Skip to main content Accessibility help
×
Home

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

  • RUSSELL RICKS (a1)

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.

Copyright

References

Hide All
[1] Adams, S. and Ballmann, W.. Amenable isometry groups of Hadamard spaces. Math. Ann. 312(1) (1998), 183195.
[2] Babillot, M.. On the mixing property for hyperbolic systems. Israel J. Math. 129 (2002), 6176.
[3] Ballmann, W.. Lectures on Spaces of Non-Positive Curvature (DMV Seminar, 25) . Birkhäuser, Basel, 1995, 112 pp; with an appendix by Misha Brin.
[4] 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. 1927.
[5] Bowen, R.. Periodic points and measures for Axiom A diffeomorphisms. Trans. Amer. Math. Soc. 154 (1971), 377397.
[6] Bowen, R.. Maximizing entropy for a hyperbolic flow. Math. Systems Theory 7(4) (1973), 300303.
[7] 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.
[8] 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.
[9] Burns, K. and Spatzier, R.. Manifolds of non-positive curvature and their buildings. Publ. Math. Inst. Hautes Études Sci.(65) (1987), 3559.
[10] Caprace, P.-E. and Sageev, M.. Rank rigidity for CAT(0) cube complexes. Geom. Funct. Anal. 21(4) (2011), 851891.
[11] Chen, S. S. and Eberlein, P.. Isometry groups of simply connected manifolds of non-positive curvature. Illinois J. Math. 24(1) (1980), 73103.
[12] Dal’bo, F.. Remarques sur le spectre des longueurs d’une surface et comptages. Bol. Soc. Brasil. Mat. (N.S.) 30(2) (1999), 199221.
[13] Eberlein, P.. Geodesic flows on negatively curved manifolds. I. Ann. of Math. (2) 95 (1972), 492510.
[14] Eberlein, P.. Geodesic flows on negatively curved manifolds. II. Trans. Amer. Math. Soc. 178 (1973), 5782.
[15] 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), 30693095.
[16] Hamenstädt, U.. Cocycles, Hausdorff measures and cross ratios. Ergod. Th. & Dynam. Sys. 17(5) (1997), 10611081.
[17] 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. 4359.
[18] Hopf, E.. Ergodic theory and the geodesic flow on surfaces of constant negative curvature. Bull. Amer. Math. Soc. (N.S.) 77 (1971), 863877.
[19] 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), 361393; hyperbolic behaviour of dynamical systems (Paris, 1990).
[20] Kim, I.. Marked length rigidity of rank one symmetric spaces and their product. Topology 40(6) (2001), 12951323.
[21] Knieper, G.. On the asymptotic geometry of non-positively curved manifolds. Geom. Funct. Anal. 7(4) (1997), 755782.
[22] Lytchak, A.. Rigidity of spherical buildings and joins. Geom. Funct. Anal. 15(3) (2005), 720752.
[23] 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.
[24] Ontaneda, P.. Some remarks on the geodesic completeness of compact non-positively curved spaces. Geom. Dedicata 104 (2004), 2535.
[25] Otal, J.-P.. Sur la géometrie symplectique de l’espace des géodésiques d’une variété à courbure négative. Rev. Mat. Iberoam. 8(3) (1992), 441456.
[26] Patterson, S. J.. The limit set of a Fuchsian group. Acta Math. 136(3–4) (1976), 241273.
[27] 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.
[28] Roblin, T.. Ergodicité et équidistribution en courbure négative. Mém. Soc. Math. Fr. (N.S.)(95) (2003), vi+96.
[29] Sullivan, D.. The density at infinity of a discrete group of hyperbolic motions. Publ. Math. Inst. Hautes Études Sci.(50) (1979), 171202.
[30] Sullivan, D.. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153(3–4) (1984), 259277.

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed