The Hydra Map and Numen Formalisms for Collatz-Type Problems
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866915793775624192 |
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| author | Siegel, Maxwell C. |
| author_facet | Siegel, Maxwell C. |
| contents | This paper details a generalization of the formalism presented in the author's 2024 paper, "The Collatz Conjecture and Non-Archimedean Spectral Theory - Part I - Arithmetic Dynamical Systems and Non-Archimedean Value Distribution Theory", to the case of Hydra maps on the ring of integers $\mathcal{O}_{K}$ of a global field $K$. In addition to recounting these definitions, background material is presented for the necessary standard material in algebraic number theory and integration and Fourier analysis with respect to the $p$-adic Haar measure. This paper is meant to serve as a technical manual for use of Hydra maps and numens in future research. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_17030 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Hydra Map and Numen Formalisms for Collatz-Type Problems Siegel, Maxwell C. Dynamical Systems Number Theory This paper details a generalization of the formalism presented in the author's 2024 paper, "The Collatz Conjecture and Non-Archimedean Spectral Theory - Part I - Arithmetic Dynamical Systems and Non-Archimedean Value Distribution Theory", to the case of Hydra maps on the ring of integers $\mathcal{O}_{K}$ of a global field $K$. In addition to recounting these definitions, background material is presented for the necessary standard material in algebraic number theory and integration and Fourier analysis with respect to the $p$-adic Haar measure. This paper is meant to serve as a technical manual for use of Hydra maps and numens in future research. |
| title | The Hydra Map and Numen Formalisms for Collatz-Type Problems |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2601.17030 |