| _version_ | 1866901598590992384 |
|---|---|
| author | okabe, arata |
| author_facet | okabe, arata |
| contents | <p>We study the Collatz conjecture through 2-adic arithmetic, thermodynamic formalism, and spectral theory of countable Markov shifts. The Syracuse map T(n) = (3n+1)/2^{v_2(3n+1)} is analyzed as a dynamical system on the 2-adic integers Z_2. We make four principal advances in this study.</p> <p>(1) Countable Markov shift structure. The inverse branches phi_s(y) = (2^s y - 1)/3, s = 1,2,3,..., generate a countable-state Markov shift (Sigma, sigma) whose alphabet is the set of step sizes {1,2,3,...}.</p> <p>(2) BIP condition. We verify that the singleton F = {1} witnesses the big images and preimages (BIP) property: for every state s >= 1, both the transitions 1 -> s and s -> 1 are admissible. This is established algebraically via the invertibility of 9 in Z_2 and confirmed computationally for s = 1, ..., 30.</p> <p>(3) Sarig's theorem on Z_2. Combining BIP with topological mixing, summable variations, and finite Gurevich pressure, we invoke Sarig (1999, 2003) to obtain a spectral gap for the Ruelle-Perron-Frobenius (RPF) operator L_gamma with potential psi_gamma = -gamma log |S| on the full space Z_2, for every gamma > 0. This yields a simple maximal eigenvalue, a unique Gibbs measure, and exponential decay of correlations.</p> <p>(4) Phase transition and certified gap. The spectral gap onset at gamma = 0 is first-order: gap(gamma, m) ≈ C(m) * gamma as gamma -> 0, with C(m) increasing in m. Computer-assisted certification using Bauer-Fike perturbation bounds with 50-digit arithmetic confirms the gap rigorously for m = 8, 10, 12. At m = 12, the eigenvalue ratio lambda_2 / lambda_1 = 1/sqrt(2), suggesting a deep connection to 2-adic structure.</p> <p>Building upon our preliminary framework, we present the results on fake cycles, the Hensel obstruction, and the Fake Cycle Vanishing Conjecture (FCVC), embedding them in the strengthened spectral framework. Together, these results provide the strongest evidence to date for uniqueness of equilibrium states in Collatz dynamics in the sense of Santana (2025).</p> <p>32 pages, 2 appendices, computational code included as ancillary files. With computational analysis by Claude (Anthropic) and Gemini (Google DeepMind).</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19587689 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Fake Cycles, Spectral Gap, and the BIP Property in Collatz Dynamics okabe, arata Collatz conjecture Markov shift Thermodynamic formalism <p>We study the Collatz conjecture through 2-adic arithmetic, thermodynamic formalism, and spectral theory of countable Markov shifts. The Syracuse map T(n) = (3n+1)/2^{v_2(3n+1)} is analyzed as a dynamical system on the 2-adic integers Z_2. We make four principal advances in this study.</p> <p>(1) Countable Markov shift structure. The inverse branches phi_s(y) = (2^s y - 1)/3, s = 1,2,3,..., generate a countable-state Markov shift (Sigma, sigma) whose alphabet is the set of step sizes {1,2,3,...}.</p> <p>(2) BIP condition. We verify that the singleton F = {1} witnesses the big images and preimages (BIP) property: for every state s >= 1, both the transitions 1 -> s and s -> 1 are admissible. This is established algebraically via the invertibility of 9 in Z_2 and confirmed computationally for s = 1, ..., 30.</p> <p>(3) Sarig's theorem on Z_2. Combining BIP with topological mixing, summable variations, and finite Gurevich pressure, we invoke Sarig (1999, 2003) to obtain a spectral gap for the Ruelle-Perron-Frobenius (RPF) operator L_gamma with potential psi_gamma = -gamma log |S| on the full space Z_2, for every gamma > 0. This yields a simple maximal eigenvalue, a unique Gibbs measure, and exponential decay of correlations.</p> <p>(4) Phase transition and certified gap. The spectral gap onset at gamma = 0 is first-order: gap(gamma, m) ≈ C(m) * gamma as gamma -> 0, with C(m) increasing in m. Computer-assisted certification using Bauer-Fike perturbation bounds with 50-digit arithmetic confirms the gap rigorously for m = 8, 10, 12. At m = 12, the eigenvalue ratio lambda_2 / lambda_1 = 1/sqrt(2), suggesting a deep connection to 2-adic structure.</p> <p>Building upon our preliminary framework, we present the results on fake cycles, the Hensel obstruction, and the Fake Cycle Vanishing Conjecture (FCVC), embedding them in the strengthened spectral framework. Together, these results provide the strongest evidence to date for uniqueness of equilibrium states in Collatz dynamics in the sense of Santana (2025).</p> <p>32 pages, 2 appendices, computational code included as ancillary files. With computational analysis by Claude (Anthropic) and Gemini (Google DeepMind).</p> |
| title | Fake Cycles, Spectral Gap, and the BIP Property in Collatz Dynamics |
| topic | Collatz conjecture Markov shift Thermodynamic formalism |
| url | https://doi.org/10.5281/zenodo.19587689 |