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Upper and lower bounds for the mass of the geodesic flow on graphs

Published online by Cambridge University Press:  01 May 1997

MICHEL COORNAERT
Affiliation:
Institut de Recherche Mathématique Avancée (Université Louis Pasteur et CNRS), 7 rue René Descartes, 67084 Strasbourg Cedex (France)
ATHANASE PAPADOPOULOS
Affiliation:
Institut de Recherche Mathématique Avancée (Université Louis Pasteur et CNRS), 7 rue René Descartes, 67084 Strasbourg Cedex (France)

Abstract

Let G be a connected locally finite simplicial graph with rk(π1(G))[ges ]2 and let T be the universal cover of G. Consider a π1(G)-invariant conformal density μ of dimension d on ∂T. The total mass function ϕμ of μ is defined on the set of vertices of G. Let |ϕμ| be its l2-norm. Let Ω be the geodesic flow space of G and mμ the invariant measure on Ω associated to μ. The main results of this paper are the following:

(i) Assume that there exists an integer k[ges ]3 such that the degree at each vertex of G is [les ]k. Then, if d>½log(k−1) and mμ(Ω)<∞, we have ϕμl2(V) and

μ|2 [les ]Cmμ(Ω),

with C= (e2d−1)/ (e2dk+1).

(ii) Assume that there exists an integer k[ges ]3 such that the degree at each vertex of G is [ges ]k. Then, if ϕμl2(V), we have d>½log(k−1) and

Cmμ(Ω) [les ]|ϕμ|2.

Type
Research Article
Copyright
Cambridge Philosophical Society 1997

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