Periodic Points and Contractive Mappings

1974 ◽  
Vol 17 (2) ◽  
pp. 209-211
Author(s):  
Tsu-Teh Hsieh ◽  
Kok-Keong Tan
Keyword(s):  
Fixed Point ◽  
Positive Integer ◽  
Periodic Orbit ◽  
Periodic Point ◽  
Periodic Points ◽  
Contractive Mappings ◽  
Finite Set

Let X be a non-empty set and f:X→X. A point x ∈ X is (i) a fixed point off f(x)=x, and (ii) a periodic point of f iff there is a positive integer N such that fN(x)=x. Also a periodic orbit of f is the (finite) set {x, f(x), f2(x),…} where x is a periodic point of f.

scholarly journals Examples of expanding maps with some special properties

1987 ◽  
Vol 36 (3) ◽  
pp. 469-474 ◽  
Author(s):  
Bau-Sen Du
Keyword(s):  
Fixed Point ◽  
Positive Integer ◽  
Periodic Point ◽  
Topological Entropy ◽  
Real Line ◽  
Periodic Points ◽  
Unit Interval ◽  
Expanding Maps ◽  
The Real ◽  
Piecewise Monotonic

Let I be the unit interval [0, 1] of the real line. For integers k ≥ 1 and n ≥ 2, we construct simple piecewise monotonic expanding maps Fk, n in C0 (I, I) with the following three properties: (1) The positive integer n is an expanding constant for Fk, n for all k; (2) The topological entropy of Fk, n is greater than or equal to log n for all k; (3) Fk, n has periodic points of least period 2k · 3, but no periodic point of least period 2k−1 (2m+1) for any positive integer m. This is in contrast to the fact that there are expanding (but not piecewise monotonic) maps in C0(I, I) with very large expanding constants which have exactly one fixed point, say, at x = 1, but no other periodic point.

scholarly journals Existence of Periodic Fixed Point Theorems in the Setting of Generalized Quasi-Metric Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Chi-Ming Chen ◽  
Erdal Karapınar ◽  
Vladimir Rakočević
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Periodic Points ◽  
Contractive Mappings

We introduce the notions of(α-ϕ-ψ)-weaker Meir-Keeler contractive mappings and(α-φ)-stronger Meir-Keeler contractive mappings. We discuss the existence of periodic points in the setting of generalized quasi-metric spaces. Our results improve, extend, and generalize several results in the literature.

scholarly journals On common fixed and periodic points of commuting functions

1998 ◽  
Vol 21 (2) ◽  
pp. 269-276 ◽  
Author(s):  
Aliasghar Alikhani-Koopaei
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Periodic Point ◽  
Periodic Points ◽  
Continuous Functions ◽  
Contractive Condition ◽  
Unique Common Fixed Point

It is known that two commuting continuous functions on an interval need not have a common fixed point. It is not known if such two functions have a common periodic point. In this paper we first give some results in this direction. We then define a new contractive condition, under which two continuous functions must have a unique common fixed point.

scholarly journals Periodic Points and Fixed Points for the Weaker(ϕ,φ)-Contractive Mappings in Complete Generalized Metric Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Chi-Ming Chen ◽  
W. Y. Sun
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Contractive Mapping ◽  
Periodic Points ◽  
Generalized Metric Spaces ◽  
Complete Metric Spaces ◽  
Contractive Mappings ◽  
Generalized Metric

We introduce the notion of weaker(ϕ,φ)-contractive mapping in complete metric spaces and prove the periodic points and fixed points for this type of contraction. Our results generalize or improve many recent fixed point theorems in the literature.

scholarly journals Generalized Contractive Mappings and Weaklyα-Admissible Pairs inG-Metric Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
N. Hussain ◽  
V. Parvaneh ◽  
S. J. Hoseini Ghoncheh
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Coincidence Point ◽  
Periodic Points ◽  
Contractive Mappings ◽  
Admissible Pairs ◽  
Weakly Contractive ◽  
Common Fixed Point Theorems

The aim of this paper is to present some coincidence and common fixed point results for generalized (ψ,φ)-contractive mappings using partially weaklyG-α-admissibility in the setup ofG-metric space. As an application of our results, periodic points of weakly contractive mappings are obtained. We also derive certain new coincidence point and common fixed point theorems in partially orderedG-metric spaces. Moreover, some examples are provided here to illustrate the usability of the obtained results.

Lefschetz numbers of periodic orbits of pseudo-Anosov homeomorphisms

1994 ◽  
Vol 115 (1) ◽  
pp. 121-132 ◽  
Author(s):  
John Guaschi
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Periodic Points ◽  
Lefschetz Numbers ◽  
Finite Set ◽  
Surface Homeomorphism

AbstractGiven a surface homeomorphism isotopic to the identity which is pseudo-Anosov relative to a finite set, we show that the sum of the Lefschetz numbers of periodic points of any period greater than one is non-negative. If this period is odd and greater than a number which depends only on the surface, the sum is zero. If we consider sequences of periods such that each element is twice that of its predecessor, then this sum is increasing beyond a certain point also depending on the surface. As a corollary, for each periodic orbit contained within the boundary of the surface there exists one of the same period contained in the interior.

scholarly journals Periodic Points of Weaker Meir-Keeler Contractive Mappings on Generalized Quasimetric Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Ing-Jer Lin ◽  
Chi-Ming Chen ◽  
Erdal Karapınar
Keyword(s):  
Periodic Point ◽  
Periodic Points ◽  
Contractive Mappings ◽  
Admissible Mapping

By using the weaker Meir-Keeler functionϕand the triangularα-admissible mappingα, we introduce the notion of(α-ϕ)-weaker Meir-Keeler contractive mappings and prove a theorem which assures the existence of a periodic point for these mappings on generalized quasimetric spaces.

scholarly journals THE SHADOW-CURVES OF THE ORBIT DIAGRAM PERMEATE THE BIFURCATION DIAGRAM, TOO

2009 ◽  
Vol 19 (09) ◽  
pp. 3017-3031 ◽  
Author(s):  
CHIP ROSS ◽  
MEREDITH ODELL ◽  
SARAH CREMER
Keyword(s):  
Fixed Point ◽  
Linear System ◽  
Bifurcation Diagram ◽  
Periodic Point ◽  
Infinite Family ◽  
Periodic Points ◽  
New Phenomena

The "Q-curves [Formula: see text] have long been observed and studied as the shadowy curves which appear illusively — not explicitly drawn — in the familiar orbit diagram of Myrberg's map fc(x) = x2 + c. We illustrate that Q-curves also appear implicitly, for a different reason, in a computer-drawn bifurcation diagram of x2 + c as well — by "bifurcation diagram" we mean the collection of all periodic points of fc (attracting, indifferent and repelling) — these collections form what we call "P-curves". We show Q-curves and P-curves intersect in one of two ways: At a superattracting periodic point on a P-curve, the infinite family of Q-curve s which intersect there are all tangent to the P-curve. At a Misiurewicz point, no tangencies occur at these intersections; the slope of the P-curve is the fixed point of a linear system whose iterates give the slopes of the Q-curves. We also introduce some new phenomena associated with c sin x illustrating briefly how its two different families of Q-curves interact with P-curves. Our algorithm for finding and plotting all periodic points (up to any reasonable period) in the bifurcation diagram is reviewed in an Appendix.

scholarly journals ON PERIODIC POINTS OF FREE INVERSE MONOID HOMOMORPHISMS

2013 ◽  
Vol 23 (08) ◽  
pp. 1789-1803 ◽  
Author(s):  
EMANUELE RODARO ◽  
PEDRO V. SILVA
Keyword(s):  
Fixed Point ◽  
Periodic Point ◽  
Periodic Points ◽  
Inverse Monoid ◽  
Finitely Generated ◽  
Context Sensitive ◽  
Context Free Language ◽  
Context Free ◽  
Free Language ◽  
Free Inverse Monoid

It is proved that the periodic point submonoid of a free inverse monoid endomorphism is always finitely generated. Using Chomsky's hierarchy of languages, we prove that the fixed point submonoid of an endomorphism of a free inverse monoid can be represented by a context-sensitive language but, in general, it cannot be represented by a context-free language.

scholarly journals On Generalized WeaklyG-Contractive Mappings in Partially OrderedG-Metric Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
A. Razani ◽  
V. Parvaneh
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Metric Space ◽  
Metric Spaces ◽  
Periodic Points ◽  
Partially Ordered ◽  
Contractive Mappings

The aim of this paper is to present some coincidence and common fixed point results for generalized weaklyG-contractive mappings in the setup of partially orderedG-metric space. We also provide an example to illustrate the results presented herein. As an application of our results, periodic points of weaklyG-contractive mappings are obtained.

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