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Arithmetic Landscape Functions of a Discrete Cat Map

  • Aryaman Chandra
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Abstract

For a permutation of a finite set, we define a landscape by summing a discounted weight over the times each point returns to itself. The landscape value is determined by the point's least period. We relate the trace of the associated resolvent to fixed-point and exact-period counts, and show that the associated determinant is the reciprocal of the Artin-Mazur dynamical zeta function. We apply this framework to the discrete cat map modulo N. For every prime p≠5, all nonzero points have period equal to the order of the cat-map matrix modulo p, yielding explicit formulas for the landscape trace and determinant. We prove a period-lifting formula at odd primes p≠5, derive exact fixed-point and period counts modulo 5a, and combine prime-power results for composite moduli using the Chinese remainder theorem.

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Cite

BibTeXRIS
BibTeX
@misc{chandra2026arithmetic,
  title = {{Arithmetic Landscape Functions of a Discrete Cat Map}},
  author = {Aryaman Chandra},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.24857},
  eprint = {2607.24857},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.24857},
}
RIS
TY  - UNPB
TI  - Arithmetic Landscape Functions of a Discrete Cat Map
AU  - Aryaman Chandra
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2607.24857
UR  - https://arxiv.org/abs/2607.24857
AB  - For a permutation of a finite set, we define a landscape by summing a discounted weight over the times each point returns to itself. The landscape value is determined by the point's least period. We relate the trace of the associated resolvent to fixed-point and exact-period counts, and show that the associated determinant is the reciprocal of the Artin-Mazur dynamical zeta function. We apply this framework to the discrete cat map modulo N. For every prime p≠5, all nonzero points have period equal to the order of the cat-map matrix modulo p, yielding explicit formulas for the landscape trace and determinant. We prove a period-lifting formula at odd primes p≠5, derive exact fixed-point and period counts modulo 5^a, and combine prime-power results for composite moduli using the Chinese remainder theorem.
ER  -