Fixed Point Property Research Articles

Weak fixed point property of order p in Banach lattices

Weak fixed point property of order p in Banach lattices

Expansions of real closed fields with the Banach fixed point property

AbstractWe study a variant of converses of the Banach fixed point theorem and its connection to tameness in expansions of a real closed field. An expansion of a real closed ordered field is said to have the Banach fixed point property when, for every locally closed definable set , if every definable contraction on has a fixed point, then is closed. Let be an expansion of a real closed field. We prove that if has an o‐minimal open core, then it has the Banach fixed point property; and if is definably complete and has the Banach fixed point property, then it has a locally o‐minimal open core.

Property $\mathrm{(NL)}$ for group actions on hyperbolic spaces (with an appendix by Alessandro Sisto)

We introduce Property \mathrm{(NL)} , which indicates that a group does not admit any (isometric) action on a hyperbolic space with loxodromic elements. In other words, such a group G can only admit elliptic or horocyclic hyperbolic actions, and consequently its poset of hyperbolic structures \mathcal{H}(G) is trivial. It turns out that many groups satisfy this property, and we initiate the formal study of this phenomenon. Of particular importance is the proof of a dynamical criterion in this paper that ensures that groups with “rich” actions on compact Hausdorff spaces have Property \mathrm{(NL)} . These include many Thompson-like groups, such as V, T and even twisted Brin–Thompson groups, which implies that every finitely generated group quasi-isometrically embeds into a finitely generated simple group with Property \mathrm{(NL)} . We also study the stability of the property under group operations and explore connections to other fixed point properties. In the appendix (by Alessandro Sisto), we describe a construction of cobounded actions on hyperbolic spaces starting from non-cobounded ones that preserves various properties of the initial action.

The topological space of Schur-concave copulas is homeomorphic to the Hilbert cube

Schur-concave copulas, a distinct subclass of copulas, have emerged as a crucial component in recent advancements of copula theory. In this paper, we employ techniques from infinite-dimensional topology to investigate the topological structure of the space of all Schur-concave copulas equipped with the uniform metric and its topological position in the (symmetric) copula space. Moreover, we show that there exists a homeomorphism from the copula space onto the Hilbert cube, which relates the symmetric copula space onto an end of the Hilbert cube and relates the Schur-concave copula space onto an end of the end of the Hilbert cube. As a consequence, the Schur-concave copula space has the fixed point property. Furthermore, we show that a subset of the family of all associative copulas is a Z-set in the Schur-concave copula space if and only if it is compact.

On the Ryll-Nardzewski fixed point property in convex uniform spaces

In this paper, we introduce the concept of a convex uniform space as a natural generalization of locally convex spaces, and a fixed point property as a semigroup formulation of the famous Ryll-Nardzewski fixed point theorem, and study its non-linear counterpart in this framework. The Nardzewski's fixed point theorem is a beautiful result in fixed point theory that ensures the existence of a common fixed point for any noncontracting semigroup of continuous affine self-mappings on a non-void weakly compact convex set in a separated locally convex space. We establish that in a separated convex uniform space satisfying a certain property (P), any noncontracting finite semigroup of contraction mappings of a non-empty bounded closed convex subset possesses a common fixed point. In case where the underlying space is admissible, we prove that this result is extendable to arbitrary noncontracting semigroups without condition (P) if we assume that the phase space is weakly compact and convex.

Explicit models of ℓ_1-preduals and the weak* fixed point property in ℓ_1

We provide a concrete isometric description of all the preduals of $\ell_1$ for which the standard basis in $\ell_1$ has a finite number of $w^*$-limit points. Then, we apply this result to give an example of an $\ell_1$-predual $X$ such that its dual $X^*$ lacks the weak$^*$ fixed point property for nonexpansive mappings (briefly, $w^*$-FPP), but $X$ does not contain an isometric copy of any hyperplane $W_{\alpha}$ of the space $c$ of convergent sequences such that $W_\alpha$ is a predual of $\ell_1$ and $W_\alpha^*$ lacks the $w^*$-FPP. This answers a question left open in the 2017 paper of the present authors.

Fixed points of $G$-monotone mappings in metric and modular spaces

Let $C$ be a bounded, closed and convex subset of a reflexive metric space with a digraph $G$ such that $G$-intervals along walks are closed and convex. In the main theorem we show that if $T\colon C\rightarrow C$ is a monotone $G$-nonexpansive mapping and there exists $c\in C$ such that $Tc\in [c,\rightarrow )_{G}$, then $T$ has a fixed point provided for each $a\in C$, $[a,a]_{G}$ has the fixed point property for nonexpansive mappings. In particular, it gives an essential generalization of the Dehaish-Khamsi theorem concerning partial orders in complete uniformly convex hyperbolic metric spaces. Some counterparts of this result for modular spaces, and for commutative families of mappings are given too.

An approximate fixed point property

A map of a metric space into itself has the approximate fixed point property (AFPP for short) if every nearly fixed point is close to some fixed point. It is proven that both linear operators acting on finite-dimensional Banach spaces and uniformly expansive linear homeomorphisms on Banach spaces exhibit the AFPP. Furthermore, an illustration is provided of a linear homeomorphism that does not satisfy the AFPP.

Quaternion weighted Schatten [formula omitted]-norm minimization for color image restoration with convergence guarantee

Over the past decades, the rank approximation issue have been extensively investigated, among which weighted nuclear norm minimization (WNNM) and weighted Schatten p-norm minimization (WSNM) are two prevailing methods and have shown great superiority in various image restoration (IR) problems. However, for the complicated color image restoration (CIR) problems, traditional WNNM/WSNM method only processes three color channels individually and fails to consider their cross-channel correlations. Very recently, a quaternion-based WNNM approach (QWNNM) has been developed to mitigate this issue, which is capable of representing the color image as a whole in the quaternion domain. Despite QWNNM’s empirical success, its convergence behavior has not been rigorously studied. The main contributions of this paper are twofold. Firstly, we extend the WSNM into quaternion domain and correspondingly propose a novel quaternion-based WSNM model (QWSNM) for tackling the CIR problems. Extensive experiments on two representative CIR tasks, including color image denoising and deblurring, demonstrate that the proposed QWSNM method performs favorably against many state-of-the-art alternatives, in both quantitative and qualitative evaluations. Secondly, we provide a preliminary theoretical convergence analysis. By modifying the quaternion alternating direction method of multipliers (QADMM) through a simple continuation strategy, we theoretically prove the fixed-point convergence property of the iterative sequences generated by QWNNM and QWSNM. The source code of our algorithm can be available at the website: https://github.com/qiuxuanzhizi/QWSNM.

The topological structures of the spaces of diagonal and opposite diagonal functions with the uniform metric

Diagonal and opposite diagonal functions play an important role in various applications of the copula theory. Denote the families of all 2-Lipschitz continuous functions from the unit closed interval to itself, diagonal functions and opposite diagonal functions, by L2(I), D and O, respectively. In this work, we study the topological structures of the sets D and O with the uniform metric d∞, and mainly show that there exist homeomorphisms H1,H2:(L2(I),d∞)→[0,1]×Q such that H1(D,d∞)=H2(O,d∞)={0}×Q, where Q=[−1,1]N is the Hilbert cube. This result allows us to deduce that the set of all (opposite) diagonal sections of (quasi-)copulas with the topology induced by the uniform metric is homeomorphic to the Hilbert cube Q, and hence has the fixed point property.

On parabolic subgroups of Artin groups

Given an Artin group AΓ, a common strategy in the study of AΓ is the reduction to parabolic subgroups whose defining graphs have small diameter, i.e., showing that AΓ has a specific property if and only if all “small” parabolic subgroups of AΓ have this property. Since “small” parabolic subgroups are the building blocks of AΓ one needs to study their behavior, in particular their intersections. The conjecture we address here says that the class of parabolic subgroups of AΓ is closed under intersection. Under the assumption that intersections of parabolic subgroups in complete Artin groups are parabolic, we show that the intersection of a complete parabolic subgroup with an arbitrary parabolic subgroup is parabolic. Further, we connect the intersection behavior of complete parabolic subgroups of AΓ to fixed point properties and to automatic continuity of AΓ using Bass–Serre theory and a generalization of the Deligne complex.

Numberings, c.e. oracles, and fixed points

The Arslanov completeness criterion says that a c.e. set A is Turing complete if and only there exists an A-computable function f without fixed points, i.e. a function f such that W f ( x ) ≠ W x for each integer x. Recently, Barendregt and Terwijn proved that the completeness criterion remains true if we replace the Gödel numbering x ↦ W x with an arbitrary precomplete computable numbering. In this paper, we prove criteria for noncomputability and highness of c.e. sets in terms of (pre)complete computable numberings and fixed point properties. We also find some precomplete and weakly precomplete numberings of arbitrary families computable relative to Turing complete and non-computable c.e. oracles respectively.

Specificity and sensitivity of the fixed-point test for binary mixture distributions

When two cognitive processes contribute to a behavioral output—each process producing a specific distribution of the behavioral variable of interest—and when the mixture proportion of these two processes varies as a function of an experimental condition, a common density point should be present in the observed distributions of the data across said conditions. In principle, one can statistically test for the presence (or absence) of a fixed point in experimental data to provide evidence in favor of (or against) the presence of a mixture of processes, whose proportions are affected by an experimental manipulation. In this paper, we provide an empirical diagnostic of this test to detect a mixture of processes. We do so using resampling of real experimental data under different scenarios, which mimic variations in the experimental design suspected to affect the sensitivity and specificity of the fixed-point test (i.e., mixture proportion, time on task, and sample size). Resampling such scenarios with real data allows us to preserve important features of data which are typically observed in real experiments while maintaining tight control over the properties of the resampled scenarios. This is of particular relevance considering such stringent assumptions underlying the fixed-point test. With this paper, we ultimately aim at validating the fixed-point property of binary mixture data and at providing some performance metrics to researchers aiming at testing the fixed-point property on their experimental data.

A Fixed Point Theorem in the Lebesgue Spaces of Variable Integrability Lp(·)

We establish a fixed point property for the Lebesgue spaces with variable exponents Lp(·), focusing on the scenario where the exponent closely approaches 1. The proof does not impose any additional conditions. In particular, our investigation centers on ρ-non-expansive mappings defined on convex subsets of Lp(·), satisfying the “condition of uniform decrease” that we define subsequently.

The persistence principle over weak interpretability logic

AbstractWe focus on the persistence principle over weak interpretability logic. Our object of study is the logic obtained by adding the persistence principle to weak interpretability logic from several perspectives. Firstly, we prove that this logic enjoys a weak version of the fixed point property. Secondly, we introduce a system of sequent calculus and prove the cut‐elimination theorem for it. As a consequence, we prove that the logic enjoys the Craig interpolation property. Thirdly, we show that the logic is the natural basis of a generalization of simplified Veltman semantics, and prove that it has the finite frame property with respect to that semantics. Finally, we prove that it is sound and complete with respect to some appropriate arithmetical semantics.

“Lion–man” game and the fixed point property in Hilbert spaces

The main goal of the paper is to show that in the setting of a Hilbert space a closed and convex K has the fixed point property for nonexpansive mappings if and only if the lion always wins the Lion–Man game played in K.

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Geometry of infinite dimensional unitary groups: Convexity and fixed points

In this article we study convexity properties of distance functions in infinite dimensional Finsler unitary groups, such as the full unitary group, the unitary Schatten perturbations of the identity and unitary groups of finite von Neumann algebras. The Finsler structures are defined by translation of different norms on the tangent space at the identity. We first prove a convexity result for the metric derived from the operator norm on the full unitary group. We also prove strong convexity results for the squared metrics in Hilbert-Schmidt unitary groups and unitary groups of finite von Neumann algebras. In both cases the tangent spaces are endowed with an inner product defined with a trace. These results are applied to fixed point properties and to quantitative metric bounds in certain rigidity problems. Radius bounds for all convexity and fixed point results are shown to be optimal.

Multilabeled and topological versions of the Hex theorem

The purpose of our paper is to give a purely combinatorial and algorithmic proof of the Hex theorem with multiple labels. Moreover we extend this result for a new class of complexes, called n-essential. We also present a topological version of the Hex theorem and explain its relation to the fixed point property, the Poincaré-Miranda theorem and the covering (Lebesgue) dimension.

Thompson-like groups, Reidemeister numbers, and fixed points

We investigate fixed-point properties of automorphisms of groups similar to Richard Thompson’s group F. Revisiting work of Gonçalves and Kochloukova, we deduce a cohomological criterion to detect infinite fixed-point sets in the abelianization, implying the so-called property R_infty . Using the Bieri–Neumann–Strebel varSigma -invariant and drawing from works of Gonçalves–Sankaran–Strebel and Zaremsky, we show that our tool applies to many F-like groups, including Stein’s group F_{2,3}, cleary’s irrational-slope group F_tau , the Lodha–Moore groups, and the braided version of F.