Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-29T02:22:17.340Z Has data issue: false hasContentIssue false
  • English
  • Français

Flat strips, Bowen–Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces

Published online by Cambridge University Press:  28 December 2015

RUSSELL RICKS
RUSSELL RICKS*
Affiliation:
University of Michigan, Ann Arbor, MI 48109, USA email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $X$ be a proper, geodesically complete CAT($0$) space under a proper, non-elementary, isometric action by a group $\unicode[STIX]{x1D6E4}$ with a rank one element. We construct a generalized Bowen–Margulis measure on the space of unit-speed parametrized geodesics of $X$ modulo the $\unicode[STIX]{x1D6E4}$-action. Although the construction of Bowen–Margulis measures for rank one non-positively curved manifolds and for CAT($-1$) spaces is well known, the construction for CAT($0$) spaces hinges on establishing a new structural result of independent interest: almost no geodesic, under the Bowen–Margulis measure, bounds a flat strip of any positive width. We also show that almost every point in $\unicode[STIX]{x2202}_{\infty }X$, under the Patterson–Sullivan measure, is isolated in the Tits metric. (For these results we assume the Bowen–Margulis measure is finite, as it is in the cocompact case.) Finally, we precisely characterize mixing when $X$ has full limit set: a finite Bowen–Margulis measure is not mixing under the geodesic flow precisely when $X$ is a tree with all edge lengths in $c\mathbb{Z}$ for some $c>0$. This characterization is new, even in the setting of CAT($-1$) spaces. More general (technical) versions of these results are also stated in the paper.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 3 , May 2017 , pp. 939 - 970
Copyright
© Cambridge University Press, 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adams, S. and Ballmann, W.. Amenable isometry groups of Hadamard spaces. Math. Ann. 312(1) (1998), 183–195.CrossRefGoogle Scholar
Babillot, M.. On the mixing property for hyperbolic systems. Israel J. Math. 129 (2002), 61–76.CrossRefGoogle Scholar
Ballmann, W.. Lectures on Spaces of Non-Positive Curvature (DMV Seminar, 25) . Birkhäuser, Basel, 1995, 112 pp; with an appendix by Misha Brin.CrossRefGoogle Scholar
Ballmann, W. and Buyalo, S.. Periodic rank one geodesics in Hadamard spaces. Geometric and Probabilistic Structures in Dynamics (Contemporary Mathematics, 469) . American Mathematical Society, Providence, RI, 2008, pp. 19–27.CrossRefGoogle Scholar
Bowen, R.. Periodic points and measures for Axiom A diffeomorphisms. Trans. Amer. Math. Soc. 154 (1971), 377–397.Google Scholar
Bowen, R.. Maximizing entropy for a hyperbolic flow. Math. Systems Theory 7(4) (1973), 300–303.CrossRefGoogle Scholar
Bridson, M. R. and Haefliger, A.. Metric Spaces of Non-Positive Curvature (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319) . Springer, Berlin, 1999, 643 pp.CrossRefGoogle Scholar
Burago, D., Burago, Y. and Ivanov, S.. A course in Metric Geometry (Graduate Studies in Mathematics, 33) . American Mathematical Society, Providence, RI, 2001, 415 pp.Google Scholar
Burns, K. and Spatzier, R.. Manifolds of non-positive curvature and their buildings. Publ. Math. Inst. Hautes Études Sci.(65) (1987), 35–59.CrossRefGoogle Scholar
Caprace, P.-E. and Sageev, M.. Rank rigidity for CAT(0) cube complexes. Geom. Funct. Anal. 21(4) (2011), 851–891.CrossRefGoogle Scholar
Chen, S. S. and Eberlein, P.. Isometry groups of simply connected manifolds of non-positive curvature. Illinois J. Math. 24(1) (1980), 73–103.CrossRefGoogle Scholar
Dal’bo, F.. Remarques sur le spectre des longueurs d’une surface et comptages. Bol. Soc. Brasil. Mat. (N.S.) 30(2) (1999), 199–221.CrossRefGoogle Scholar
Eberlein, P.. Geodesic flows on negatively curved manifolds. I. Ann. of Math. (2) 95 (1972), 492–510.CrossRefGoogle Scholar
Eberlein, P.. Geodesic flows on negatively curved manifolds. II. Trans. Amer. Math. Soc. 178 (1973), 57–82.CrossRefGoogle Scholar
Guralnik, D. P. and Swenson, E. L.. A ‘transversal’ for minimal invariant sets in the boundary of a CAT(0) group. Trans. Amer. Math. Soc. 365(6) (2013), 3069–3095.CrossRefGoogle Scholar
Hamenstädt, U.. Cocycles, Hausdorff measures and cross ratios. Ergod. Th. & Dynam. Sys. 17(5) (1997), 1061–1081.CrossRefGoogle Scholar
Hamenstädt, U.. Rank-one isometries of proper CAT(0)-spaces. Discrete Groups and Geometric Structures (Contemporary Mathematics, 501) . American Mathematical Society, Providence, RI, 2009, pp. 43–59.CrossRefGoogle Scholar
Hopf, E.. Ergodic theory and the geodesic flow on surfaces of constant negative curvature. Bull. Amer. Math. Soc. (N.S.) 77 (1971), 863–877.CrossRefGoogle Scholar
Kaimanovich, V. A.. Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds. Ann. Inst. H. Poincaré Phys. Théor. 53(4) (1990), 361–393; hyperbolic behaviour of dynamical systems (Paris, 1990).Google Scholar
Kim, I.. Marked length rigidity of rank one symmetric spaces and their product. Topology 40(6) (2001), 1295–1323.CrossRefGoogle Scholar
Knieper, G.. On the asymptotic geometry of non-positively curved manifolds. Geom. Funct. Anal. 7(4) (1997), 755–782.CrossRefGoogle Scholar
Lytchak, A.. Rigidity of spherical buildings and joins. Geom. Funct. Anal. 15(3) (2005), 720–752.CrossRefGoogle Scholar
Margulis, G. A.. On Some Aspects of the Theory of Anosov Systems (Springer Monographs in Mathematics) . Springer, Berlin, 2004, 139 pp; with a survey by Richard Sharp: Periodic orbits of hyperbolic flows, Translated from the Russian by Valentina Vladimirovna Szulikowska.CrossRefGoogle Scholar
Ontaneda, P.. Some remarks on the geodesic completeness of compact non-positively curved spaces. Geom. Dedicata 104 (2004), 25–35.CrossRefGoogle Scholar
Otal, J.-P.. Sur la géometrie symplectique de l’espace des géodésiques d’une variété à courbure négative. Rev. Mat. Iberoam. 8(3) (1992), 441–456.CrossRefGoogle Scholar
Patterson, S. J.. The limit set of a Fuchsian group. Acta Math. 136(3–4) (1976), 241–273.CrossRefGoogle Scholar
Ricks, R.. Flat strips, Bowen–Margulis measures, and mixing of the geodesic flow for rank one $\text{CAT}(0)$ spaces. PhD Thesis, University of Michigan, 2015.CrossRefGoogle Scholar
Roblin, T.. Ergodicité et équidistribution en courbure négative. Mém. Soc. Math. Fr. (N.S.)(95) (2003), vi+96.Google Scholar
Sullivan, D.. The density at infinity of a discrete group of hyperbolic motions. Publ. Math. Inst. Hautes Études Sci.(50) (1979), 171–202.CrossRefGoogle Scholar
Sullivan, D.. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153(3–4) (1984), 259–277.CrossRefGoogle Scholar