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Fixed point theorems for metric spaces with a conical geodesic bicombing

Published online by Cambridge University Press:  14 February 2017

GIULIANO BASSO
GIULIANO BASSO*
Affiliation:
Mathematik Departement, ETH Zürich, Rämistrasse 101, 8092 Zürich, Schweiz email [email protected]
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Abstract

We derive two fixed point theorems for a class of metric spaces that includes all Banach spaces and all complete Busemann spaces. We obtain our results by the use of a $1$-Lipschitz barycenter construction and an existence result for invariant Radon probability measures. Furthermore, we construct a bounded complete Busemann space that admits an isometry without fixed points.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 5 , August 2018 , pp. 1642 - 1657
Copyright
© Cambridge University Press, 2017 

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References

Adamski, W.. On the existence of invariant probability measures. Israel J. Math. 65(1) (1989), 79–95.Google Scholar
Bader, U., Gelander, T. and Monod, N.. A fixed point theorem for L 1 spaces. Invent. Math. 189(1) (2012), 143–148.CrossRefGoogle Scholar
Bridson, M. R. and Haefliger, A.. Metric Spaces of Non-positive Curvature. Vol. 319. Springer, Berlin, 1999.Google Scholar
Descombes, D.. Asymptotic rank of spaces with bicombings. Math. Z. (2016), 1–14. doi:10.1007/s00209-016-1680-3.Google Scholar
Descombes, D. and Lang, U.. Convex geodesic bicombings and hyperbolicity. Geom. Dedicata 177(1) (2015), 367–384.Google Scholar
Edelstein, M.. On non-expansive mappings of Banach spaces. Math. Proc. Cambridge Philos. Soc. 60(7) (1964), 439–447.Google Scholar
Edwards, D. A.. On the Kantorovich–Rubinstein theorem. Exp. Math. 29(4) (2011), 387–398.CrossRefGoogle Scholar
Epstein, D. B. A., Paterson, M. S., Cannon, J. W., Holt, D. F., Levy, S. V. and Thurston, W. P.. Word Processing in Groups. A. K. Peters, Ltd, Natick, MA, 1992.Google Scholar
Es-Sahib, A. and Heinich, H.. Barycentre canonique pour un espace métrique à courbure négative. Séminaire de Probabilités XXXIII. Springer, Berlin, 1999, pp. 355–370.Google Scholar
Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, NJ, 2014.Google Scholar
Itoh, S.. Some fixed point theorems in metric spaces. Fund. Math. 102(2) (1979), 109–117.Google Scholar
Johnson, W. B. and Lindenstrauss, J.. Handbook of the Geometry of Banach Spaces. Vol. 1. Elsevier Science, Amsterdam, 2001.Google Scholar
Lang, U.. Injective hulls of certain discrete metric spaces and groups. J. Topol. Anal. 5(03) (2013), 297–331.Google Scholar
Leuştean, L.. Nonexpansive iterations in uniformly convex W-hyperbolic spaces. Nonlinear Anal. 513 (2010), 193–209.Google Scholar
Mankiewicz, P.. Extension of isometries in normed linear spaces. Bull. Acad. Polon. Sci. 20(5) (1972), 367–370.Google Scholar
Molchanov, I.. Theory of Random Sets. Springer, London, 2006.Google Scholar
Navas, A.. An L 1 ergodic theorem with values in a non-positively curved space via a canonical barycenter map. Ergod. Th. & Dynam. Sys. 33(4) (2013), 609–623.Google Scholar
Oxtoby, J. C. and Ulam, S.. On the existence of a measure invariant under a transformation. Ann. of Math. (2) 40(3) (1939), 560–566.CrossRefGoogle Scholar
Papadopoulos, A.. Metric Spaces, Convexity and Nonpositive Curvature, 2nd edn. European Mathematical Society, Zurich, 2014.Google Scholar
Ryll-Nardzewski, C.. On fixed points of semigroups of endomorphisms of linear spaces. Proc. Fifth Berkeley Symp. on Mathematical Statistics and Probability. Vol. 2. University of California Press, Berkeley, CA, 1967.Google Scholar
Sturm, K.-T.. Probability measures on metric spaces of nonpositive curvature. Contemp. Math. 338 (2003), 357–390.Google Scholar
Sucheston, L.. On existence of finite invariant measures. Math. Z. 86(4) (1964), 327–336.Google Scholar
Topsøe, F.. Compactness in spaces of measures. Studia Math. 3(36) (1970), 195–212.Google Scholar
Villani, C.. Topics in Optimal Transportation. American Mathematical Society, Providence, RI, 2003.Google Scholar
Villani, C.. Optimal Transport: Old and New. Springer, Berlin, 2009.Google Scholar