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

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 ϕ_μ of μ is defined on the set of vertices of G. Let |ϕ_μ| be its l²-norm. Let Ω be the geodesic flow space of G and m_μ 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 m_μ(Ω)<∞, we have ϕ_μ∈l²(V) and

|ϕ_μ|²[les ]Cm_μ(Ω),

with C=(e^2d−1)/(e^2d−k+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 ϕ_μ∈l²(V), we have d>½log(k−1) and

Cm_μ(Ω)[les ]|ϕ_μ|².