scholarly journals Stabilizers of fixed point classes and Nielsen numbers of $n$-valued maps

2017 ◽  
Vol 24 (4) ◽  
pp. 523-535 ◽  
Author(s):  
Robert F. Brown ◽  
Junzheng Nan
Keyword(s):  
Fixed Point ◽  
Nielsen Numbers

scholarly journals FIXED POINTS AND COINCIDENCES IN TORUS BUNDLES

2011 ◽  
Vol 03 (02) ◽  
pp. 177-212 ◽  
Author(s):  
ULRICH KOSCHORKE
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Careful Analysis ◽  
Natural Generalization ◽  
Homotopy Classes ◽  
Nielsen Numbers ◽  
Coincidence Theory ◽  
Torus Bundles ◽  
Odd Order ◽  
Minimum Numbers

Minimum numbers of fixed points or of coincidence components (realized by maps in given homotopy classes) are the principal objects of study in topological fixed point and coincidence theory. In this paper, we investigate fiberwise analoga and present a general approach e.g. to the question when two maps can be deformed until they are coincidence free. Our method involves normal bordism theory, a certain pathspace EB and a natural generalization of Nielsen numbers. As an illustration we determine the minimum numbers for all maps between torus bundles of arbitrary (possibly different) dimensions over spheres and, in particular, over the unit circle. Our results are based on a careful analysis of the geometry of generic coincidence manifolds. They also allow a simple algebraic description in terms of the Reidemeister invariant (a certain selfmap of an abelian group) and its orbit behavior (e.g. the number of odd order orbits which capture certain nonorientability phenomena). We carry out several explicit sample computations, e.g. for fixed points in (S1)2-bundles. In particular, we obtain existence criteria for fixed point free fiberwise maps.

scholarly journals Averaging Formula for Nielsen Numbers

2005 ◽  
Vol 178 ◽  
pp. 37-53 ◽  
Author(s):  
Seung Won Kim ◽  
Jong Bum lee ◽  
Kyung Bai Lee
Keyword(s):  
Fixed Point ◽  
Nielsen Numbers ◽  
Regular Covering ◽  
Continuous Map ◽  
Continuous Maps ◽  
Point Class

We prove that the averaging formula for Nielsen numbers holds for continuous maps on infra-nilmanifolds: Let M be an infra-nilmanifold and ƒ: M → M be a continuous map. Suppose MK is a regular covering of M which is a compact nilmanifold with π1(MK = K. Assume that f*(K) ⊂ K. Then ƒ has a lifting . We prove a question raised by McCord, which is for any with an essential fixed point class, fix =1. As a consequence, we obtain the following averaging formula for Nielsen numbers

Relative Nielsen Numbers, Braids and Periodic Segments

2017 ◽  
Vol 17 (3) ◽  
Author(s):  
Klaudiusz Wójcik
Keyword(s):  
Fixed Point ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Nielsen Numbers ◽  
Nielsen Fixed Point Theory

AbstractThe aim of this paper is to establish a connection between the method of period segments and the relative Nielsen fixed point theory. We prove that if

Mental equilibrium: Identifying fixed-point attractors in the stream of thought

2003 ◽  
Author(s):  
Robin R. Vallacher ◽  
Andrzej Nowak ◽  
Matthew Rockloff
Keyword(s):  
Fixed Point

Bifurcation in a Simple Model of the Cardiovascular System

2000 ◽  
Vol 39 (02) ◽  
pp. 118-121 ◽  
Author(s):  
S. Akselrod ◽  
S. Eyal
Keyword(s):  
Nonlinear Dynamics ◽  
Blood Pressure ◽  
Heart Rate ◽  
Fixed Point ◽  
Cardiovascular System ◽  
Modified Model ◽  
Mayer Waves ◽  
Sustained Oscillations ◽  
Human Cardiovascular System ◽  
Physiological Values

Abstract:A simple nonlinear beat-to-beat model of the human cardiovascular system has been studied. The model, introduced by DeBoer et al. was a simplified linearized version. We present a modified model which allows to investigate the nonlinear dynamics of the cardiovascular system. We found that an increase in the -sympathetic gain, via a Hopf bifurcation, leads to sustained oscillations both in heart rate and blood pressure variables at about 0.1 Hz (Mayer waves). Similar oscillations were observed when increasing the -sympathetic gain or decreasing the vagal gain. Further changes of the gains, even beyond reasonable physiological values, did not reveal another bifurcation. The dynamics observed were thus either fixed point or limit cycle. Introducing respiration into the model showed entrainment between the respiration frequency and the Mayer waves.

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