Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-20T11:21:29.816Z Has data issue: false hasContentIssue false
  • English
  • Français

Recurrence and pressure for group extensions

Published online by Cambridge University Press:  11 August 2014

JOHANNES JAERISCH
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
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the thermodynamic formalism for recurrent potentials on group extensions of countable Markov shifts. Our main result characterizes recurrent potentials depending only on the base space, in terms of the existence of a conservative product measure and a homomorphism from the group into the multiplicative group of real numbers. We deduce that, for a recurrent potential depending only on the base space, the group is necessarily amenable. Moreover, we give equivalent conditions for the base pressure and the skew product pressure to coincide. Finally, we apply our results to analyse the Poincaré series of Kleinian groups and the cogrowth of group presentations.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 1 , February 2016 , pp. 108 - 126
Copyright
© Cambridge University Press, 2014 

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

Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50). American Mathematical Society, Providence, RI, 1997.CrossRefGoogle Scholar
Aaronson, J. and Denker, M.. On exact group extensions. Sankhyā Ser. A 62(3) (2000), 339–349.Google Scholar
Aaronson, J. and Denker, M.. Group extensions of Gibbs–Markov maps. Probab. Theory Related Fields 123(1) (2002), 38–40.CrossRefGoogle Scholar
Aaronson, J., Denker, M. and Urbański, M.. Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. Amer. Math. Soc. 337(2) (1993), 495–548.CrossRefGoogle Scholar
Aaronson, J. and Weiss, B.. On Herman’s theorem for ergodic, amenable group extensions of endomorphisms. Ergod. Th. & Dynam. Sys. 24(5) (2004), 1283–1293.CrossRefGoogle Scholar
Beardon, A. F.. The Geometry of Discrete Groups (Graduate Texts in Mathematics, 91). Springer, New York, 1995, corrected reprint of the 1983 original.Google Scholar
Bergman, G. M.. An Invitation to General Algebra and Universal Constructions. Henry Helson, Berkeley, CA, 1998.Google Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin, 1975.CrossRefGoogle Scholar
Bowen, R.. Hausdorff dimension of quasicircles. Inst. Hautes Études Sci. Publ. Math. 50 (1979), 11–25.CrossRefGoogle Scholar
Brooks, R.. The bottom of the spectrum of a Riemannian covering. J. reine angew. Math. 357 (1985), 101–114.Google Scholar
Cohen, J. M.. Cogrowth and amenability of discrete groups. J. Funct. Anal. 48(3) (1982), 301–309.CrossRefGoogle Scholar
Dudley, R. M.. Random walks on abelian groups. Proc. Amer. Math. Soc. 13 (1962), 447–450.CrossRefGoogle Scholar
Grigorchuk, R. I.. Symmetrical random walks on discrete groups. Multicomponent Random Systems (Advances in Probability and Related Topics, 6). Dekker, New York, 1980, pp. 285–325.Google Scholar
Gromov, M.. Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math.(53) (1981), 53–73.CrossRefGoogle Scholar
Gurevič, B. M.. Topological entropy of a countable Markov chain. Dokl. Akad. Nauk SSSR 187 (1969), 715–718.Google Scholar
Gurevič, B. M.. Shift entropy and Markov measures in the space of paths of a countable graph. Dokl. Akad. Nauk SSSR 192 (1970), 963–965.Google Scholar
Jaerisch, J.. Thermodynamic formalism for group-extended Markov systems with applications to Fuchsian groups, PhD Thesis, University of Bremen, 2011, http://d-nb.info/1011939185/34.Google Scholar
Jaerisch, J.. A lower bound for the exponent of convergence of normal subgroups of Kleinian groups. J. Geom. Anal. (2013), online first, http://dx.doi.org/10.1007/s12220-013-9427-4.Google Scholar
Jaerisch, J.. Fractal models for normal subgroups of Schottky groups. Trans. Amer. Math. Soc. (2014), published online May 2014, http://www.ams.org/journals/tran/0000-000-00/S0002-9947-2014-06095-9/, 33 pages.CrossRefGoogle Scholar
Jaerisch, J.. Group-extended Markov systems, amenability, and the Perron–Frobenius operator. Proc. Amer. Math. Soc. (2014), accepted for publication.CrossRefGoogle Scholar
Jaerisch, J.. Conformal fractals for normal subgroups of free groups. Conform. Geom. Dyn. 18 (2014), 31–55.CrossRefGoogle Scholar
Jaerisch, J., Kesseböhmer, M. and Lamei, S.. Induced topological pressure for countable state Markov shifts. Stoch. Dyn. 14(2) (2014), 1–31.CrossRefGoogle Scholar
Kesten, H.. Full Banach mean values on countable groups. Math. Scand. 7 (1959), 146–156.CrossRefGoogle Scholar
Kesten, H.. Symmetric random walks on groups. Trans. Amer. Math. Soc. 92 (1959), 336–354.CrossRefGoogle Scholar
Kesten, H.. The Martin boundary of recurrent random walks on countable groups. Proc. Fifth Berkeley Sympos. Mathematical Statistics and Probability (Berkeley, CA, 1965–1966) (Vol. II: Contributions to Probability Theory, Part 2). University of California Press, Berkeley, CA, 1967, pp. 51–74.Google Scholar
Maskit, B.. Kleinian groups. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences, 287). Springer, Berlin, 1988.Google Scholar
Matsuzaki, K. and Yabuki, Y.. The Patterson–Sullivan measure and proper conjugation for Kleinian groups of divergence type. Ergod. Th. & Dynam. Sys. 29(2) (2009), 657–665.CrossRefGoogle Scholar
Mauldin, R. D. and Urbański, M.. Graph Directed Markov Systems (Cambridge Tracts in Mathematics, 148). Cambridge University Press, Cambridge, 2003.CrossRefGoogle Scholar
von Neumann, J.. Zur allgemeinen Theorie des Masses. Fund. Math. 13 (1929), 73–116.CrossRefGoogle Scholar
Nicholls, P. J.. The Ergodic Theory of Discrete Groups (London Mathematical Society Lecture Note Series, 143). Cambridge University Press, Cambridge, 1989.CrossRefGoogle Scholar
Patterson, S. J.. The limit set of a Fuchsian group. Acta Math. 136(3–4) (1976), 241–273.CrossRefGoogle Scholar
Pólya, G.. Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz. Math. Ann. 84(1–2) (1921), 149–160.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
Rees, M.. Divergence type of some subgroups of finitely generated Fuchsian groups. Ergod. Th. & Dynam. Sys. 1(2) (1981), 209–221.CrossRefGoogle Scholar
Ruelle, D.. Statistical mechanics of a one-dimensional lattice gas. Comm. Math. Phys. 9 (1968), 267–278.CrossRefGoogle Scholar
Ruelle, D.. Statistical Mechanics: Rigorous Results. W. A. Benjamin, New York–Amsterdam, 1969.Google Scholar
Sarig, O. M.. Thermodynamic formalism for countable Markov shifts. Ergod. Th. & Dynam. Sys. 19(6) (1999), 1565–1593.CrossRefGoogle Scholar
Sarig, O. M.. Thermodynamic formalism for null recurrent potentials. Israel J. Math. 121 (2001), 285–311.CrossRefGoogle Scholar
Sarig, O. M.. Existence of Gibbs measures for countable Markov shifts. Proc. Amer. Math. Soc. 131(6) (2003), 1751–1758 (electronic).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
Stadlbauer, M.. An extension of Kesten’s criterion for amenability to topological Markov chains. Adv. Math. 235 (2013), 450–468.CrossRefGoogle Scholar
Varopoulos, N. T.. Théorie du potentiel sur des groupes et des variétés. C. R. Acad. Sci. Paris Sér. I Math. 302(6) (1986), 203–205.Google Scholar
Vere-Jones, D.. Geometric ergodicity in denumerable Markov chains. Quart. J. Math. Oxford Ser. 13(2) (1962), 7–28.CrossRefGoogle Scholar
Woess, W.. Random walks on infinite graphs and groups—a survey on selected topics. Bull. Lond. Math. Soc. 26(1) (1994), 1–60.CrossRefGoogle Scholar
Woess, W.. Topological groups and recurrence of quasitransitive graphs. Rend. Semin. Mat. Fis. Milano 64 (1994), 185–213.CrossRefGoogle Scholar
Woess, W.. Random Walks on Infinite Graphs and Groups (Cambridge Tracts in Mathematics, 138). Cambridge University Press, Cambridge, 2000.CrossRefGoogle Scholar
Zimmer, R. J.. Amenable ergodic group actions and an application to Poisson boundaries of random walks. J. Funct. Anal. 27(3) (1978), 350–372.CrossRefGoogle Scholar