this new Nielsen type number and the Nielsen type relative essential orbit number.…”

“…Nielsen fixed point theory has been extended to a Nielsen type theory of periodic orbits [7,III.3]. Much has been introduced about the Nielsen number of fibre map [2,3,7,8,12]. Heath [3] defined a Nielsen type number for fibre preserving maps which are essentially fibre uniform, and exhibited connections between it and ordinary Nielsen numbers.…”