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The conformal measures of a normal subgroup of a cocompact Fuchsian group

Part of: Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  28 September 2020

OFER SHWARTZ
OFER SHWARTZ*
Affiliation:
Technion – Israel Institute of Technology, Haifa, Israel (e-mail: ofer12@yahoo.com)
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Abstract

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In this paper we study the conformal measures of a normal subgroup of a cocompact Fuchsian group. In particular, we relate the extremal conformal measures to the eigenmeasures of a suitable Ruelle operator. Using Ancona’s theorem, adapted to the Ruelle operator setting, we show that if the group of deck transformations G is hyperbolic then the extremal conformal measures and the hyperbolic boundary of G coincide. We then interpret these results in terms of the asymptotic behavior of cutting sequences of geodesics on a regular cover of a compact hyperbolic surface.

Keywords

conformal measuresregular coversFuchsian groupsRuelle operatorMartin boundary

MSC classification

Primary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 9 , September 2021 , pp. 2845 - 2878
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

References

Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50). American Mathematical Society, Providence, RI, 1997.CrossRefGoogle Scholar
Adler, R. and Flartto, L.. Geodesic flows, interval maps, and symbolic dynamics. Bull. Amer. Math. Soc. (N.S.) 25(2) (1991), 229–334.CrossRefGoogle Scholar
Ancona, A.. Negatively curved manifolds, elliptic operators, and the Martin boundary. Ann. Math. 125(3) (1987), 495–536.CrossRefGoogle Scholar
Ancona, A.. Positive harmonic functions and hyperbolicity. Potential Theory Surveys and Problems. Springer, New York, NY, 1988, pp. 1–23 Google Scholar
Babillot, M.. On the classification of invariant measures for horosphere foliations on nilpotent covers of negatively curved manifolds. Random Walks and Geometry. Walter de Gruyter, Berlin, 2004, pp. 319–335.Google Scholar
Bedford, T., Keane, M. and Series, C.. Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces. Oxford University Press, Oxford, UK, 1991.Google Scholar
Birman, J. S. and Series, C.. Dehn’s algorithm revisited, with applications to simple curves on surfaces. Combinatorial Group Theory and Topology (Utah, 1984). Ed. Gersten, D. M. and Stallings, J. R.. Princeton University Press, Princeton, NJ, 1987, pp. 451–478.CrossRefGoogle Scholar
Bispo, S. R. P. and Stadlbauer, M.. The Martin boundary of an extension by a hyperbolic group. Preprint, 2020, arXiv:2005.03723. Google Scholar
Blachère, S., Haïssinsky, P. and Mathieu, P.. Harmonic measures versus quasiconformal measures for hyperbolic groups. Ann. Sci. Éc. Norm. Supér. (4) 44(4) (2011), 683–721.CrossRefGoogle Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, New York, NY, 1975.CrossRefGoogle Scholar
Bowen, R. and Series, C.. Markov maps associated with Fuchsian groups. Publ. Math. Inst. Hautes Études Sci. 50(1) (1979), 153–170.CrossRefGoogle Scholar
Burger, M.. Horocycle flow on geometrically finite surfaces. Duke Math. J. 61(3) (1990), 779–803.CrossRefGoogle Scholar
Constantine, D., Lafont, J. F. and Thompson, D. J.. Strong symbolic dynamics for geodesic flow on CAT $(-1)$ spaces and other metric Anosov flows. J. Éc. Polytech. Math. 7 (2020), 201–231.CrossRefGoogle Scholar
Coornaert, M.. Mesures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens de Gromov. Pacific J. Math. 159(2) (1993), 241–270.CrossRefGoogle Scholar
Coulon, R., Dal’Bo, F., and Sambusetti, A.. Growth gap in hyperbolic groups and amenability. Geom. Funct. Anal. 28(5) (2018), 1260–1320.CrossRefGoogle Scholar
Dani, S. G.. Invariant measures of horospherical flows on noncompact homogeneous spaces. Invent. Math. 47(2) (1978), 101–138.CrossRefGoogle Scholar
de La Harpe, P.. Topics in Geometric Group Theory. University of Chicago Press, Chicago, IL, 2000.Google Scholar
Denker, M. and Urbański, M.. On the existence of conformal measures. Trans. Amer. Math. Soc. 328(2) (1991), 563–587.CrossRefGoogle Scholar
Dougall, R. and Sharp, R.. Amenability, critical exponents of subgroups and growth of closed geodesics. Math. Ann. 365(3-4) (2016), 1359–1377.CrossRefGoogle Scholar
Eberlein, P.. Geodesic flow in certain manifolds without conjugate points. Trans. Amer. Math. Soc. 167 (1972), 151–170.CrossRefGoogle Scholar
Falk, K., Matsuzaki, K. and Stratmann, B.. Checking atomicity of conformal ending measures for Kleinian groups. Conform. Geom. Dyn. 14(8) (2010), 167–183.CrossRefGoogle Scholar
Furstenberg, H.. The unique ergodigity of the horocycle flow. Recent Advances in Topological Dynamics. Springer, New York, NY, 1973, pp. 95–115.CrossRefGoogle Scholar
Ghys, É. and de La Harpe, P.. Sur les groupes hyperboliques d’apres Mikhael Gromov (Progress in Mathematics, 83). Birkhäuser Boston, Inc., Boston, 1990.CrossRefGoogle Scholar
Gouëzel, S.. Local limit theorem for symmetric random walks in Gromov-hyperbolic groups. J. Amer. Math. Soc. 27(3) (2014), 893–928.CrossRefGoogle Scholar
Gouëzel, S. and Lalley, S. P.. Random walks on co-compact Fuchsian groups. Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), 131–175.CrossRefGoogle Scholar
Jaerisch, J.. Group-extended Markov systems, amenability, and the Perron-Frobenius operator. Proc. Amer. Math. Soc. 143(1) (2015), 289–300.CrossRefGoogle Scholar
Kaimanovich, V. A.. Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces. J. Reine Angew. Math. 455 (1994), 57–104.Google Scholar
Kaimanovich, V. A.. Ergodic properties of the horocycle flow and classification of Fuchsian groups. J. Dyn. Control Syst. 6(1) (2000), 21–56.CrossRefGoogle Scholar
Karlsson, A. and Ledrappier, F.. On laws of large numbers for random walks. Ann. Probab. 34(5) (2006), 1693–1706.CrossRefGoogle Scholar
Karlsson, A. and Margulis, G. A.. A multiplicative ergodic theorem and nonpositively curved spaces. Commun. Math. Phys. 208(1) (1999), 107–123.CrossRefGoogle Scholar
Karpelevič, F. I.. The geometry of geodesics and the eigenfunctions of the Beltrami-Laplace operator on symmetric space. Trans. Moscow Math. Soc. (1965), 51–199.Google Scholar
Landesberg, O. and Lindenstrauss, E.. On Radon measures invariant under horospherical flows on geometrically infinite quotients. Preprint, 2019, arXiv:1910.08956.Google Scholar
Ledrappier, F. and Sarig, O.. Invariant measures for the horocycle flow on periodic hyperbolic surfaces. Israel J. Math. 160(1) (2007), 281–315.CrossRefGoogle Scholar
Lyons, T. and Sullivan, D.. Function theory, random paths and covering spaces. J. Differential Geom. 19(2) (1984), 299–323.CrossRefGoogle Scholar
Mauldin, D. and Urbański, M.. Gibbs states on the symbolic space over an infinite alphabet. Israel J. Math. 125(1) (2001), 93–130.CrossRefGoogle Scholar
Nicholls, P. J.. The Ergodic Theory of Discrete Groups (London Mathematical Society Lecture Note Series, 143). Cambridge University Press, Cambridge, MA, 1989.CrossRefGoogle Scholar
Patterson, S. J.. The limit set of a Fuchsian group. Acta Math. 136(1) (1976), 241–273.CrossRefGoogle Scholar
Rees, M.. Checking ergodicity of some geodesic flows with infinite Gibbs measure. Ergod. Th. & Dynam. Sys. 1(1) (1981), 107–133.CrossRefGoogle Scholar
Roblin, T.. Ergodicité et équidistribution en courbure négative. Société Mathématique de France, Paris. 2003.CrossRefGoogle Scholar
Roblin, T.. Comportement harmonique des densités conformes et frontière de Martin. Bull. Soc. Math. France 139(1) (2011), 97–127.CrossRefGoogle Scholar
Sarig, O.. Thermodynamic formalism for countable Markov shifts. Ergod. Th. & Dynam. Sys. 19(06) (1999), 1565–1593.CrossRefGoogle Scholar
Sarig, O.. Thermodynamic formalism for null recurrent potentials. Israel J. Math. 121(1) (2001), 285–311.CrossRefGoogle Scholar
Sarig, O.. The horocycle flow and the Laplacian on hyperbolic surfaces of infinite genus. Geom. Funct. Anal. 19(6) (2010), 1757–1812.CrossRefGoogle Scholar
Sarig, O.. Thermodynamic formalism for countable Markov shifts. Proc. Sympos. Pure Math. 89 (2015), 81–117.CrossRefGoogle Scholar
Sarig, O. and Schapira, B.. The generic points for the horocycle flow on a class of hyperbolic surfaces with infinite genus. Int. Math. Res. Not. IMRN 2008 (2008), rnn086.CrossRefGoogle Scholar
Series, C.. The infinite word problem and limit sets in Fuchsian groups. Ergod. Th. & Dynam. Sys. 1(3) (1981), 337–360.CrossRefGoogle Scholar
Series, C.. Martin boundaries of random walks on Fuchsian groups. Israel J. Math. 44(3) (1983), 221–242.CrossRefGoogle Scholar
Series, C.. Geometrical Markov coding of geodesics on surfaces of constant negative curvature. Ergod. Th. & Dynam. Sys. 6(4) (1986), 601–625.CrossRefGoogle Scholar
Shwartz, O.. Thermodynamic formalism for transient potential functions. Commun. Math. Phys. 366(2) (2019), 737–779.CrossRefGoogle Scholar
Stadlbauer, M.. An extension of Kesten’s criterion for amenability to topological Markov chains. Adv. Math. 235 (2013), 450–468.CrossRefGoogle Scholar
Stadlbauer, M.. On conformal measures and harmonic functions for group extensions. New Trends in One-dimensional Dynamics. Springer, New York, NY, 2016.Google Scholar
Sullivan, D.. The density at infinity of a discrete group of hyperbolic motions. Publ. Math. Inst. Hautes Études Sci. 50(1) (1979), 171–202.CrossRefGoogle Scholar
Sullivan, D.. Related aspects of positivity in Riemannian geometry. J. Differential Geom. 25(3) (1987), 327–351.CrossRefGoogle Scholar
Woess, W.. Random Walks on Infinite Graphs and Groups (Cambridge Tracts in Mathematics, 138). Cambridge University Press, Cambridge, MA, 2000.CrossRefGoogle Scholar