A fixed point theorem in Euclidean buildings
The author considers non-expanding self-mappings of Euclidean buildings \(X\) that are the composition of a decent isometry \(\mathcal{F}\) of \(X\) with a weakly contracting map \(\Phi\) that moves every point a fixed distance towards a specified point at infinity. The decency condition, which is inspired by the related notion of decent isocrystals, means that \(\mathcal{F}\) is an isometry of \(X\) such that the set of points Min\((\mathcal{F})\) at which the minimal distance \(\inf\{\mathrm{dist}(x,\mathcal{F}x) : x \in X\}\) is attained is a nonempty locally compact subbuilding of \(X\) and satisfies a certain additional condition. The latter extra condition is automatically satisfied when the spherical building at infinity is thick. The main results are various conditions on a certain numerical value in \(\{-\infty\}\cup\mathbb{R}\), associated with \(\mathcal{F}\) and \(\Phi\) and defined via Busemann functions, that describes when the set of fixed points of \(\mathcal{F}\circ\Phi\) is nonempty or bounded or is a singleton. The overall strategy of the proof is similar to that of the main theorem of \textit{A. Balser} [Can. Math. Bull. 49, No. 3, 321--336 (2006; Zbl 1114.53030)]. Two special cases, when \(X\) is a Euclidean affine space and when \(X\) is a product of Euclidean buildings are dealt with first. The main part of the proof is the verification that if one has a non-positive value then \(\mathcal{F}\circ\Phi\) has bounded orbits. The author states that his study was motivated by a theorem of Laffaille in \(p\)-adic Hodge theory and that his result can be applied to generalize this theorem [the author, ``Mazur's inequality and Laffaille's theorem'', Preprint, \url{arXiv:1509.00574}].
scientific article; zbMATH DE number 6854776
scientific article