On the stability of the penalty function for the $\mathbb{Z}^2$-hard square shift
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916851041173504 |
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| author | Oguri, Chihiro Shinoda, Mao |
| author_facet | Oguri, Chihiro Shinoda, Mao |
| contents | We investigate the stability of maximizing measures for a penalty function of a two-dimensional subshift of finite type, building on the work of Gonschorowski et al. \cite{GQS}. In the one-dimensional case, such measures remain stable under Lipschitz perturbations for any subshift of finite type. However, instability arises for a penalty function of the Robinson tiling, which is a two-dimensional subshift of finite type with no periodic points and zero entropy. This raises the question of whether stability persists in two-dimensional subshifts of finite type with positive topological entropy. In this paper, we address this question by studying a nearest-neighbor subshift of finite type satisfying the single-site fillability property. Our main theorem establishes that, in contrast to previous results, a penalty function of such a subshift of finite type remains stable under Lipschitz perturbations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_06958 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the stability of the penalty function for the $\mathbb{Z}^2$-hard square shift Oguri, Chihiro Shinoda, Mao Dynamical Systems 37B51, 37B10, 37B25 We investigate the stability of maximizing measures for a penalty function of a two-dimensional subshift of finite type, building on the work of Gonschorowski et al. \cite{GQS}. In the one-dimensional case, such measures remain stable under Lipschitz perturbations for any subshift of finite type. However, instability arises for a penalty function of the Robinson tiling, which is a two-dimensional subshift of finite type with no periodic points and zero entropy. This raises the question of whether stability persists in two-dimensional subshifts of finite type with positive topological entropy. In this paper, we address this question by studying a nearest-neighbor subshift of finite type satisfying the single-site fillability property. Our main theorem establishes that, in contrast to previous results, a penalty function of such a subshift of finite type remains stable under Lipschitz perturbations. |
| title | On the stability of the penalty function for the $\mathbb{Z}^2$-hard square shift |
| topic | Dynamical Systems 37B51, 37B10, 37B25 |
| url | https://arxiv.org/abs/2503.06958 |