Ihara zeta functions for periodic simple graphs

A trace on fractal graphs and the Ihara zeta function

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, …

Ihara's zeta function for periodic graphs and its approximation in the amenable case

In this paper, we give a more direct proof of the results by Clair and Mokhtari-Sharghi on the zeta functions of periodic graphs. In particular, using appropriate operator-algebraic techniques, we establish a determinant formula in this context and examine its consequences for the Ihara zeta function. Moreover, we answer in the affirmative one of the questions raised by Grigorchuk and Zuk. Accordingly, we show that the zeta function of a periodic graph with an amenable group action is the limit of the zeta functions of a suitable sequence of finite subgraphs.

The Ihara Zeta function for infinite graphs

We put forward the concept of measure graphs. These are (possibly uncountable) graphs equipped with an action of a groupoid and a measure invariant under this action. Examples include finite graphs, periodic graphs, graphings and percolation graphs. Making use of Connes' non-commutative integration theory we construct a Zeta function and present a determinant formula for it. We further introduce a notion of weak convergence of measure graphs and show that our construction is compatible with it. The approximation of the Ihara Zeta function via the normalized version on finite graphs in the sens…

Zeta functions for infinite graphs and functional equations

The definitions and main properties of the Ihara and Bartholdi zeta functions for infinite graphs are reviewed. The general question of the validity of a functional equation is discussed, and various possible solutions are proposed.

虚二次域的类数ζ函数

Class number zeta function of imaginary quadratic fields

本文引入一个通过类数计数虚二次数域的ζ函数，证明该函数仅依赖于八次单位根，从而得到素数类数的下界。方法基于动力系统周期点的研究。

Continuous orbit equivalence of topological Markov shifts and dynamical zeta functions

For continuously orbit equivalent one-sided topological Markov shifts $(X_A,σ_A)$ and $(X_B,σ_B)$, their eventually periodic points and cocycle functions are studied. As a result we directly construct an isomorphism between their ordered cohomology groups $(\bar{H}^A, \bar{H}^A_+)$ and $(\bar{H}^B, \bar{H}^B_+)$. We also show that the cocycle functions for the continuous orbit equivalences give rise to positive elements of the ordered cohomology, so that the the zeta functions of continuously orbit equivalent topological Markov shifts are related. The set of Borel measures is shown to be invar…