Duke Mathematical Journal

Harmonicity of Gibbs measures

Chris Connell and Roman Muchnik

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Abstract

We show that any continuous measure ν in the class of a generalized Gibbs stream on the boundary of a CAT(κ) group G arises as a harmonic measure for a random walk on G. Under an additional mild hypothesis on G and for ν, Hölder equivalent to a Gibbs measure, we show that (G,ν) arises as a Poisson boundary for a random walk on G. We also prove a new approximation theorem for general metric measure spaces giving quite flexible conditions for a set of functions to be a positive basis for the cone of positive continuous functions

Article information

Source
Duke Math. J., Volume 137, Number 3 (2007), 461-509.

Dates
First available in Project Euclid: 6 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1175865518

Digital Object Identifier
doi:10.1215/S0012-7094-07-13732-3

Mathematical Reviews number (MathSciNet)
MR2309151

Zentralblatt MATH identifier
1133.60032

Subjects
Primary: 60J50: Boundary theory 20F67: Hyperbolic groups and nonpositively curved groups 37A35: Entropy and other invariants, isomorphism, classification 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)

Citation

Connell, Chris; Muchnik, Roman. Harmonicity of Gibbs measures. Duke Math. J. 137 (2007), no. 3, 461--509. doi:10.1215/S0012-7094-07-13732-3. https://projecteuclid.org/euclid.dmj/1175865518


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References

  • U. Bader and R. Muchnik, Irreducible $L^2$-representations of CAT($-$1) groups, preprint, 2005.
  • F. F. Bonsall and D. Walsh, Vanishing $l\sp 1$-sums of the Poisson kernel, and sums with positive coefficients, Proc. Edinburgh Math. Soc. (2) 32 (1989), 431--447.
  • M. Bourdon, Structure conforme au bord et flot géodésique d'un $\rm CAT(-1)$-espace, Enseign. Math. (2) 41 (1995), 63--102.
  • —, Sur le birapport au bord des $\rmCAT(-1)$-espaces, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 95--104.
  • R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math. 470, Springer, Berlin, 1975.
  • C. Connell and R. Muchnik, Harmonicity of quasiconformal measures and Poisson boundaries of hyperbolic spaces, to appear in Geom. Funct. Anal., preprint.
  • M. Coornaert, Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov, Pacific J. Math. 159 (1993), 241--270.
  • M. Coornaert and A. Papadopoulos, Sur une formule de transformation pour les densités conformes au bord des CAT$(-1)$-espaces, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 1231--1236.
  • A. Furman, ``Random walks on groups and random transformations'' in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, 931--1014.
  • H. Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. (2) 77 (1963), 335--386.
  • —, Poisson boundaries and envelopes of discrete groups, Bull. Amer. Math. Soc. 73 (1967), 350--356.
  • —, ``Random walks and discrete subgroups of Lie groups'' in Advances in Probability and Related Topics, Vol. 1, Dekker, New York, 1971, 1--63.
  • S. J. Gardiner, Representation of continuous functions as sums of Green functions, Proc. Amer. Math. Soc. 124 (1996), 1149--1157.
  • A. Gorodnik, Uniform distribution of orbits of lattices on spaces of frames, Duke Math. J. 122 (2004), 549--589.
  • U. HamenstäDt, Harmonic measures for compact negatively curved manifolds, Acta Math. 178 (1997), 39--107.
  • W. K. Hayman and T. J. Lyons, Bases for positive continuous functions, J. London Math. Soc. (2) 42 (1990), 292--308.
  • V. A. Kaimanovich, ``Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds'' in Hyperbolic Behaviour of Dynamical Systems (Paris, 1990), Ann. Inst. H. Poincaré Phys. Théor. 53 (1990), 361--393.
  • —, The Poisson boundary of hyperbolic groups, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), 59--64.
  • —, The Poisson formula for groups with hyperbolic properties, Ann. of Math. (2) 152 (2000), 659--692.
  • —, Poisson boundary of discrete groups, unpublished survey, 2003.
  • —, Differentiability of Gibbs states on negatively curved manifolds, unpublished survey, 2004.
  • V. A. Kaimanovich and H. Masur, The Poisson boundary of the mapping class group, Invent. Math. 125 (1996), 221--264.
  • —, The Poisson boundary of Teichmüller space, J. Funct. Anal. 156 (1998), 301--332.
  • V. A. KaĭManovich [Kaimanovich] and A. M. Vershik, Random walks on discrete groups: Boundary and entropy, Ann. Probab. 11 (1983), 457--490.
  • S. P. Lalley, Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits, Acta Math. 163 (1989), 1--55.
  • I. Mineyev, Metric conformal structures and hyperbolic dimension, preprint, 2005.
  • R. Muchnik, A note on stationarity of spherical measures, Israel J. Math. 152 (2006), 271--283.
  • —, Mixing and irreducibility of $l^2$ on the boundaries of CAT$(-1)$ spaces, preprint, 2005.
  • S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), 241--273.
  • D. Ruelle, Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics, Encyclopedia Math. Appl. 5, Addison-Wesley, Reading, Mass., 1978.
  • D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171--202.