Group-extended Markov systems, amenability, and the Perron-Frobenius operator

Jaerisch, Johannes

doi:10.1090/S0002-9939-2014-12237-4

Group-extended Markov systems, amenability, and the Perron-Frobenius operator
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Proc. Amer. Math. Soc. 143 (2015), 289-300 Request permission

Abstract:

We characterise amenability of a countable group in terms of the spectral radius of the Perron-Frobenius operator associated to a group extension of a countable Markov shift and a Hölder continuous potential. This extends a result of Day for random walks and recent work of Stadlbauer for dynamical systems. Moreover, we show that if the potential satisfies a symmetry condition with respect to the group extension, then the logarithm of the spectral radius of the Perron-Frobenius operator is given by the Gurevič pressure of the potential.

References

Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989
Mahlon Marsh Day, Convolutions, means, and spectra, Illinois J. Math. 8 (1964), 100–111. MR 159230
Peter Gerl, Random walks on graphs with a strong isoperimetric property, J. Theoret. Probab. 1 (1988), no. 2, 171–187. MR 938257, DOI 10.1007/BF01046933
B. M. Gurevič, Topological entropy of a countable Markov chain, Dokl. Akad. Nauk SSSR 187 (1969), 715–718 (Russian). MR 0263162
B. M. Gurevič, Shift entropy and Markov measures in the space of paths of a countable graph, Dokl. Akad. Nauk SSSR 192 (1970), 963–965 (Russian). MR 0268356
Johannes Jaerisch, Fractal models for normal subgroups of Schottky groups, Trans. Amer. Math. Soc., to appear. DOI 10.1090/S0002-9947-2014-06095-9.
—, Recurrence and pressure for group extensions, Ergodic Theory Dynam. Systems, to appear.
—, Thermodynamic formalism for group-extended Markov systems with applications to Fuchsian groups, Doctoral Dissertation at the University Bremen (2011).
Harry Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 146–156. MR 112053, DOI 10.7146/math.scand.a-10568
Harry Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959), 336–354. MR 109367, DOI 10.1090/S0002-9947-1959-0109367-6
R. Daniel Mauldin and Mariusz Urbański, Graph directed Markov systems, Cambridge Tracts in Mathematics, vol. 148, Cambridge University Press, Cambridge, 2003. Geometry and dynamics of limit sets. MR 2003772, DOI 10.1017/CBO9780511543050
Ronald Ortner and Wolfgang Woess, Non-backtracking random walks and cogrowth of graphs, Canad. J. Math. 59 (2007), no. 4, 828–844. MR 2338235, DOI 10.4153/CJM-2007-035-1
Jean-Paul Pier, Amenable locally compact groups, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 767264
Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
David Ruelle, Statistical mechanics: Rigorous results, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0289084
Omri M. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems 19 (1999), no. 6, 1565–1593. MR 1738951, DOI 10.1017/S0143385799146820
Omri M. Sarig, Thermodynamic formalism for null recurrent potentials, Israel J. Math. 121 (2001), 285–311. MR 1818392, DOI 10.1007/BF02802508
Omri Sarig, Existence of Gibbs measures for countable Markov shifts, Proc. Amer. Math. Soc. 131 (2003), no. 6, 1751–1758. MR 1955261, DOI 10.1090/S0002-9939-03-06927-2
Richard Sharp, Critical exponents for groups of isometries, Geom. Dedicata 125 (2007), 63–74. MR 2322540, DOI 10.1007/s10711-007-9137-9
Manuel Stadlbauer, An extension of Kesten’s criterion for amenability to topological Markov chains, Adv. Math. 235 (2013), 450–468. MR 3010065, DOI 10.1016/j.aim.2012.12.004