arXiv subjects

Bas Lemmens

Publications and source records attributed to Bas Lemmens.

At least 19 recordsLinked to original sources

arXiv2025

Carath\'eodory distance-preserving maps between bounded symmetric domains

We study the rigidity of maps between bounded symmetric domains that preserve the Carath\'eodory/Kobayashi distance. We show that such maps are only possible when the rank of the co-domain is at least as great as that of the domain. When the ranks are equal, and the domain is irreducible, we prove that the map is either holomorphic or antiholomorphic. In the holomorphic case, we show that the map is in fact a triple homomorphism, under the additional assumption that the origin is mapped to the origin. We exploit the large-scale geometry of the Carath\'eodory distance and use the horocompactification and Gromov product to obtain these results without requiring any smoothness assumptions on the maps.

math.CV↗

arXiv2025

On the structure of the dimension spectrum for continued fraction expansions

We analyse the dimension spectrum of continued fractions expansions with coefficients restricted to infinite subsets of $ \mathbb{N}$. We prove that the set of powers $P_q=\{q^n\colon n\in \mathbb{N}\}$ has full dimension spectrum for each integer $q\geq 2$, answering a question by Chousionis, Leykekhman and Urba\'nski. On the other hand, we show that the dimension spectrum for $P^*_q=\{q^n\colon n\in \mathbb{N}\}\cup\{1\}$ has many gaps and regions where it is nowhere dense. We also investigate the case where $A$ is generated by a monomial, $M_q=\{n^q\colon n\in\mathbb{N}\}$. For $M_q$ we prove that the dimension spectrum is full for $q\in\{1,2,3,4,5\}$, and it has a gap for each $q\geq 6$. Furthermore we show for $q\in\{6,7,8\}$ that the dimension spectrum of $M_q$ is the disjoint union of two nontrivial closed intervals, and it is the disjoint union of three nontrivial closed intervals for $q \in\{9,10\}$. For $q\geq 11$ we show that the dimension spectrum of $M_q$ consists of finitely many disjoint nontrivial closed intervals. The results concerning $M_q$ extend existing results for $q=1$ and $q=2$. In our analysis we employ Perron-Frobenius (transfer) operators, and numerical tools developed by Falk and Nussbaum that give rigorous estimates for the Hausdorff dimension for continued fractions expansions.

math.NT↗

arXiv2025

Infinite dimensional symmetric cones and gauge-reversing maps

The famous Koecher-Vinberg theorem characterises the finite dimensional formally real Jordan algebras among the finite dimensional order unit spaces as the ones that have a symmetric cone. An alternative characterisation of symmetric cones was obtained by Walsh who showed that the symmetric cones correspond exactly to the finite dimensional order unit spaces for which there exists a gauge-reversing map from the interior of the cone to itself. In this paper we prove an infinite dimensional version of this characterisation of symmetric cones.

math.FA↗

arXiv2022

Horofunctions and metric compactification of noncompact Hermitian symmetric spaces

Given a Hermitian symmetric space $M$ of noncompact type, we give a complete description of the horofunctions in the metric compactification of $M$ with respect to the Carath\'eodory distance, via the realisation of $M$ as the open unit ball $D$ of a Banach space $(V,\|\cdot\|)$ equipped with a Jordan structure, called a $\mathrm{JB}^*$-triple. The Carath\'eodory distance $\rho$ on $D$ has a Finsler structure. It is the integrated distance of the Carath\'eodory differential metric, and the norm $\|\cdot\|$ in the realisation is the Carath\'eodory norm with respect to the origin $0\in D$. We also identify the horofunctions of the metric compactification of $(V,\|\cdot\|)$ and relate its geometry and global topology to the closed dual unit ball (i.e., the polar of $D$). Moreover, we show that the exponential map $\exp_0 \colon V \longrightarrow D$ at $0\in D$ extends to a homeomorphism between the metric compactifications of $(V,\|\cdot\|)$ and $(D,\rho)$, preserving the geometric structure. Consequently, the metric compactification of $M$ admits a concrete realisation as the closed dual unit ball of $(V,\|\cdot\|)$.

math.DG↗

arXiv2022

Horofunction compactifications of symmetric cones under Finsler distances

In this paper we consider symmetric cones as symmetric spaces equipped with invariant Finsler distances, namely the Thompson distance and the Hilbert distance. We establish a correspondence between the horofunction compactification of a symmetric cone $A_+^\circ$ under these invariant Finsler distances and the horofunction compactification of the normed space in the tangent bundle. More precisely, for the Thompson distance on $A^\circ_+$ we show that the exponential map extends as a homeomorphism between the horofunction compactification of the normed space in the tangent bundle, which is a JB-algebra, and the horofunction compactification of $A_+^\circ$. We give a complete characteristation of the Thompson distance horofunctions and provide an explicit extension of the exponential map. Analogues results are established for the Hilbert distance on the projective cone $PA_+^\circ$. The analysis yields a geometric description of the horofunction compactifications of these spaces in terms of the facial structure of the closed unit ball of the dual norm of the norm in the tangent space.

math.MG↗

arXiv2022

Horofunction compactifications and duality

We study the global topology of the horofunction compactification of smooth manifolds with a Finsler distance. The main goal is to show, for certain classes of these spaces, that the horofunction compactification is naturally homeomorphic to the closed unit ball of the dual norm of the norm in the tangent space (at the base point) that generates the Finsler distance. We construct explicit homeomorphisms for a variety of spaces in three settings: bounded convex domains in $\mathbb{C}^n$ with the Kobayashi distance, Hilbert geometries, and finite dimensional normed spaces. For the spaces under consideration, the horofunction boundary has an intrinsic partition into so called parts. The natural connection with the dual norm arises through the fact that the homeomorphism maps each part in the horofunction boundary onto the relative interior of a boundary face of the dual unit ball. For normed spaces the connection between the global topology of the horofunction boundary and the dual norm was suggested by Kapovich and Leeb. We confirm this connection for Euclidean Jordan algebras equipped with the spectral norm.

math.GT↗

arXiv2021

A metric version of Poincaré's theorem concerning biholomorphic inequivalence of domains

We show that if $Y_j\subset \mathbb{C}^{n_j}$ is a bounded strongly convex domain with $C^3$-boundary for $j=1,\dots,q$, and $X_j\subset \mathbb{C}^{m_j}$ is a bounded convex domain for $j=1,\ldots,p$, then the product domain $\prod_{j=1}^p X_j\subset \mathbb{C}^m$ cannot be isometrically embedded into $\prod_{j=1}^q Y_j\subset \mathbb{C}^n$ under the Kobayashi distance, if $p>q$. This result generalises Poincaré's theorem which says that there is no biholomorphic map from the polydisc onto the Euclidean ball in $\mathbb{C}^n$ for $n\geq 2$. The method of proof only relies on the metric geometry of the spaces and will be derived from a result for products of proper geodesic metric spaces with the sup-metric. In fact, the main goal of the paper is to establish a general criterion, in terms of certain asymptotic geometric properties of the individual metric spaces, that yields an obstruction for the existence of an isometric embedding between product metric spaces.

math.CV↗

arXiv2019

On the linearity of order-isomorphisms

A basic problem in the theory of partially ordered vector spaces is to characterise those cones on which every order-isomorphism is linear. We show that this is the case for every Archimedean cone that equals the inf-sup hull of the sum of its engaged extreme rays. This condition is milder than existing ones and is satisfied by, for example, the cone of positive operators in the space of bounded self-adjoint operators on a Hilbert space. We also give a general form of order-isomorphisms on the inf-sup hull of the sum of all extreme rays of the cone, which extends results of Artstein-Avidan and Slomka to infinite dimensional partially ordered vector spaces, and prove the linearity of homogeneous order-isomorphisms in a variety of new settings.

math.FA↗

arXiv2018

On the complexity of detecting positive eigenvectors of nonlinear cone maps

In recent work with Lins and Nussbaum the first author gave an algorithm that can detect the existence of a positive eigenvector for order-preserving homogeneous maps on the standard positive cone. The main goal of this paper is to determine the minimum number of iterations this algorithm requires. It is known that this number is equal to the illumination number of the unit ball of the variation norm. In this paper we determine its illumination number, and hence provide a sharp lower bound for the running time of the algorithm.

math.OC↗

arXiv2017

Monotone dynamical systems with dense periodic points

In this paper we prove a recent conjecture by M. Hirsch, which says that if $(f,Ω)$ is a discrete time monotone dynamical system, with $f\colon Ω\toΩ$ a homeomorphism on an open connected subset of a finite dimensional vector space, and the periodic points of $f$ are dense in $Ω$, then $f$ is periodic.

math.DS↗

arXiv2017

Hilbert and Thompson isometries on cones in JB-algebras

Hilbert's and Thompson's metric spaces on the interior of cones in JB-algebras are important examples of symmetric Finsler spaces. In this paper we characterize the Hilbert's metric isometries on the interiors of cones in JBW-algebras, and the Thompson's metric isometries on the interiors of cones in JB-algebras. These characterizations generalize work by Bosché on the Hilbert and Thompson isometries on symmetric cones, and work by Hatori and Molnár on the Thompson isometries on the cone of positive self-adjoint elements in a unital $C^*$-algebra. To obtain the results we develop a variety of new geometric and Jordan algebraic techniques.

math.MG↗

arXiv2017

Isometries of infinite dimensional Hilbert geometries

In this paper we extend results by De la Harpe concerning the isometries of strictly convex Hilbert geometries, and the characterisation of the isometry groups of Hilbert geometries on finite dimensional simplices, to infinite dimensions. The proofs rely on a mix of geometric and functional analytic methods.

math.MG↗

arXiv2017

An order theoretic characterization of spin factors

The famous Koecher-Vinberg theorem characterizes the Euclidean Jordan algebras among the finite dimensional order unit spaces as the ones that have a symmetric cone. Recently Walsh gave an alternative characterization of the Euclidean Jordan algebras. He showed that the Euclidean Jordan algebras correspond to the finite dimensional order unit spaces $(V,C,u)$ for which there exists a bijective map $g\colon C^\circ\to C^\circ$ with the property that $g$ is antihomogeneous, i.e., $g(λx) =λ^{-1}g(x)$ for all $λ>0$ and $x\in C^\circ$, and $g$ is an order-antimorphism, i.e., $x\leq_C y$ if and only if $g(y)\leq_C g(x)$. In this paper we make a first step towards extending this order theoretic characterization to infinite dimensional JB-algebras. We show that if $(V,C,u)$ is a complete order unit space with a strictly convex cone and $\dim V\geq 3$, then there exists a bijective antihomogeneous order-antimorphism $g\colon C^\circ\to C^\circ$ if and only if $(V,C,u)$ is a spin factor.

math.FA↗

arXiv2016

Detecting fixed points of nonexpansive maps by illuminating the unit ball

We give necessary and sufficient conditions for a nonexpansive map on a finite dimensional normed space to have a nonempty, bounded set of fixed points. Among other results we show that if $f : V \rightarrow V$ is a nonexpansive map on a finite dimensional normed space $V$, then the fixed point set of $f$ is nonempty and bounded if and only if there exist $w_1, \ldots , w_m$ in $V$ such that $\{f(w_i) - w_i : i = 1, \ldots, m \}$ illuminates the unit ball. This yields a numerical procedure for detecting fixed points of nonexpansive maps on finite dimensional spaces. We also discuss applications of this procedure to certain nonlinear eigenvalue problems arising in game theory and mathematical biology.

math.FA↗

arXiv2016

Denjoy-Wolff theorems for Hilbert's and Thompson's metric spaces

We study the dynamics of fixed point free mappings on the interior of a normal, closed cone in a Banach space that are nonexpansive with respect to Hilbert's metric or Thompson's metric. We establish several Denjoy-Wolff type theorems that confirm conjectures by Karlsson and Nussbaum for an important class of nonexpansive mappings. We also extend and put into a broader perspective results by Gaubert and Vigeral concerning the linear escape rate of such nonexpansive mappings.

math.DS↗

arXiv2015

Midpoints for Thompson's metric on symmetric cones

We characterise the affine span of the midpoints sets for Thompson's metric on symmetric cones in terms of a translation of the zero-component of the Peirce decomposition of an idempotent. As a consequence we derive an explicit formula for the dimension of the affine span of the midpoints sets in case the associated Euclidean Jordan algebra is simple.

math.MG↗

arXiv2014

Bands in partially ordered vector spaces with order unit

In an Archimedean directed partially ordered vector space $X$ one can define the concept of a band in terms of disjointness. Bands can be studied by using a vector lattice cover $Y$ of $X$. If $X$ has an order unit, $Y$ can be represented as $C(Ω)$, where $Ω$ is a compact Hausdorff space. We characterize bands in $X$, and their disjoint complements, in terms of subsets of $Ω$. We also analyze two methods to extend bands in $X$ to $C(Ω)$ and show how the carriers of a band and its extensions are related. We use the results to show that in each $n$-dimensional partially ordered vector space with a closed generating cone, the number of bands is bounded by $\frac{1}{4}2^{2^n}$ for $n\geq 2$. We also construct examples of $(n+1)$-dimensional partially ordered vector spaces with ${2n\choose n}+2$ bands. This shows that there are $n$-dimensional partially ordered vector spaces that have more bands than an $n$-dimensional Archimedean vector lattice when $n\geq 4$.

math.FA↗

arXiv2013

Unique geodesics for Thompson's metric

In this paper a geometric characterization of the unique geodesics in Thompson's metric spaces is presented. This characterization is used to prove a variety of other geometric results. Firstly, it will be shown that there exists a unique Thompson's metric geodesic connecting $x$ and $y$ in the cone of positive self-adjoint elements in a unital $C^*$-algebra if, and only if, the spectrum of $x^{-1/2}yx^{-1/2}$ is contained in $\{1/β,β\}$ for some $β\geq 1$. A similar result will be established for symmetric cones. Secondly, it will be shown that if $C^\circ$ is the interior of a finite-dimensional closed cone $C$, then the Thompson's metric space $(C^\circ,d_C)$ can be quasi-isometrically embedded into a finite-dimensional normed space if, and only if, $C$ is a polyhedral cone. Moreover, $(C^\circ,d_C)$ is isometric to a finite-dimensional normed space if, and only if, $C$ is a simplicial cone. It will also be shown that if $C^\circ$ is the interior of a strictly convex cone $C$ with $3\leq \dim C<\infty$, then every Thompson's metric isometry is projectively linear.

math.MG↗