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Residues of weighted dynamical Ihara zeta functions on finite regular graphs

  • Guendalina Palmirotta
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Abstract

A weighted dynamical version Zf 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 Zf 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.

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Cite

BibTeXRIS
BibTeX
@misc{palmirotta2026residues,
  title = {{Residues of weighted dynamical Ihara zeta functions on finite regular graphs}},
  author = {Guendalina Palmirotta},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12548},
  eprint = {2609.12548},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12548},
}
RIS
TY  - UNPB
TI  - Residues of weighted dynamical Ihara zeta functions on finite regular graphs
AU  - Guendalina Palmirotta
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12548
UR  - https://arxiv.org/abs/2609.12548
AB  - A weighted dynamical version 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 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.
ER  -