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Equivalence of boundary measures on covering trees of finite graphs

Published online by Cambridge University Press:  19 September 2008

Russell Lyons
Russell Lyons
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN 47405-5701, USA
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Abstract

Let T be the universal covering tree of a finite graph, G. By analogy with an open problem concerning negatively curved covering manifolds, Kaimanovich asked when two of the three natural measure classes on ∂T can coincide, the three measures being harmonic measure, the Patterson measure, and visibility measure. We provide an almost complete answer and discuss related issues. The answer is quite surprising in some cases.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 3 , September 1994 , pp. 575 - 597
Copyright
Copyright © Cambridge University Press 1994

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References

REFERENCES

[BS]Berretti, Alberto and Sokal, Alan D.. New Monte Carlo method for the self-avoiding walk. J. Stat. Phys. 40 (1985), 483–531.CrossRefGoogle Scholar
[C]Coornaert, Michel. Mesures de Patterson-Sullivan sur le bord d'unespace hyperbolique au sens de Gromov. Pacific J. Math. 159 (1994), 241–270.CrossRefGoogle Scholar
[H]Hamenstadt, Ursula. Harmonic measure and Bowen—Margulis measure are almost always singular. Preprint.Google Scholar
[Kai]Kaimanovich, Vadim A.. Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds. Ann. Inst. H. Poincaré, Physique Théorique 53 (1990), 361–393.Google Scholar
[Kat1]Katok, Anatole. Entropy and closed geodesies. Ergod. Th. & Dynam. Sys. 2 (1982), 339–365.CrossRefGoogle Scholar
[Kat2]Katok, Anatole. Four applications of conformal equivalence to geometry and dynamics. Ergod. Th. & Dynam. Sys. 8* (1988), 139–152.Google Scholar
[KO]Karlin, Samuel & Ost, Friedemann. Some monotonicity properties of Schur powers of matrices and related inequalities. Linear Algebra Appl. 68 (1985), 47–65.CrossRefGoogle Scholar
[Kr]Krug, Joachim. Surface structure of random aggregates on the Cayley tree. J. Phys. A: Math. Gen. 21 (1988), 4637–4647.CrossRefGoogle Scholar
[La]Lagarias, Jeffrey. Personal communication (1993).Google Scholar
[LS]Lawler, Gregory F. & Sokal, Alan D.. Bounds on the L2 spectrum for Markov chains and Markov processes: a generalization of Cheeger's inequality. Trans. Amer. Math. Soc. 309 (1988), 557–580.Google Scholar
[Le1]Ledrappier, Francois. Propriété de Poisson et courbure négative. C.R. Acad. Sci. Paris 305 (1987), 191–194.Google Scholar
[Le2]Ledrappier, Francois. Harmonic measures and Bowen-Margulis measures. Israel J. Math. 71 (1990), 275–287.CrossRefGoogle Scholar
[Ly]Lyons, Russell. Random walks and percolation on trees. Ann. Prob. 18 (1990), 931–958.CrossRefGoogle Scholar
[S]Seneta, E.. Non-negative Matrices and Markov Chains. 2nd ed.Springer: New York, 1981.CrossRefGoogle Scholar
[W]Walters, Peter. An Introduction to Ergodic Theory. Springer: New York, 1982.CrossRefGoogle Scholar
[Y]Yue, C. B.. Contribution to Sullivan's conjecture. Preprint.Google Scholar