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 $\{1/\beta,\beta\}$ for some $\beta\geq 1$. 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.Comment: 30 page

On the dynamics of sup-norm non-expansive maps

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 3n and improves the previously known bounds. Further, we consider a special class of sup-norm non-expansive maps, namely topical functions. For topical functions $f\colon\mathbb{R}^n\to\mathbb{R}^n$ Gunawardena and Sparrow have conjectured that the optimal upper bound for the periods of periodic points is $\big(\begin{smallmatrix}n\\ \lfloor n/2\rfloor\end{smallmatrix}\big)$. We give a proof of this conjecture. To obtain the results we use combinatorial and geometric arguments. In particular, we analyse the cardinality of anti-chains in certain partially ordered sets

A Metric Version of Poincaré’s Theorem Concerning Biholomorphic Inequivalence of Domains

We show that if Yj⊂Cnj is a bounded strongly convex domain with C3-boundary for j=1,…,q, and Xj⊂Cmj is a bounded convex domain for j=1,…,p, then the product domain ∏pj=1Xj⊂Cm cannot be isometrically embedded into ∏qj=1Yj⊂Cn 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 Cn for n≥2. The method of proof only relies on the metric geometry of the spaces and will be derived from a more general 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

Midpoints for Thompson's metric on symmetric cones

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]

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 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

On the complexity of detecting 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