A trace on fractal graphs and the Ihara zeta function

Abstract

Starting with Ihara's work in 1968, there has been a growing interest in the study of zeta functions of finite graphs, by Sunada, Hashimoto, Bass, Stark and Terras, Mizuno and Sato, to name just a few authors. Then, Clair and Mokhtari-Sharghi have studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms. The main formula in all these treatments establishes a connection between the zeta function, originally defined as an infinite product, and the Laplacian of the graph. In this article, we consider a different class of infinite graphs. They are fractal graphs, i.e. they enjoy a self-similarity property. We define a zeta function for these graphs and, using the machinery of operator algebras, we prove a determinant formula, which relates the zeta function with the Laplacian of the graph. We also prove functional equations, and a formula which allows approximation of the zeta function by the zeta functions of finite subgraphs.

30 pages, 5 figures. v3: minor corrections, to appear on Transactions AMS

References in corpus (5)

• Ihara's zeta function for periodic graphs and its approximation in the amenable case
• Dimensions and spectral triples for fractals in R^N
• Aperiodic order, integrated density of states and the continuous algebras of John von Neumann
• A semicontinuous trace for almost local operators on an open manifold
• A C*-algebra of geometric operators on self-similar CW-complexes. Novikov-Shubin and L^2-Betti numbers

Cited by in corpus (6)

• Operator theory of electrical resistance networks
• Ihara's zeta function for periodic graphs and its approximation in the amenable case
• Applications of Automata and Graphs: Labeling Operators in Hilbert Space II
• A C*-algebra of geometric operators on self-similar CW-complexes. Novikov-Shubin and L^2-Betti numbers
• Zeta functions for infinite graphs and functional equations
• Fourier duality for fractal measures with affine scales