Bas Lemmens

University of Kent | KENT · School of Mathematics, Statistics and Actuarial Science

Given a Hermitian symmetric space M of noncompact type, we show, among other things, that the metric compactification of M with respect to its Carathéodory distance is homeomorphic to a closed ball in its tangent space. We first give a complete description of the horofunctions in the compactification of M via the realisation of M as the open unit b...

In this paper we consider symmetric cones equipped with invariant Finsler distances, namely the Thompson distance and the Hilbert distance. We give a complete characterisation of the horofunctions of the symmetric cone \(A_+^\circ\) under the Thompson distance and establish a correspondence between the horofunction compactification of \(A_+^\circ\)...

We study the global topology and geometry of the horofunction compactification of classes of symmetric spaces under Finsler distances in three settings: bounded symmetric domains of the form $$B=B_1\times \cdots \times B_r$$ B = B 1 × ⋯ × B r , where $$B_i$$ B i is an open Euclidean ball in $${\mathbb {C}}^{n_i}$$ C n i , with the Kobayashi distanc...

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...

We show that if $$Y_j\subset \mathbb {C}^{n_j}$$ Y j ⊂ C n j is a bounded strongly convex domain with $$C^3$$ C 3 -boundary for $$j=1,\dots ,q$$ j = 1 , ⋯ , q , and $$X_j\subset \mathbb {C}^{m_j}$$ X j ⊂ C m j is a bounded convex domain for $$j=1,\ldots ,p$$ j = 1 , … , p , then the product domain $$\prod _{j=1}^p X_j\subset \mathbb {C}^m$$ ∏ j = 1...

In this paper we establish a correspondence between the horofunction compactification of a symmetric cone $A_+^\circ$ under certain Finsler distances and the horofunction compactification of the normed space in the tangent bundle. More precisely, we consider the Thompson distance on $A^\circ_+$ and the Hilbert distance on the projective cone $PA_+^...

We study the global topology of the horofunction compactification of proper geodesic metric spaces with a Finsler structure. 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) tha...

We show that If $D\subset \mathbb{C}^n$ is a bounded strongly convex domain with $C^3$ boundary, and $X\subset \mathbb{C}^m$ and $Y\subset \mathbb{C}^k$ are bounded convex domains, then $X\times Y$ cannot be isometrically embedded into $D$ under the Kobayashi distance. This result generalises Poincar\'e's theorem which says that there is no biholom...

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 co...

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...

If ℝ ⁿ is partially ordered by a generating closed cone K, then (ℝ ⁿ , K) is a pre-Riesz space. We show for a disjointness preserving bijection T on (ℝ ⁿ , K) that the inverse of T is also disjointness preserving. We prove that for T there is k ∈ 𝒫(b) such that T k is band preserving, where b is the number of bands in (ℝ ⁿ , K), and 𝒫(b) the set of...

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 co...

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 illum...

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

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 space...

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 ther...

We characterise the affine span of the midpoints sets, M (x, y), 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 M (x, y) in case the associated Euclidean Jordan algebra is sim...

In this note, it is shown that the maximum number of pairwise touching translates of an n -simplex is at least n+3 for n=7 , and for all n≥5 such that n≡1mod4 . The current best known lower bound for general n is n+2 . For n=2 k −1 and k≥2 , we will also present an alternative construction to give n+2 touching simplices using Hadamard matrices.

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 a...

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.

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 $C(\Omega)$, where $\Omega$ is a compact Hausdorff space. We characterize bands in $X$, and their disjoint c...

We show that the Riesz completion of an Archimedean partially ordered vector space $X$ with unit can be represented as a norm dense Riesz subspace of the smallest functional representation of $X.$ This yields a convenient way to find the Riesz completion. To illustrate the method, the Riesz completions of spaces ordered by Lorentz cones, cones...

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^...

This is a survey article concerning applications of Hilbert's metric in the analysis and dynamics of linear and nonlinear mappings on cones. It will appear as a chapter in the "Handbook of Hilbert geometry", ed. G. Besson, A. Papadopoulos and M. Troyanov, European Mathematical Society Publishing House, Z\"urich.

In the past several decades the classical Perron–Frobenius theory for nonnegative matrices has been extended to obtain remarkably precise and beautiful results for classes of nonlinear maps. This nonlinear Perron–Frobenius theory has found significant uses in computer science, mathematical biology, game theory and the study of dynamical systems. Th...

This paper deals with the iterative behavior of nonexpansive mappings on Hilbert's metric spaces (X, d X). We show that if (X, d X) is strictly convex and does not contain a hyperbolic plane, then for each nonexpansive mapping, with a fixed point in X, all orbits converge to periodic orbits. In addition, we prove that if X is an open 2-simplex, the...

This paper concerns the question whether the cone spectral radius of a continuous compact order-preserving homogenous map on a closed cone in Banach space depends continuously on the map. Using the fixed point index we show that if there exist points not in the cone spectrum arbitrarily close to the cone spectral radius, then the cone spectral radi...

We study the stable behaviour of discrete dynamical systems where the map is convex and monotone with respect to the standard positive cone. The notion of tangential stability for fixed points and periodic points is introduced, which is weaker than Lyapunov stability. Among others we show that the set of tangentially stable fixed points is isomorph...

This paper concerns the dynamics of non-expansive maps on strictly convex finite dimensional normed spaces. By using results of Edelstein and Lyubich, we show that if X = (ℝ n , ∥ · ∥) is strictly convex and X has no 1-complemented Euclidean plane, then every bounded orbit of a non-expansive map f: X → X, converges to a periodic orbit. By putting e...

We show that the isometry group of a polyhedral Hilbert geometry coincides with its group of collineations (projectivities) if and only if the polyhedron is not an n-simplex with n>=2. Moreover, we determine the isometry group of the Hilbert geometry on the n-simplex, and find that it has the collineation group as an index-two subgroup. These resul...

There is a long-standing conjecture of Nussbaum which asserts that every finite set in Rn on which a cyclic group of sup-norm isometries acts transitively contains at most 2n points. The existing evidence supporting Nussbaum’s conjecture only uses abelian properties of the group. It has therefore been suggested that Nussbaum’s conjecture might hold...

In this paper we study one-complemented subspaces of Minkowski spaces. The main objective is to examine norms on Rn for which every one-complemented subspace has a block basis, i.e., a basis of vectors with mutually disjoint supports. We introduce a collection of norms on Rn and show that, for these norms, each one-complemented subspace has a block...

Several recent results concerning the dynamics of order preserving (sub)homogeneous maps on polyhedral cones are reviewed. These results were obtained by the author in collaboration with M. Akian, S. Gaubert and R. Nussbaum [Math. Proc. Camb. Philos. Soc. 140, No. 1, 157–176 (2006; Zbl 1101.37032)], M. Scheutzow [Ergodic Theory Dyn. Syst. 25, No. 3...

We investigate the iterative behaviour of continuous order preserving subhomogeneous maps $f: K\,{\rightarrow}\, K$, where $K$ is a polyhedral cone in a finite dimensional vector space. We show that each bounded orbit of $f$ converges to a periodic orbit and, moreover, the period of each periodic point of $f$ is bounded by \[ \beta_N = \max_{q+r+s=...

We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces $\ell_p(X)$, where $X$ is a Banach space with a 1-unconditional basis and $p \in (1,2)\cup (2,\infty)$. If the norm of $X$ is twice continuously differentiable and satisfies certain conditions connecting the norm and the notion of disjointness with resp...

We present several results for the periods of periodic points of sup-norm non-expansive maps. In particular, we show that the period of each periodic point of a sup-norm non-expansive map $f\colon M\to M$, where $M\subset \mathbb{R}^n$, is at most $\max_k\, 2^k \big(\begin{smallmatrix}n\\ k\end{smallmatrix}\big)$. This upper bound is smaller than 3...

Let ℝ+n be the standard closed positive cone in ℝn and let Γ(ℝ+n) be the set of integers p ≥ 1 for which there exists a continuous, order preserving, subhomogeneous map f: ℝ+n → ℝ +n, which has a periodic point with period p. It has been shown by Akian, Gaubert, Lemmens, and Nussbaum that Γ(ℝ +n) is contained in the set B(n) consisting of those p ≥...

This book contains a collection of survey papers by leading researchers in ergodic theory, low-dimensional and topological dynamics and it comprises nine chapters on a range of important topics. These include: the role and usefulness of ultrafilters in ergodic theory, topological dynamics and Ramsey theory; topological aspects of kneading theory to...

In this paper we will examine the asymptotic behaviour of the iterates of linear maps that are nonexpansive (contractive) with respect to a classical p-norm on . As a main result it will be shown that if 1⩽p⩽∞ and p≠2, there exists an integer q⩾1 such that the sequence (Akqx)k is convergent for each . Moreover the integer q is the order, or twice t...

In this paper the set of minimal periods of periodic points of 1-norm nonexpansive maps f:\mathbbRn®\mathbbRnf:\mathbb{R}^n\rightarrow\mathbb{R}^n is studied. This set is denoted by R(n). The main goal is to present a characterization of R(n) by arithmetical and combinatorial constraints. More precisely, it is shown that R(n)=Q¢(2n)R(n)=Q'(2n)...

In this paper several results concerning the periodic points of 1-norm non-expansive maps will be presented. In particular, we will examine the set $R(n)$, which consists of integers $p\geq 1$ such that there exist a 1-norm nonexpansive map $f{:}\ \mathbb{R}^n\rightarrow\mathbb{R}^n$ and a periodic point of $f$ of minimal period $p$. The principal...

If D is a subset of Rn and f: D → D is an l1-norm nonexpansive map, then it is known that every bounded orbit of f approaches a periodic orbit. Moreover, the minimal period of each periodic point of f is bounded by n! 2m, where m = 2n − 1. In this paper we shall describe two different procedures to construct periodic orbits of l1-norm nonexpansive...

For ℓ1-norm and sup-norm nonexpansive maps it is known that bounded orbits approach periodic orbits. Moreover the minimal period of a periodic point of such a map has an a priori upper bound that only depends on the dimension and the given norm. We shall show that the question whether for a given positive integer p there exists an ℓ1-norm (or sup-n...