Dr Bas Lemmens

Article

• Lemmens, B., Roelands, M. and Wortel, M. (2019). Hilbert and Thompson isometries on cones in JB-algebras. Mathematische Zeitschrift [Online] 292:1511-1547. Available at: https://doi.org/10.1007/s00209-018-2144-8.
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  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 Bosche on the Hilbert and Thompson isometries on symmetric cones, and work by Hatori and Molnar 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.
• Kalauch, A., Lemmens, B. and van Gaans, O. (2019). Inverses of disjointness preserving operators in finite dimensional pre-Riesz spaces. Quaestiones Mathematicae [Online] 42:423-430. Available at: https://doi.org/10.2989/16073606.2018.1451405.
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• Lemmens, B., Roelands, M. and Wortel, M. (2018). Isometries of infinite dimensional Hilbert geometries. Journal of Topology and Analysis [Online] 10:941-959. Available at: http://dx.doi.org/10.1142/S1793525318500255.
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  In this paper we extend two classical results 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.
• Lemmens, B., van Gaans, O. and Van Imhoff, H. (2018). Monotone dynamical systems with dense periodic points. Journal of Differential Equations [Online] 265:5709-5715. Available at: http://doi.org/10.1016/j.jde.2018.07.012.
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  n this paper we prove a recent conjecture by M. Hirsch, which says that if is a discrete time monotone dynamical system, with a homeomorphism on an open connected subset of a finite dimensional vector space, and the periodic points of f are dense in ?, then f is periodic.
• Lemmens, B. and van Gaans, O. (2018). Order structures, Jordan algebras, and geometry. Nieuw Archief voor Wiskunde [Online] June:111-114. Available at: http://www.nieuwarchief.nl/serie5/toonnummer.php?deel=19&nummer=2&taal=0.
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• Lemmens, B. and White, L. (2018). On the complexity of detecting eigenvectors of nonlinear cone maps. Involve: A Journal of Mathematics [Online] 12:141-150. Available at: https://doi.org/10.2140/involve.2019.12.141.
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• Lemmens, B., Lins, B. and Nussbaum, R. (2018). Detecting fixed points of nonexpansive maps by illuminating the unit ball. Israel Journal of Mathematics [Online] 224:231-262. Available at: https://doi.org/10.1007/s11856-018-1641-0.
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  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?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 ,…,w m in V such that {f(w i )?w i :i=1,…,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.
• Lemmens, B., Lins, B., Nussbaum, R. and Wortel, M. (2018). Denjoy-Wolff theorems for Hilbert’s and Thompson’s metric spaces. Journal d’Analyse Mathematique [Online] 134:671-718. Available at: https://doi.org/10.1007/s11854-018-0022-2.
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  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.
• Lemmens, B., Roelands, M. and Van Imhoff, H. (2017). An order theoretic characterization of spin factors. The Quarterly Journal of Mathematics [Online] 68:1001-1017. Available at: https://doi.org/10.1093/qmath/hax010.
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  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 \(g : C° → C°\) with the property that \(g\) is antihomogeneous, that is, \(g (\lambda x) = \lambda^{-1}g(x)\) for all \(\lambda > 0\) and \( x \in C°\), and \(g\) is an order-antimorphism, that is, \(x ≤ c\space y\) if and only if \(g(y) ≤ g (x)\). 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 \(dim V ≥ 3\), then there exists a bijective antihomogeneous order-antimorphism \(g : C° → C°\) if and only if \((V, C, u)\) is a spin factor.
• Lemmens, B. and Roelands, M. (2016). Midpoints for Thompson’s metric on symmetric cones. Osaka Journal of Mathematics [Online] 54:197-208. Available at: http://projecteuclid.org/euclid.ojm/1488531790.
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  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 simple. In particular, we find for \(A\) and \(B\) in the cone positive definite Hermitian matrices that \(dim(aff M(A,B)) = q^2\), where \(q\) is the number of eigenvalues \(\mu\) of \(A^{-1}B\), counting multiplicities, such that \(\mu ≠ max\{\lambda_+(A^{-1}B),\lambda_-(A^{-1}B)^{-1}\},\) where \(\lambda_+(A^{-1}B) := max\{\lambda:\lambda \in \sigma(A^{-1}B)\}\) and \(\lambda_-(A^{-1}B) := min\{\lambda:\lambda \in \sigma(A^{-1}B)\}\). These results extend work by Y. Lim [18].
• Lemmens, B., Kalauch, A. and van Gaans, O. (2015). Bands in partially ordered vector spaces with order unit. Positivity [Online] 9:489-511. Available at: http://link.springer.com/article/10.1007%2Fs11117-014-0311-7.
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  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(?), where ? is a compact Hausdorff space. We characterize bands in X, and their disjoint complements, in terms of subsets of ?. We also analyze two methods to extend bands in X to C(?) 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 (1/4)2^(2^n) for n?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?4.
• Lemmens, B. and Parsons, C. (2015). On the number of pairwise touching simplices. Involve: A Journal of Mathematics [Online] 8:513-520. Available at: http://dx.doi.org/10.2140/involve.2015.8.513.
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  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.
• Lemmens, B. and Roelands, M. (2015). Unique geodesics for Thompson’s metric. Annales de l’Institut Fourier (Grenoble) [Online] 65:315-348. Available at: https://doi.org/10.5802/aif.2932.
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  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 \(\{1/\beta,\beta\}\) for some \(\beta ≥ 1\). A similar result will be established for symmetric cones. Secondly, it will be shown that if \(C^°\) is the interior of a finite-dimensional closed cone \(C\), then the Thompson's metric space \((C^°,d_C)\) can be quasi-isometrically embedded into a finite-dimensional normed space if, and only if, \(C\) is a polyhedral cone. Moreover, \((C^°,d_C)\) is isometric to a finite-dimensional normed space if, and only if, \(C\) is a simplicial cone. It will also be shown that if \(C^°\) is the interior of a strictly convex cone \(C\) with \(3 ≤ dim \space C ≤ \infty\), then every Thompson's metric isometry is projectively linear.
• Lemmens, B., van Gaans, O. and Kalauch, A. (2014). Riesz completions, functional representations and anti-lattices. Positivity [Online] 18:201-218. Available at: http://www.springerlink.com/content/102984/.
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  We show that the Riesz completion of an Archimedean partially or- dered vector space X with unit can be represented as a norm dense Riesz subspace of the smallest functional representation of X. This yields a con- venient way to compute the Riesz completion. To illustrate the method, the Riesz completions of spaces ordered by Lorentz cones, cones of sym- metric positive semi-definite matrices, and polyhedral cones are deter- mined. We use the representation to investigate the existence of non- trivial disjoint elements and link the absence of such elements to the no- tion of anti-lattice. One of the results is a geometric condition on the dual cone of a finite dimensional partially ordered vector space X that ensures that X is an anti-lattice.
• Lemmens, B. and Nussbaum, R. (2013). Continuity of the cone spectral radius. Proceedings of the American Mathematical Society [Online] 141:2741-2754. Available at: http://dx.doi.org/10.1090/S0002-9939-2013-11520-0.
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  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 radius is continuous. An example is presented showing that continuity may fail, if this condition does not hold. We also analyze the cone spectrum of continuous order-preserving homogeneous maps on finite dimensional closed cones. In particular, we prove that for each polyhedral cone with m faces, the cone spectrum contains at most m-1 elements, and this upper bound is sharp for each polyhedral cone. Moreover, for each non-polyhedral cone, there exist maps whose cone spectrum contains a countably infinite number of distinct points.
• Lemmens, B. (2011). Non-expansive mappings on Hilbert’s metric spaces. Topological Methods in Nonlinear Analysis [Online] 38:45-58. Available at: http://www.tmna.ncu.pl/htmls/archives/vol-38-1.html.
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  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, then the optimal upper bound for the periods of periodic points of nonexpansive mappings on (X, d(X)) is 6. The results have applications in the analysis of nonlinear mappings on cones, and extend work by Nussbaum and others.
• Lemmens, B., Akian, M. and Gaubert, S. (2011). Stability and convergence in discrete convex monotone dynamical systems. Journal of Fixed Point Theory and Applications [Online] 9:295-325. Available at: http://dx.doi.org/10.1007/s11784-011-0052-1.
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  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 isomorphic to a convex inf-semilattice, and a criterion is given for the existence of a unique tangentially stable fixed point. We also show that periods of tangentially stable periodic points are orders of permutations on n letters, where n is the dimension of the underlying space, and a sufficient condition for global convergence to periodic orbits is presented.
• Lemmens, B. and Walsh, C. (2011). Isometries of polyhedral Hilbert geometries. Topology and Analysis [Online] 3:213-241. Available at: http://dx.doi.org/10.1142/S1793525311000520.
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  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 for all n ? 2, and find that it has the collineation group as an index-two subgroup. The results confirm several conjectures of P. de la Harpe for the class of polyhedral Hilbert geometries.
• Lemmens, B. and van Gaans, O. (2009). Dynamics of non-expansive maps on strictly convex normed spaces. Israel Journal of Mathematics [Online] 171:425-445. Available at: http://dx.doi.org/10.1007/s11856-009-0057-2.
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  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 = (a"e (n) , ayen center dot ayen) 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 extra assumptions on the derivatives of the norm, we also show that the period of each periodic point of a non-expansive map f: X -> X is the order, or, twice the order of a permutation on n letters. This last result generalizes a theorem of Sine, who proved it for a"" (p) (n) where 1 < p < a and p not equal 2. To obtain the results we analyze the ranges of non-expansive projections, the geometry of 1-complemented subspaces, and linear isometries on 1-complemented subspaces.
• Lemmens, B., Scheutzow, M. and Sparrow, C. (2007). Transitive actions of finite abelian groups of sup-norm isometries. European Journal of Combinatorics [Online] 28:1163-1179. Available at: http://dx.doi.org/10.1016/j.ejc.2006.02.003.
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  There is a long-standing conjecture of Nussbaum which asserts that every finite set in R-n on which a cyclic group of sup-norm isometries acts transitively contains at most 2(n) 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 more generally for abelian groups of sup-norm isometries. This paper provides evidence supporting this stronger conjecture. Among other results, we show that it, Gamma is an abelian group of sup-norm isometrics that acts transitively on a finite set X in R-n and Gamma contains no anticlockwise additive chains, then X has at most 2(n) points.
• Lemmens, B. (2007). Variations of a combinatorial problem on finite sets. Elemente der Mathematik [Online] 62:59-67. Available at: http://dx.doi.org/10.4171/EM/55.
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• Lemmens, B., van Gaans, O. and Randrianantoanina, B. (2007). Second derivatives of norms and contractive complementation in vector valued spaces. Studia Mathematica [Online] 179:149-166. Available at: http://dx.doi.org/10.4064/sm179-2-3.
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  We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces l(p)(X), where X is a Banach space with a 1-unconditional basis and p is an element of (1,2) boolean OR (2, infinity). If the norm of X is twice continuously differentiable and satisfies certain conditions connecting the norm and the notion of disjointness with respect to the basis, then we prove that every 1-complemented subspace of l(p)(X) admits a basis of mutually disjoint elements. Moreover, we show that every contractive projection is then an averaging operator. We apply our results to the space l(p)(l(q)) with p,q is an element of (1,2) boolean OR (2, infinity) and obtain a complete characterization of its 1-complemented subspaces.
• Lemmens, B., Akian, M., Gaubert, S. and Nussbaum, R. (2006). Iteration of order preserving subhomogeneous maps on a cone. Mathematical Proceedings of the Cambridge Philosophical Society [Online] 140:157-176. Available at: http://dx.doi.org.chain.kent.ac.uk/10.1017/S0305004105008832.
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  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 \[ \beta_N = \max_{q+r+s=N}\frac{N!}{q!r!s!}= \frac{N!}{\big\lfloor\frac{N}{3}\big\rfloor!\big\lfloor\frac{N\,{+}\,1}{3}\big\rfloor! \big\lfloor\frac{N\,{+}\,2}{3}\big\rfloor!}\sim \frac{3^{N+1}\sqrt{3}}{2\pi N}, \] where $N$ is the number of facets of the polyhedral cone. By constructing examples on the standard positive cone in $\mathbb{R}^n$, we show that the upper bound is asymptotically sharp.
• Lemmens, B. and Sparrow, C. (2006). A note on periodic points of order preserving subhomogeneous maps. Proc. Amer. Math. Soc. [Online] 134:513-517. Available at: http://www.ams.org.chain.kent.ac.uk/journals/proc/2006-134-05/S0002-9939-05-08390-5/home.html.
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• Lemmens, B. and van Gaans, O. (2006). On one-complemented subspaces of Minkowski spaces with smooth Riesz norms. Rocky Mountain Journal of Mathematics [Online] 6:1937-1955. Available at: http://projecteuclid.org.chain.kent.ac.uk/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.rmjm/1181069354.
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• Lemmens, B. and Scheutzow, M. (2005). On the dynamics of sup-norm nonexpansive maps. Ergodic Theory Dynam. Systems [Online] 25:861-871. Available at: http://journals.cambridge.org.chain.kent.ac.uk/action/displayAbstract?fromPage=online&aid=303934.
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• Lemmens, B. (2003). Periods of periodic points of 1-norm nonexpansive maps. Mathematical Proceedings of the Cambridge Philosophical Society [Online] 135:165-180. Available at: http://dx.doi.org/10.1017/S0305004103006741.
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  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 problem is to find a characterization of the set $R(n)$ in terms of arithmetical and combinatorial constraints. This problem was posed in [12, section 4]. We shall present here a significant step towards such a characterization. In fact, we shall introduce for each $n\in\mathbb{N}$ a set $T(n)$ that is determined by arithmetical and combinatorial constraints only, and prove that $R(n)\subset T(n)$ for all $n\in \mathbb{N}$. Moreover, we will see that $R(n)=T(n)$ for $n=1,2,3,4,6,7$, and 10, but it remains an open problem whether the sets $R(n)$ and $T(n)$ are equal for all $n\in \mathbb{N}$.
• Lemmens, B. and van Gaans, O. (2003). Iteration of linear p-norm nonexpansive maps. Linear Algebra and its Applications [Online] 371:265-276. Available at: http://www.sciencedirect.com.chain.kent.ac.uk/science/article/pii/S0024379503004543.
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• Lemmens, B. and Scheutzow, M. (2003). A characterization of the periods of periodic points of 1-norm nonexpansive maps. Selecta Mathematica - New Series [Online] 9:557-578. Available at: http://www.springerlink.com.chain.kent.ac.uk/content/q7gt8fjj1e5f63n9/.
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• Lemmens, B. and van Gaans, O. (2003). Periods of order-preserving nonexpansive maps on strictly convex normed spaces. Journal of Nonlinear Convex Analysis [Online] 4:356-363. Available at: http://www.ybook.co.jp/jnca.html.
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• Lemmens, B., Nussbaum, R. and Verduyn Lunel, S. (2001). Lower and upper bounds for omega-limit sets of nonexpansive maps. Indagationes Mathematicae (New Series) [Online] 12:191-211. Available at: http://www.sciencedirect.com.chain.kent.ac.uk/science/article/pii/S0019357701800252.
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• Lemmens, B. (1999). Integral rigid sets and periods of nonexpansive maps. Indagationes Mathematicae (New Series) [Online] 10:437-447. Available at: http://www.sciencedirect.com.chain.kent.ac.uk/science/article/pii/S0019357799800342.
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Book

• Lemmens, B. and Nussbaum, R. (2012). Nonlinear Perron-Frobenius Theory. [Online]. Vol. 189. Cambridge University Press. Available at: http://www.cambridge.org/gb/knowledge/isbn/item6510133/Nonlinear%20Perron–Frobenius%20Theory/?site_locale=en_GB.
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  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. This is the first comprehensive and unified introduction to nonlinear Perron–Frobenius theory suitable for graduate students and researchers entering the field for the first time. It acquaints the reader with recent developments and provides a guide to challenging open problems. To enhance accessibility, the focus is on finite dimensional nonlinear Perron–Frobenius theory, but pointers are provided to infinite dimensional results. Prerequisites are little more than basic real analysis and topology.

Book section

• Lemmens, B. and Nussbaum, R. (2014). Birkhoff’s version of Hilbert’s metric and applications. In: Papadopoulos, A. and Troyanov, M. eds. Handbook of Hilbert Geometry. European Math. Soc., pp. 275-303. Available at: http://www.ems-ph.org/books/book_articles.php?proj_nr=184.
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  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. A. Papadopoulos and M. Troyanov, European Mathematical Society Publishing House, Z\"urich.
• Lemmens, B. (2003). Periodic points of nonexpansive maps: a survey. In: Kolyada, S. and Bezuglyi, S. eds. Topics in Dynamics and Ergodic Theory. Cambridge, UK: Cambridge University Press, pp. 125-144. Available at: https://doi.org/10.1017/CBO9780511546716.009.
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Conference or workshop item

• Lemmens, B. (2006). Nonlinear Perron-Frobenius theory and dynamics of cone maps. In: Springer Berlin Heidelberg, pp. 399-406. Available at: http://www.springer.com.chain.kent.ac.uk/series/642.
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Edited journal

• Farenick, D., Lemmens, B., Van Barel, M. and Vandebril, R. eds. (2018). Proceedings of the 20th Conference of the International Linear Algebra Society in Leuven 2016. Linear Algebra and its Applications [Online] 542:1-3. Available at: https://doi.org/10.1016/j.laa.2017.09.015.
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Forthcoming

• Lemmens, B., Van Imhoff, H. and van Gaans, O. (2019). On the linearity of order-isomorphisms. Canadian journal of mathematics.
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  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.