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Conformal Geometry and Dynamics

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Conformal fractals for normal subgroups of free groups


Author: Johannes Jaerisch
Journal: Conform. Geom. Dyn. 18 (2014), 31-55
MSC (2010): Primary 37C45, 30F40; Secondary 37C85, 43A07
DOI: https://doi.org/10.1090/S1088-4173-2014-00263-X
Published electronically: March 7, 2014
MathSciNet review: 3175016
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Abstract: We investigate subsets of a multifractal decomposition of the limit set of a conformal graph directed Markov system which is constructed from the Cayley graph of the free group $ F_{d}$ with at least two generators. The subsets we consider are parametrised by a normal subgroup $ N$ of $ F_{d}$ and mimic the radial limit set of a Kleinian group. Our main results show that, regarding the Hausdorff dimension of these sets, various results for Kleinian groups can be generalised. Namely, under certain natural symmetry assumptions on the multifractal decomposition, we prove that, for a subset parametrised by $ N$, the Hausdorff dimension is maximal if and only if $ F_{d}/N$ is amenable and that the dimension is greater than half of the maximal value. We also give a criterion for amenability via the divergence of the Poincaré series of $ N$. Our results are applied to the Lyapunov spectrum for normal subgroups of Kleinian groups of Schottky type.


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  • [Bea95] Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1995. Corrected reprint of the 1983 original. MR 1393195 (97d:22011)
  • [BJ97] Christopher J. Bishop and Peter W. Jones, Hausdorff dimension and Kleinian groups, Acta Math. 179 (1997), no. 1, 1-39. MR 1484767 (98k:22043), https://doi.org/10.1007/BF02392718
  • [Bow75] Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin, 1975. MR 0442989 (56 #1364)
  • [Bro85] Robert Brooks, The bottom of the spectrum of a Riemannian covering, J. Reine Angew. Math. 357 (1985), 101-114. MR 783536 (86h:58138), https://doi.org/10.1515/crll.1985.357.101
  • [BTMT12] Petra Bonfert-Taylor, Katsuhiko Matsuzaki, and Edward C. Taylor, Large and small covers of a hyperbolic manifold, J. Geom. Anal. 22 (2012), no. 2, 455-470. MR 2891734, https://doi.org/10.1007/s12220-010-9204-6
  • [Coh82] Joel M. Cohen, Cogrowth and amenability of discrete groups, J. Funct. Anal. 48 (1982), no. 3, 301-309. MR 678175 (85e:43004), https://doi.org/10.1016/0022-1236(82)90090-8
  • [Day49] M. M. Day, Means on semigroups and groups, Bull. Amer. Math. Soc. 55 (1949), 1054-1055.
  • [FS04] Kurt Falk and Bernd O. Stratmann, Remarks on Hausdorff dimensions for transient limit sets of Kleinian groups, Tohoku Math. J. (2) 56 (2004), no. 4, 571-582. MR 2097162 (2005g:30053)
  • [Gri80] R. I. Grigorchuk, Symmetrical random walks on discrete groups, Multicomponent random systems, Adv. Probab. Related Topics, vol. 6, Dekker, New York, 1980, pp. 285-325. MR 599539 (83k:60016)
  • [Jae11] Johannes Jaerisch, Thermodynamic Formalism for Group-Extended Markov Systems with Applications to Fuchsian Groups, Ph.D. thesis, University Bremen, http://d-nb.info/1011939185/34, 2011.
  • [Jae12] -, Recurrence and pressure for group extensions, preprint available at http://arxiv.org/abs/1205.4490 (2012).
  • [Jae13] -, A lower bound for the exponent of convergence of normal subgroups of Kleinian groups, J. Geom. Anal. (accepted for publication), doi: 10.1007/s12220-013-9427-4 (2013).
  • [Jae14a] -, Fractal models for normal subgroups of Schottky groups, Trans. Amer. Math. Soc. (accepted for publication), preprint available at http://arxiv.org/abs/1106.0026 (2014).
  • [Jae14b] -, Group-extended Markov systems, amenability, and the Perron-Frobenius operator, Proc. Amer. Math. Soc. (accepted for publication), preprint available at http://arxiv.org/abs/1205.5126 (2014).
  • [JK11] Johannes Jaerisch and Marc Kesseböhmer, Regularity of multifractal spectra of conformal iterated function systems, Trans. Amer. Math. Soc. 363 (2011), no. 1, 313-330. MR 2719683 (2012b:37064), https://doi.org/10.1090/S0002-9947-2010-05326-7
  • [JKL14] Johannes Jaerisch, Marc Kesseböhmer, and Sanaz Lamei, Induced topological pressure for countable state Markov shifts, Stoch. Dyn. 14 (2014), no. 2, (to appear) doi: 10.1142/S0219493713500160.
  • [KMS12] Marc Kesseböhmer, Sara Munday, and Bernd O. Stratmann, Strong renewal theorems and Lyapunov spectra for $ \alpha $-Farey and $ \alpha $-Lüroth systems, Ergodic Theory Dynam. Systems 32 (2012), no. 3, 989-1017. MR 2995653, https://doi.org/10.1017/S0143385711000186
  • [Mas88] Bernard Maskit, Kleinian groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135 (90a:30132)
  • [MT98] Katsuhiko Matsuzaki and Masahiko Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1998. Oxford Science Publications. MR 1638795 (99g:30055)
  • [MU96] R. Daniel Mauldin and Mariusz Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3) 73 (1996), no. 1, 105-154. MR 1387085 (97c:28020), https://doi.org/10.1112/plms/s3-73.1.105
  • [MU03] R. Daniel Mauldin and Mariusz Urbański, Graph directed Markov systems, Cambridge Tracts in Mathematics, vol. 148, Cambridge, 2003.
  • [MY09] Katsuhiko Matsuzaki and Yasuhiro Yabuki, The Patterson-Sullivan measure and proper conjugation for Kleinian groups of divergence type, Ergodic Theory Dynam. Systems 29 (2009), no. 2, 657-665. MR 2486788 (2010h:37097), https://doi.org/10.1017/S0143385708080267
  • [Neu29] J. v. Neumann, Zur allgemeinen Theorie des Masses, Fund. Math. 13 (1929), 73-116.
  • [Nic89] Peter J. Nicholls, The ergodic theory of discrete groups, London Mathematical Society Lecture Note Series, vol. 143, Cambridge University Press, Cambridge, 1989. MR 1041575 (91i:58104)
  • [PW97] Yakov Pesin and Howard Weiss, A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions, J. Statist. Phys. 86 (1997), no. 1-2, 233-275. MR 1435198 (97m:58118), https://doi.org/10.1007/BF02180206
  • [Ree81a] Mary Rees, Checking ergodicity of some geodesic flows with infinite Gibbs measure, Ergodic Theory Dynamical Systems 1 (1981), no. 1, 107-133. MR 627791 (83g:58037)
  • [Ree81b] Mary Rees, Divergence type of some subgroups of finitely generated Fuchsian groups, Ergodic Theory Dynamical Systems 1 (1981), no. 2, 209-221. MR 661820 (83i:58061)
  • [Rob05] Thomas Roblin, Un théorème de Fatou pour les densités conformes avec applications aux revêtements galoisiens en courbure négative, Israel J. Math. 147 (2005), 333-357 (French, with French summary). MR 2166367 (2006i:37065), https://doi.org/10.1007/BF02785371
  • [Roc70] R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683 (43 #445)
  • [RU08] Mario Roy and Mariusz Urbański, Real analyticity of Hausdorff dimension for higher dimensional hyperbolic graph directed Markov systems, Math. Z. 260 (2008), no. 1, 153-175. MR 2413348 (2009m:37064), https://doi.org/10.1007/s00209-007-0267-4
  • [Rue69] David Ruelle, Statistical mechanics: Rigorous results, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0289084 (44 #6279)
  • [Rue78] David Ruelle, Thermodynamic formalism, Encyclopedia of Mathematics and its Applications, vol. 5, Addison-Wesley Publishing Co., Reading, Mass., 1978. The mathematical structures of classical equilibrium statistical mechanics; With a foreword by Giovanni Gallavotti and Gian-Carlo Rota. MR 511655 (80g:82017)
  • [Sar99] Omri M. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems 19 (1999), no. 6, 1565-1593. MR 1738951 (2000m:37009), https://doi.org/10.1017/S0143385799146820
  • [Sar01] Omri M. Sarig, Thermodynamic formalism for null recurrent potentials, Israel J. Math. 121 (2001), 285-311. MR 1818392 (2001m:37059), https://doi.org/10.1007/BF02802508
  • [Sar03] Omri Sarig, Existence of Gibbs measures for countable Markov shifts, Proc. Amer. Math. Soc. 131 (2003), no. 6, 1751-1758 (electronic). MR 1955261 (2004b:37056), https://doi.org/10.1090/S0002-9939-03-06927-2
  • [Sch99] Jörg Schmeling, On the completeness of multifractal spectra, Ergodic Theory Dynam. Systems 19 (1999), no. 6, 1595-1616. MR 1738952 (2000k:37009), https://doi.org/10.1017/S0143385799151988
  • [Ser81] Caroline Series, The infinite word problem and limit sets in Fuchsian groups, Ergodic Theory Dynamical Systems 1 (1981), no. 3, 337-360 (1982). MR 662473 (84d:30084)
  • [Sta13] Manuel Stadlbauer, An extension of Kesten's criterion for amenability to topological Markov chains, Adv. Math. 235 (2013), 450-468. MR 3010065, https://doi.org/10.1016/j.aim.2012.12.004
  • [Str06] Bernd O. Stratmann, Fractal geometry on hyperbolic manifolds, Non-Euclidean geometries, Math. Appl. (N. Y.), vol. 581, Springer, New York, 2006, pp. 227-247. MR 2191250 (2006g:37038), https://doi.org/10.1007/0-387-29555-0_12

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Additional Information

Johannes Jaerisch
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka, 560-0043 Japan
Email: jaerisch@cr.math.sci.osaka-u.ac.jp

DOI: https://doi.org/10.1090/S1088-4173-2014-00263-X
Keywords: Kleinian groups, exponent of convergence, normal subgroups, amenability, conformal graph directed Markov systems
Received by editor(s): June 20, 2013
Published electronically: March 7, 2014
Additional Notes: The author was supported by the research fellowship JA 2145/1-1 of the German Research Foundation (DFG)
Article copyright: © Copyright 2014 American Mathematical Society

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