Examples and Questions in the Theory of Fixed Point Sets

1979 ◽  
Vol 31 (5) ◽  
pp. 1017-1032 ◽  
Author(s):  
John R. Martin ◽  
Sam B. Nadler
Keyword(s):  
Fixed Point ◽  
Continuous Function ◽  
Banach Spaces ◽  
Metric Spaces ◽  
Compact Manifolds ◽  
Invariance Property ◽  
Point Sets ◽  
Fixed Point Set ◽  
Point Set ◽  
Fixed Point Sets

All spaces considered in this paper will be metric spaces. A subset A of a space X is called a fixed point set of X if there is a map (i.e., continuous function) ƒ: X → X such that ƒ(x) = x if and only if x ∈ A. In [22] L. E. Ward, Jr. defines a space X to have the complete invariance property (CIP) provided that each of the nonempty closed subsets of X is a fixed point set of X. The problem of determining fixed point sets of spaces has been investigated in [14] through [20] and [22]. Some spaces known to have CIP are n-cells[15], dendrites [20], convex subsets of Banach spaces [22], compact manifolds without boundary [16], and a class of polyhedra which includes all compact triangulable manifolds with or without boundary [18].

scholarly journals A Note on Fixed Point Sets and Wedges

1980 ◽  
Vol 23 (4) ◽  
pp. 453-455 ◽  
Author(s):  
John R. Martin ◽  
Sam B. Nadler
Keyword(s):  
Fixed Point ◽  
Closed Subset ◽  
Invariance Property ◽  
Point Sets ◽  
Fixed Point Set ◽  
Nonempty Closed Subset ◽  
One Dimensional ◽  
Point Set ◽  
Fixed Point Sets

A space Z is said to have the complete invariance property (CIP) provided that every nonempty closed subset of Z is the fixed point set of some continuous self-mapping of Z. In this paper it is shown that there exists a one-dimensional contractible planar continuum having CIP whose wedge with itself at a specified point is contractible, planar, and does not have CIP.

scholarly journals Fixed poin sets in digital topology, 1

2020 ◽  
Vol 21 (1) ◽  
pp. 87 ◽  
Author(s):  
Laurence Boxer ◽  
P. Christopher Staecker
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Digital Images ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Point Sets ◽  
Fixed Point Set ◽  
Complete Computation ◽  
Point Set ◽  
Fixed Point Sets

<p>In this paper, we examine some properties of the fixed point set of a digitally continuous function. The digital setting requires new methods that are not analogous to those of classical topological fixed point theory, and we obtain results that often differ greatly from standard results in classical topology.</p><p>We introduce several measures related to fixed points for continuous self-maps on digital images, and study their properties. Perhaps the most important of these is the fixed point spectrum F(X) of a digital image: that is, the set of all numbers that can appear as the number of fixed points for some continuous self-map. We give a complete computation of F(C<sub>n</sub>) where C<sub>n</sub> is the digital cycle of n points. For other digital images, we show that, if X has at least 4 points, then F(X) always contains the numbers 0, 1, 2, 3, and the cardinality of X. We give several examples, including C<sub>n</sub>, in which F(X) does not equal {0, 1, . . . , #X}.</p><p>We examine how fixed point sets are affected by rigidity, retraction, deformation retraction, and the formation of wedges and Cartesian products. We also study how fixed point sets in digital images can be arranged; e.g., for some digital images the fixed point set is always connected.</p>

scholarly journals INVOLUTIONS OF GRAPH LINK EXTERIORS WHOSE FIXED POINT SETS ARE CLOSED SURFACES

2010 ◽  
Vol 81 (2) ◽  
pp. 298-303 ◽  
Author(s):  
TORU IKEDA
Keyword(s):  
Fixed Point ◽  
Lower Estimate ◽  
Closed Surface ◽  
Point Sets ◽  
Fixed Point Set ◽  
Closed Surfaces ◽  
Point Set ◽  
Graph Link ◽  
Fixed Point Sets

AbstractA link L in S3 possibly admits an involution of the exterior E(L) with fixed point set a closed surface, which is not extendable to an involution of S3. In this paper, we focus on the case of graph links and show that the genus of the surface provides a lower estimate of the number of link components.

scholarly journals Infinitely Periodic Knots

1985 ◽  
Vol 37 (1) ◽  
pp. 17-28 ◽  
Author(s):  
Erica Flapan
Keyword(s):  
Fixed Point ◽  
Point Sets ◽  
Fixed Point Set ◽  
Finite Group Actions ◽  
Three Sphere ◽  
Knot Invariant ◽  
Periodic Transformation ◽  
Point Set ◽  
Fixed Point Sets

One aspect of the study of 3-manifolds is to determine what finite group actions a given manifold has. Some important questions that one can ask about these actions on a given manifold are: What periods could they have? and, what sets of points may be fixed by the action? In the case of periodic transformations of homology spheres, Smith [18] classified the types of fixed point sets which could occur. For homology 3-spheres the fixed point set will be ∅, S0, S1, or S2. Fox [4] looked at periodic transformations of the three sphere which leave a knot invariant and, using Smith's classification of fixed point sets, determined that there were eight types of transformations according to how the fixed point set met the knot. For convenience we shall say a knot is (a, b)-periodic if there is a periodic transformation of S3 leaving the knot invariant with fixed point set homeomorphic to a and with the fixed point set meeting the knot in a set homeomorphic to b.

Fixed-point sets of smooth actions on spheres

2007 ◽  
Vol 1 (1) ◽  
pp. 95-128
Author(s):  
Masaharu Morimoto
Keyword(s):  
Fixed Point ◽  
Finite Groups ◽  
Basic Problem ◽  
Point Sets ◽  
Fixed Point Set ◽  
Surgery Theory ◽  
Smooth Action ◽  
Point Set ◽  
Current Article ◽  
Fixed Point Sets

AbstractGiven a group, it is a basic problem to determine which manifolds can occur as a fixed-point set of a smooth action of this group on a sphere. The current article answers this problem for a family of finite groups including perfect groups and nilpotent Oliver groups. We obtain the answer as an application of a new deleting and inserting theorem which is formulated to delete (or insert) fixed-point sets from (or to) disks with smooth actions of finite groups. One of the keys to the proof is an equivariant interpretation of the surgery theory of S. E. Cappell and J. L. Shaneson, for obtaining homology equivalences.

scholarly journals Fixed point sets in digital topology, 2

2020 ◽  
Vol 21 (1) ◽  
pp. 111
Author(s):  
Laurence Boxer
Keyword(s):  
Fixed Point ◽  
Digital Images ◽  
Digital Topology ◽  
Point Sets ◽  
Fixed Point Set ◽  
Point Set ◽  
Fixed Point Sets

We continue the work of [10], studying properties of digital images determined by fixed point invariants. We introduce pointed versions of invariants that were introduced in [10]. We introduce freezing sets and cold sets to show how the existence of a fixed point set for a continuous self-map restricts the map on the complement of the fixed point set.

On fixed point sets of wavelet induced isomorphisms

2014 ◽  
Vol 12 (06) ◽  
pp. 1450042 ◽  
Author(s):  
Dania Masood ◽  
Pooja Singh
Keyword(s):  
Fixed Point ◽  
Point Sets ◽  
Fixed Point Set ◽  
Point Set ◽  
Fixed Point Sets

In this paper, we show that [½, 1) is not a fixed point set of any wavelet induced isomorphism. The same holds for [0, ½). This settles Problem 1 posed in the paper "On wavelet induced isomorphisms" by Singh in Int. J. Wavelets Multiresolut. Inf. Process.8 (2010) 359–371. Further, we obtain that every other subinterval of [0, 1) of measure ½ is a fixed point set of some wavelet induced isomorphism.

scholarly journals Approximation of a Common Element of the Fixed Point Sets of Multivalued Strictly Pseudocontractive-Type Mappings and the Set of Solutions of an Equilibrium Problem in Hilbert Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
F. O. Isiogugu
Keyword(s):  
Fixed Point ◽  
Equilibrium Problem ◽  
Hilbert Spaces ◽  
Contemporary Literature ◽  
Common Element ◽  
Point Sets ◽  
Fixed Point Set ◽  
Strict Fixed Point ◽  
Point Set ◽  
Fixed Point Sets

The strong convergence of a hybrid algorithm to a common element of the fixed point sets of multivalued strictly pseudocontractive-type mappings and the set of solutions of an equilibrium problem in Hilbert spaces is obtained using a strict fixed point set condition. The obtained results improve, complement, and extend the results on multivalued and single-valued mappings in the contemporary literature.

scholarly journals Product involutions with0or1-dimensional fixed point sets on orientable handlebodies

1994 ◽  
Vol 17 (3) ◽  
pp. 457-462
Author(s):  
Roger B. Nelson
Keyword(s):  
Fixed Point ◽  
Necessary Conditions ◽  
Point Sets ◽  
Fixed Point Set ◽  
Point Set ◽  
Fixed Point Sets

Lethbe an involution with a0or1-dimensional fixed point set on an orientable handlebodyM. We show that obvious necessary conditions for fiberingMasA×Iso thath=τ×rwithτan involution ofAandrreflection about the midpoint ofIalso turn out to be sufficient. We also show that such a “product” involution is determined by its fixed point set.

scholarly journals On convergence theorems for single-valued and multi-valued mappings in p-uniformly convex metric spaces

2021 ◽  
Vol 37 (3) ◽  
pp. 513-527
Author(s):  
JENJIRA PUIWONG ◽  
◽  
SATIT SAEJUNG ◽  
◽  
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Nonexpansive Mappings ◽  
Iterative Sequence ◽  
Uniformly Convex ◽  
Fixed Point Set ◽  
Convergence Theorems ◽  
Convex Metric Spaces ◽  
Strict Fixed Point ◽  
Point Set

We prove ∆-convergence and strong convergence theorems of an iterative sequence generated by the Ishikawa’s method to a fixed point of a single-valued quasi-nonexpansive mappings in p-uniformly convex metric spaces without assuming the metric convexity assumption. As a consequence of our single-valued version, we obtain a result for multi-valued mappings by showing that every multi-valued quasi-nonexpansive mapping taking compact values admits a quasi-nonexpansive selection whose fixed-point set of the selection is equal to the strict fixed-point set of the multi-valued mapping. In particular, we immediately obtain all of the convergence theorems of Laokul and Panyanak [Laokul, T.; Panyanak, B. A generalization of the (CN) inequality and its applications. Carpathian J. Math. 36 (2020), no. 1, 81–90] and we show that some of their assumptions are superfluous.

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