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On the geodesic flow on CAT(0) spaces

Published online by Cambridge University Press:  15 August 2019

CHARALAMPOS CHARITOS
Affiliation:
Mathematics Laboratory, Agricultural University of Athens, 11855Athens, Greece email bakis@aua.gr, papadoperakis@aua.gr, get@aegean.gr
IOANNIS PAPADOPERAKIS
Affiliation:
Mathematics Laboratory, Agricultural University of Athens, 11855Athens, Greece email bakis@aua.gr, papadoperakis@aua.gr, get@aegean.gr
GEORGIOS TSAPOGAS
Affiliation:
Mathematics Laboratory, Agricultural University of Athens, 11855Athens, Greece email bakis@aua.gr, papadoperakis@aua.gr, get@aegean.gr
Corresponding

Abstract

Under certain assumptions on CAT(0) spaces, we show that the geodesic flow is topologically mixing. In particular, the Bowen–Margulis’ measure finiteness assumption used by Ricks [Flat strips, Bowen–Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces. Ergod. Th. & Dynam. Sys. 37 (2017), 939–970] is removed. We also construct examples of CAT(0) spaces that do not admit finite Bowen–Margulis measure.

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Original Article
Copyright
© Cambridge University Press, 2019

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References

Anosov, D.V.. Geodesic Flows on Closed Riemannian Manifolds with Negative Curvature (Proceedings Stelkov Institute of Mathematics, 90). American Mathematical Society, Providence, RI, 1969.Google Scholar
Ballmann, W.. Lectures on Spaces of Non Positive Curvature. Birkhäuser, Basel, 1995.CrossRefGoogle Scholar
Bridson, M. and Haefliger, A.. Metric Spaces of Non-Positive Curvature (Grundlehren der Mathematischen Wissenschaften, 319). Springer, Berlin, 1999.CrossRefGoogle Scholar
Charitos, Ch. and Papadoperakis, I.. On the geometry of Hyperbolic surfaces with a conical singularity. Ann. Global Anal. Geom. 23(4) (2003), 323357.CrossRefGoogle Scholar
Charitos, C. and Tsapogas, G.. Topological mixing in CAT(-1)-spaces. Trans. Amer. Math. Soc. 178 (2001), 235264.CrossRefGoogle Scholar
Charitos, Ch., Papadoperakis, I. and Tsapogas, G.. The geometry of Euclidean surfaces with conical singularities. Math. Z. 284(3) (2016), 10731087.CrossRefGoogle Scholar
Coornaert, M.. Sur les groupes proprement discontinus d’isometries des espaces hyperboliques au sens de Gromov. Thèse de ULP, Publication de IRMA, Strasburg, 1990.Google Scholar
Coornaert, M., Delzant, T. and Papadopoulos, A.. Géometrie et Théorie des Groupes (Lecture Notes in Mathematics, 1441). Springer, Berlin, 1980.Google Scholar
Dal’bo, F., Peigni, M., Picaud, J.-C. and Sambusetti, A.. Convergence and counting in infinite measure. Ann. Inst. Fourier (Grenoble) 67(2) (2017), 483520.CrossRefGoogle Scholar
Eberlein, P.. Geodesic flows on negatively curved manifolds II. Trans. Amer. Math. Soc. 178 (1973), 5782.CrossRefGoogle Scholar
Eberlein, P. and O’Neill, B.. Visibility manifolds. Pacific J. Math. 46 (1973), 45109.CrossRefGoogle Scholar
Gromov, M.. Hyperbolic groups. Essays in Group Theory (MSRI Publications, 8). Springer, Berlin, 1987, pp. 75263.CrossRefGoogle Scholar
Paulin, F.. Constructions of hyperbolic groups via hyperbolization of polyhedra. Group Theory from a Geometrical Viewpoint, ICTP, Trieste, Italy, March 26–April 6, 1990. Eds. Ghys, E. and Haefliger, A.. World Scientific, Italy, 1991.Google Scholar
Ricks, R.. Flat strips, Bowen–Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces. Ergod. Th. & Dynam. Sys. 37 (2017), 939970.CrossRefGoogle Scholar
Troyanov, M.. Les surfaces euclidiennes a singularites coniques. Enseign. Math. (2) 32(1–2) (1986), 7994.Google Scholar

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