Sectional category and The Fixed Point Property

2019

In this work we exhibit an unexpected connection between sectional category theory and the fixed point property. On the one hand, a topological space X is said to have the fixed point property (FPP) if, for every continuous self-map f of X, there is a point x of X such that f(x) = x. On the other hand, for a continuous surjection p : E → B, the standard sectional number secop(p) is the minimal cardinality of open covers {Ui} of B such that each Ui admits a continuous local section for p. Let F (X, k) denote the configuration space of k ordered distinct points in X and consider the natural projection πk,1 : F (X, k) → X. We demonstrate that a space X has the FPP if and only if secop(π2,1) = 2. This characterization connects a standard problem in fixed point theory to current research trends in topological robotics.

Topological Methods in Nonlinear Analysis, 1995

2007

We use the ideas of Lusternik–Schnirelmann theory to describe the set of fixed points of certain homotopy equivalences of a general space. In fact, we extend Lusternik–Schnirelmann theory to pairs (φ, f), where φ is a homotopy equivalence of a topological space X and where f : X → R is a continuous function satisfying f(φ(x)) < f(x) unless φ(x) = x; in addition, the pair (φ, f) is supposed to satisfy a discrete analogue of the Palais–Smale condition. In order to estimate the number of fixed points of φ in a subset of X, we consider different relative categories. Moreover, the theory is carried out in an equivariant setting.

Topology and its Applications, 1982

2007

We present some fixed point theorems for planar maps which satisfy a property of path-expansion along a certain direction. We also show some links between these fixed point theorems and other recent results about covering relations and topological horseshoes.

arXiv: Dynamical Systems, 2018

Certain notions of expansivity and shadowing were defined on topological spaces which are dynamical properties and generalize the usual definitions. A Topologically Anosov homeomorphism is a homeomorphism with such properties. We exhibit explicit examples of Topologically Anosov homeomorphisms on the plane. Our main result is a fixed point theorem for orientation preserving Topologically Anosov plane homeomorphisms.

Archiv der Mathematik, 2012

We show that, for any n = 2, most orientation preserving homeomorphisms of the sphere S 2n have a Cantor set of fixed points. In other words, the set of such homeomorphisms that do not have a Cantor set of fixed points is of the first Baire category within the set of all homeomorphisms. Similarly, most orientation reversing homeomorphisms of the sphere S 2n+1 have a Cantor set of fixed points for any n = 0. More generally, suppose that M is a compact manifold of dimension > 1 and = 4 and H is an open set of homeomorphisms h : M → M such that all elements of H have at least one fixed point. Then we show that most elements of H have a Cantor set of fixed points.

New class of homeomorphisms named as sαrw-homeomorphism and sαrw*-homeomorphism are explored & elaborated. Few basic properties are inspected. Their relations with some existing homeomorphisms in topological spaces are studied.

Topological Methods in Nonlinear Analysis, 2002

The Lefschetz Fixed Point Theorem for compact absorbing contraction morphisms (CAC-morphisms) of retracts of open subsets in admissible spaces in the sense of Klee is proved. Moreover, the relative version of the Lefschetz Fixed Point Theorem and the Lefschetz Periodic Theorem are considered. Additionally, a full classification of morphisms with compact attractors in the non-metric case is obtained. 1. Vietoris mappings; admissibility in the sense of Klee We are interested in theory of homology such that Vietoris theorem is satisfied for any topological space. In this paper we use a definition ofČech theory of homology with compact carriers and coefficients in the field of rationals Q given in [15] (see also [18]). A space X is acyclic if: (a) X is non-empty, (b) H q (X) = 0 for every q ≥ 1 and (c) H 0 (X) ≈ Q. A continuous mapping f : X → Y of Hausdorff topological spaces X and Y is called perfect if f is closed and for every y ∈ Y a set f −1 (y) is compact. Definition 1.1. A mapping of pair of spaces p: (Γ, Γ 0) → (X, X 0) is called Vietoris mapping provided it is a perfect surjection such that a set p −1 (x) is acyclic for any x ∈ X and Γ 0 = p −1 (X 0).

2010

There have appeared many generalizations of the Kakutani-Fan-Glicksberg fixed point theorem. Motivated by these generalizations we introduce the concept of fixed point property for a pair (T , C) of classes of compact Hausdorff topological spaces; section properties and minimax inequalities are given.