Abstract
The notion of a SAT action introduced by Jaworski was inspired by the proximal properties of the Poisson boundaries of random walks obtained by Furstenberg and by the definition of an approximatively transitive action due to Connes and Woods. We introduce a slightly stronger condition SAT* and establish several general ergodic properties of SAT* actions. This notion turns out to be quite helpful for studying ergodic properties of the horosphere foliation (particular case: the horocycle flow) on a quotient of a CAT(-1) space by a discrete group of isometries G. We look at the “intermediate covers” determined by normal subgroups H ⊲ G. Under the condition that the boundary action of G is SAT* we establish conservativity of the horosphere foliation on these intermediate covers and prove that its ergodic components are in one-to-one correspondence with the ergodic components of the boundary action of H. In particular, in this situation ergodicity of the horosphere foliation is equivalent to ergodicity of the associated boundary action. These results are applicable to several well known classes of measures at infinity (harmonic, Gibbs, conformai).
The author gratefully acknowledges a partial support from the Austrian-French program “Amadeus”.
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References
W. Ballmann, M. Gromov, V. Schroeder, Manifolds of nonpositive curvature, Birkhäuser, Boston, Mass., 1985.
M. Bourdon, Structure conforme au bord et flot géodésique d’un CAT(-1)-espace, L’Ens. Math. 41 (1995), 63–102.
M. Burger, S. Mozes, CAT(-1)-spaces, divergence groups and their commensurators, J. Amer. Math. Soc. 9 (1996), 57–93.
A. Connes, E. J. Woods, Approximatively transitive flows and ITPFI factors, Ergod. Th. Dynam. Syst. 5 (1985), 203–236.
F. Dal’bo, Remarques sur le spectre des longueurs d’une surface et comptages, Bol. Soc. Bras. Mat. 30 (1999), 199–221.
J. Feldman, C. C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), 289–324.
H. Furstenberg, Boundary theory and stochastic processes on homogeneous spaces, Proc. Symp. Pure Math. 26 (1973), 193–229.
H. Furstenberg, The unique ergodicity of the horocycle flow, Springer Lecture Notes in Math. 318 (1973), 95–115.
R. I. Grigorchuk, V. A. Kaimanovich, T. Smirnova-Nagnibeda, in preparation.
M. Gromov, Hyperbolic groups, MSRI Publ. 8 (1987), 75–263.
U. Hamenstädt, Ergodic properties of Gibbs measures on nilpotent covers, preprint, 2001.
W. Jaworski, Strongly approximately transitive group actions, the Choquet-Deny theorem, and polynomial growth, Pacific J. Math. 165 (1994), 115–129.
W. Jaworski, Random walks on almost connected locally compact groups: boundary and convergence, J. Analyse Math. 74 (1998), 235–273.
V. A. Kaimanovich, Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces, J. Reine Angew. Math. 455 (1994), 57–103.
V. A. Kaimanovich, The Poisson boundary of covering Markov operators, Israel J. Math. 89 (1995), 77–134.
V. A. Kaimanovich, The Poisson boundary of polycyclic groups, Probability measures on groups and related structures, XI, 182–195, World Sci. Publishing, River Edge, NJ, 1995.
V. A. Kaimanovich, Ergodic properties of the horocycle flow and classification of Fuchsian groups, J. Dynam. Control Systems 6 (2000), 21–56.
V. A. Kaimanovich, K. Schmidt, Ergodicity of cocycles. I. General theory, preprint, 2000.
V. A. Kaimanovich, K. Schmidt, Ergodicity of cocycles. II. Geometric applications, preprint, 2001.
V. A. Kaimanovich, W. Woess, The Dirichlet problem at infinity for random walks on graphs with a strong isoperimetric inequality, Probab. Theory Related Fields 91 (1992), 445–466.
U. Krengel, Ergodic Theorems, de Gruyter, Berlin, 1985.
T. Lyons, D. Sullivan, Function theory, random paths and covering spaces, J. Differential Geom. 19 (1984), 299–323.
C. C. Moore, C. Schochet, Global analysis on foliated spaces, Springer, New York, 1988.
T. Roblin, Sur la théorie ergodique des groupes discrets en géométrie hyperbolique, thèse, Université Paris Sud, 1999.
V. A. Rokhlin, On the fundamental ideas of measure theory (Russian), Mat. Sbornik N. S. 25(67) (1949), 107–150.
K. Schmidt, Cocycles of ergodic transformation groups, McMillan (India), New Delhi, 1977.
D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Publ. Math. IHES 50 (1979), 171–202.
D. Sullivan, Discrete conformai groups and measurable dynamics, Bull. Amer. Math. Soc. 6 (1982), 57–73.
P. Tukia, Conservative actions and the horospheric limit set, Ann. Acad. Sci. Fenn. Math. 22 (1997), 387–394.
J. A. Veiling, K. Matsuzaki, The action at infinity of conservative groups of hyperbolic motions need not have atoms, Ergod. Th. Dynam. Syst. 11 (1991), 577–582.
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Kaimanovich, V.A. (2002). SAT Actions and Ergodic Properties of the Horosphere Foliation. In: Burger, M., Iozzi, A. (eds) Rigidity in Dynamics and Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04743-9_13
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DOI: https://doi.org/10.1007/978-3-662-04743-9_13
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