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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2509.05815 |
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| _version_ | 1866918136811356160 |
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| author | Nowak-Kępczyk, Małgorzata |
| author_facet | Nowak-Kępczyk, Małgorzata |
| contents | We study discrete Laplacians on two-dimensional lattices under modular iterations, focusing on the emergence of nontrivial large-scale patterns. While purely binary or constant modular sequences quickly collapse into strict periodicity, the insertion of a single non-binary step k yields qualitatively new behavior. Through extensive computer-assisted exploration we identify a taxonomy of long-lived figures - rugs, quasi-carpets, and carpets - whose occurrence depends systematically on seed symmetry, neighborhood mask, and sequence structure. In particular, we show that mixed families of the form [2,k,2 to power s] can stabilize high-density carpets beyond the universal decay time characteristic of binary dynamics.
Our approach combines algebraic replication laws with large-scale simulations and density tracking, producing both theoretical conditions (periodicity via Lucas theorem, non-overlap criteria) and experimental evidence of persistent quasi-aperiodic architectures. The results highlight how minimal modifications in discrete local rules generate unexpectedly rich multiscale geometries, bridging rigorous analysis with computer-assisted discovery. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_05815 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stabilization and Regaining Periodicity in Modular Laplacian Dynamics Nowak-Kępczyk, Małgorzata Dynamical Systems 37B10, 05C75, 52C23 We study discrete Laplacians on two-dimensional lattices under modular iterations, focusing on the emergence of nontrivial large-scale patterns. While purely binary or constant modular sequences quickly collapse into strict periodicity, the insertion of a single non-binary step k yields qualitatively new behavior. Through extensive computer-assisted exploration we identify a taxonomy of long-lived figures - rugs, quasi-carpets, and carpets - whose occurrence depends systematically on seed symmetry, neighborhood mask, and sequence structure. In particular, we show that mixed families of the form [2,k,2 to power s] can stabilize high-density carpets beyond the universal decay time characteristic of binary dynamics. Our approach combines algebraic replication laws with large-scale simulations and density tracking, producing both theoretical conditions (periodicity via Lucas theorem, non-overlap criteria) and experimental evidence of persistent quasi-aperiodic architectures. The results highlight how minimal modifications in discrete local rules generate unexpectedly rich multiscale geometries, bridging rigorous analysis with computer-assisted discovery. |
| title | Stabilization and Regaining Periodicity in Modular Laplacian Dynamics |
| topic | Dynamical Systems 37B10, 05C75, 52C23 |
| url | https://arxiv.org/abs/2509.05815 |