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

scholarly journals Twist Periodic Orbits for Continuous Maps of the Eight Space

2013 ◽  
Vol 6 ◽  
pp. 4-8
Author(s):  
Iftichar Mudhar Talb Al-Shraa
Keyword(s):  
Fixed Point ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Continuous Map ◽  
Continuous Maps

Let g be a continuous map from 8 to itself has a fixed point at (0,0), we prove that g has a twist periodic orbit if there is a rational rotation number.

scholarly journals Twist periodic orbits and topological entropy for continuous maps of the circle of degree one which have a fixed point

1985 ◽  
Vol 5 (4) ◽  
pp. 501-517 ◽  
Author(s):  
Lluís Alsedà ◽  
Jaume Llibre ◽  
Michał Misiurewicz ◽  
Carles Simó
Keyword(s):  
Fixed Point ◽  
Lower Bound ◽  
Periodic Orbit ◽  
Topological Entropy ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Rotation Interval ◽  
Continuous Map ◽  
Continuous Maps

AbstractLet f be a continuous map from the circle into itself of degree one, having a periodic orbit of rotation number p/q ≠ 0. If (p, q) = 1 then we prove that f has a twist periodic orbit of period q and rotation number p/q (i.e. a periodic orbit which behaves as a rotation of the circle with angle 2πp/q). Also, for this map we give the best lower bound of the topological entropy as a function of the rotation interval if one of the endpoints of the interval is an integer.

scholarly journals Averaging Formula for Nielsen Numbers of Maps on Infra-Solvmanifolds of Type (R)

2009 ◽  
Vol 196 ◽  
pp. 117-134 ◽  
Author(s):  
Jong Bum Lee ◽  
Kyung Bai Lee
Keyword(s):  
Holonomy Group ◽  
Nielsen Numbers ◽  
Continuous Map ◽  
Continuous Maps

We prove that the averaging formula for Nielsen numbers holds for continuous maps on infra-solvmanifolds of type (R): Let M be such a manifold with holonomy group Ψ and let f: M → M be a continuous map. The averaging formula for Nielsen numbersis proved. This is a workable formula for the difficult number N(f).

Perturbation of the fixed point problem for continuous mappings

2020 ◽  
pp. 241-249
Author(s):  
Zukhra T. Zhukovskaya ◽  
Sergey E. Zhukovskiy
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Continuous Mapping ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Bounded Subset ◽  
Completely Continuous ◽  
Continuous Map ◽  
Continuous Mappings

We consider the problem of a double fixed point of pairs of continuous mappings defined on a convex closed bounded subset of a Banach space. It is shown that if one of the mappings is completely continuous and the other is continuous, then the property of the existence of fixed points is stable under contracting perturbations of the mappings. We obtain estimates for the distance from a given pair of points to double fixed points of perturbed mappings. We consider the problem of a fixed point of a completely continuous mapping on a convex closed bounded subset of a Banach space. It is shown that the property of the existence of a fixed point of a completely continuous map is stable under contracting perturbations. Estimates of the distance from a given point to a fixed point are obtained. As an application of the obtained results, the solvability of a difference equation of a special type is proved.

scholarly journals Diagonals of separately continuous maps with values in box products

2018 ◽  
Vol 6 (1) ◽  
pp. 26-33
Author(s):  
Olena Karlova ◽  
Volodymyr Mykhaylyuk
Keyword(s):  
Connected Space ◽  
Continuous Map ◽  
Continuous Maps ◽  
Baire One ◽  
Metrizable Spaces

Abstract We prove that if X is a paracompact connected space and Z = ∏s∈S Zs is a product of a family of equiconnected metrizable spaces endowed with the box topology, then for every Baire-one map g : X → Z there exists a separately continuous map f : X2 → Z such that f (x, x) = g(x) for all x ∈ X.

Fixed Point Principles for Cones of a Linear Normed Space

1980 ◽  
Vol 32 (6) ◽  
pp. 1372-1381 ◽  
Author(s):  
Gilles Fournier
Keyword(s):  
Fixed Point ◽  
Functional Differential Equations ◽  
Compact Attractor ◽  
Completely Continuous ◽  
Continuous Maps ◽  
Compression And Expansion ◽  
Existence Of Periodic Solutions ◽  
Functional Differential ◽  
Generalized Lefschetz Number ◽  
Fixed Point Principles

In [8] and [9], Krasnosel'skiĭ proved several fundamental fixed point principles for operators leaving invariant a cone in a Banach space. In [11], Nussbaum extended one of the results, the theorem about compression and expansion of a cone, to condensing maps and he applied this theorem to prove the existence of periodic solutions of nonlinear autonomous functional differential equations.Nussbaum's proof makes an essential use of the difficult Zabreiko and Krasnosel'skiĭ, and Steinlein (mod p)-theorem for the fixed point index [13 -16]. In [6], Fournier and Peitgen proved two different versions of this theorem for completely continuous maps each one being sufficient for Nussbaum's applications. The proofs of these two theorems are much less involved and, although they are different, they make use of the same easier generalized Lefschetz number calculations (see [12] for (mod p) and [5] for compact attractor).

Spaces on which every Continuous Map into a given Space is Constant

1986 ◽  
Vol 38 (6) ◽  
pp. 1281-1298 ◽  
Author(s):  
S. Iliadis ◽  
V. Tzannes
Keyword(s):  
Topological Spaces ◽  
Real Numbers ◽  
Continuous Map ◽  
Countable Space ◽  
Continuous Maps ◽  
Usual Topology ◽  
The Given ◽  
Connected Spaces

This paper is concerned with topological spaces whose continuous maps into a given space R are constant, as well as with spaces having this property locally. We call these spaces R-monolithic and locally R-monolithic, respectively.Spaces with such properties have been considered in [1], [5]-[7], [10], [11], [22], [28], [31], where with the exception of [10], the given space R is the set of real-numbers with the usual topology. Obviously, for a countable space, connectedness is equivalent to the property that every continuous real-valued map is constant. Countable connected (locally connected) spaces have been constructed in papers [2]-[4], [8], [9], [11]-[21], [23]-[26], [30].

A Remark on Invariant Scrambled Sets

2012 ◽  
Vol 204-208 ◽  
pp. 4776-4779
Author(s):  
Lin Huang ◽  
Huo Yun Wang ◽  
Hong Ying Wu
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Periodic Point ◽  
Metric Space ◽  
Cantor Sets ◽  
Countable Union ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Isolated Points ◽  
Weakly Mixing

By a dynamical system we mean a compact metric space together with a continuous map . This article is devoted to study of invariant scrambled sets. A dynamical system is a periodically adsorbing system if there exists a fixed point and a periodic point such that and are dense in . We show that every topological weakly mixing and periodically adsorbing system contains an invariant and dense Mycielski scrambled set for some , where has no isolated points. A subset is a Myceilski set if it is a countable union of Cantor sets.

scholarly journals A CHARACTERIZATION OF THE SET Ω (f)\ ω (f) FOR CONTINUOUS MAPS OF THE INTERVAL WITH ZERO TOPOLOGICAL ENTROPY

1995 ◽  
Vol 05 (05) ◽  
pp. 1433-1435
Author(s):  
F. BALIBREA ◽  
J. SMÍTAL
Keyword(s):  
Topological Entropy ◽  
Minimal Set ◽  
Continuous Map ◽  
Continuous Maps ◽  
Limit Points ◽  
Infinite Set ◽  
Nonwandering Points

We give a characterization of the set of nonwandering points of a continuous map f of the interval with zero topological entropy, attracted to a single (infinite) minimal set Q. We show that such a map f can have a unique infinite minimal set Q and an infinite set B ⊂ Ω (f)\ ω (f) (of nonwandering points that are not ω-limit points) attracted to Q and such that B has infinite intersections with infinitely many disjoint orbits of f.

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