有限正则图上加权动力Ihara zeta函数的留数
Residues of weighted dynamical Ihara zeta functions on finite regular graphs
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中文总结 AI 辅助
本文在有限正则图上构造加权动力Ihara zeta函数，证明其亚纯延拓并计算留数，将其与不变Ruelle分布关联，推导出Patterson-Sullivan和Wigner公式，为局部对称空间情形提供有限图类比。
中文摘要 AI 辅助
在连通有限正则图上提出了经典Ihara zeta函数的加权动力版本$\mathbf{Z}_f$，该函数以加权周期轨道数据表示。我们建立了其亚纯延拓，其极点由作用于适当Banach空间上的相关非回溯转移算子的共振给出，并计算了其留数。在简单谱参数处，$\mathbf{Z}_f$的留数被识别为图相空间上的不变Ruelle分布。将此结果与Arends-Palmirotta获得的关系相结合，我们进一步推导出Patterson-Sullivan和Wigner留数公式。这为秩一局部对称空间上测地流的加权动力zeta函数的留数公式提供了有限图类比，其精神源于Schütte-Barkhofen-Weich。
英文摘要
A weighted dynamical version $\mathbf{Z}_f$ of the classical Ihara zeta function is presented on connected finite regular graphs, expressed in terms of weighted periodic orbit data. We establish its meromorphic continuation, with poles given by the resonances of the associated non-backtracking transfer operator acting on a suitable Banach space, and compute its residues. At simple spectral parameters, the residue of $\mathbf{Z}_f$ is identified with the invariant Ruelle distribution on the graph phase space. Combining this result with a relation obtained by Arends-Palmirotta, we further derive Patterson-Sullivan and Wigner residue formulae. This provides a finite-graph analogue of residue formulae for weighted dynamical zeta functions for geodesic flow on rank one locally symmetric spaces, in the spirit of Schütte-Barkhofen-Weich.
发表机构
- Universität Paderborn（帕德博恩大学）
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