Minimal Volume Entropy on Graphs

Among the normalized metrics on a graph, we show the existence and the uniqueness of an entropy-minimizing metric, and give explicit formulas for the minimal volume entropy and the metric realizing it. Parmi les distances normalisées sur un graphe, nous montrons l’existence et l’unicité d’une distance qui minimise l’entropie, et nous donnons des formules explicites pour l’entropie volumique minimale et la distance qui la réalise. 1.Introduction Let (X, g) be a compact connected Riemannian manifold of nonpositive curvature. It was shown by A. Manning [Man] that the topological entropy htop(g) of the geodesic flow is equal to the volume entropy hvol(g) of the manifold hvol(g) = lim r→∞ 1 r log(vol(B(x, r))), where B(x, r) is the ball of radius r centered at some point x in the universal cover X̃ of X. In [BCG], G. Besson, G. Courtois and S. Gallot proved that if X has dimension at least three and carries a rank one locally symmetric metric g0, then for every Riemannian metric g such that vol(X, g) = vol(X, g0), the inequality hvol(g) ≥ hvol(g0) holds, with equality if and only if g is isometric to g0. This solved a conjecture mainly due to M. Gromov [Gro], which had been proved earlier by A. Katok [Kat] for metrics in the conformal class of the hyperbolic metric on a compact orientable surface. In this paper, we are interested in the analogous problem for finite graphs, endowed with metrics obtained by varying the length `(e) of the edges e ∈ EX of a graph X. Regular and biregular trees, as rank one buildings (see [BT]), are non-archimedian analogs of rank one symmetric spaces, and they carry many lattices (see [BL] for instance). As in the case of Riemannian manifolds, it is well known that the volume entropy of a finite metric graph is equal to the topological entropy of the geodesic flow on its universal cover (see [Gui]) and also equal to the critical exponent of its fundamental group acting on its universal covering tree ([Bou]).