• Communications of the Korean Mathematical Society
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• v.25 no.2
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• pp.263-271
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• 2010

The Reidemeister orbit set plays a crucial role in the Nielsen type theory of periodic orbits, such as the Reidemeister set does in Nielsen fixed point theory. In this paper, using Heath and You's methods on Nielsen type numbers, we show that these numbers can be evaluated by the set of essential orbit classes under suitable hypotheses, and obtain some formulas in some special cases.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.2
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• pp.341-347
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• 2012

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 our work on Reidemeister orbit sets, we improve our theorem for formulae for Nielsen type essential orbit numbers.

• Communications of the Korean Mathematical Society
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• v.11 no.1
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• pp.245-252
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• 1996

The relative Nielsen number N(f;X,A) was introduced in 1986. It gives us a better, and ideally sharp, lower bound for the minimum number MF[f;X,A] of fixed points in the homotopy class of the map $f;(X,A) \to (X,A)$. Similarly, we also can think about the Nielsen map $f:(X,A) \to (X,A)$. Similarly, we also can be think about the Nielsen root theory. In this paper, we introduce a relative root Nielsen number N(f;X,A,c) of $f:(X,A) \to (Y,B)$ and show some basic properties.

• The Journal of Natural Sciences
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• v.14 no.1
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• pp.1-6
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• 2004

In this paper, we calculate the Nielsen numbers of periods for maps on the Klein bottle.

• Journal of the Korean Mathematical Society
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• v.45 no.5
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• pp.1483-1503
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• 2008

Let f : M ${\rightarrow}$ M be a self-map on the Klein bottle M. We compute the Lefschetz number and the Nielsen number of f by using the infra-nilmanifold structure of the Klein bottle and the averaging formulas for the Lefschetz numbers and the Nielsen numbers of maps on infra-nilmanifolds. For each positive integer n, we provide an explicit algorithm for a complete computation of the Nielsen type numbers $NP_n(f)$ and $N{\Phi}_{n}(f)\;of\;f^{n}$.

• Communications of the Korean Mathematical Society
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• v.21 no.1
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• pp.193-209
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• 2006

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. In this paper, extending Cardona and Wong's work on relative Reidemeister numbers, we show that the Reidemeister orbit numbers can be used to calculate the relative essential orbit numbers. We also apply the relative Reidemeister orbit number to study periodic orbits of fibre preserving maps.

• Journal of the Chungcheong Mathematical Society
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• v.27 no.4
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• pp.721-734
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• 2014

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 our work on Reidemeister orbit sets, we obtain algebraic results such as addition formulae for irreducible Reidemeister orbit sets. Similar formulae for Nielsen type irreducible essential orbit numbers are also proved for fibre preserving maps.

• Honam Mathematical Journal
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• v.24 no.1
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• pp.75-86
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• 2002

A Nielsen number $\bar{N}(f:X-A)$ is a homotopy invariant lower bound for the number of fixed points on X-A where X is a compact connected polyhedron and A is a connected subpolyhedron of X. This number is extended to Nielsen type numbers $\bar{NP_{n}}(f:X-A)$ of least period n and $\bar{N{\phi}_{n}}(f:X-A)$ of the nth iterate on X-A where the subpolyhedron A of a compact connected polyhedron X is no longer path connected.

• Journal of the Korean Mathematical Society
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• v.45 no.3
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• pp.883-902
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• 2008

The main results concern with the self maps on the Klein bottle. We obtain the Reidemeister numbers and the Nielsen numbers for all self maps on the Klein bottle. In terms of the Nielsen numbers of their iterates, we totally determine the minimal sets of periods for all homotopy classes of self maps on the Klein bottle.

• Bulletin of the Korean Mathematical Society
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• v.33 no.2
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• pp.205-211
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• 1996

Throughout this paper, we will work in the category of compact connected manifolds and continuous maps. If $X_1, X_2$ are manifolds, $f, g: X_1 \to X_2$ are maps, then a point $x \in X_1$ is a coincidence of f and g iff f(x) = g(x) ; the set of all such points is denoted by Coin(f,g).