Topological Stability and Shadowing in Uniform Transformation Semigroups Modulo an Ideal
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915460474208256 |
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| author | Shirazi, F. Ayatollah Zadeh Hakimi, E. Hosseini, A. Tajbakhsh, Kh. |
| author_facet | Shirazi, F. Ayatollah Zadeh Hakimi, E. Hosseini, A. Tajbakhsh, Kh. |
| contents | In this paper, we introduce and analyze several key dynamical properties-namely shadowing modulo an ideal, expansivity modulo an ideal, and topological stability modulo an ideal-within the framework of uniform transformation semigroups. Given an ideal $\mathcal I$ on semigroup $T$, we investigate the interplay between these properties in compact Hausdorff transformation semigroup $(T,X,\mathfrak{X})$. Our main result establishes that if a compact Hausdorff transformation semigroup exhibits the shadowing property modulo $\mathcal I$ and is expansive modulo $\mathcal I$ then it is also topologically stable modulo $\mathcal I$. This extends known stability theorems in classical topological dynamics to the setting of ideal-constrained dynamics. Additionally, we explore the relationship between shadowing modulo $\mathcal I$ and the conventional shadowing property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17257 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Topological Stability and Shadowing in Uniform Transformation Semigroups Modulo an Ideal Shirazi, F. Ayatollah Zadeh Hakimi, E. Hosseini, A. Tajbakhsh, Kh. Dynamical Systems Geometric Topology 54H15, 37B52 In this paper, we introduce and analyze several key dynamical properties-namely shadowing modulo an ideal, expansivity modulo an ideal, and topological stability modulo an ideal-within the framework of uniform transformation semigroups. Given an ideal $\mathcal I$ on semigroup $T$, we investigate the interplay between these properties in compact Hausdorff transformation semigroup $(T,X,\mathfrak{X})$. Our main result establishes that if a compact Hausdorff transformation semigroup exhibits the shadowing property modulo $\mathcal I$ and is expansive modulo $\mathcal I$ then it is also topologically stable modulo $\mathcal I$. This extends known stability theorems in classical topological dynamics to the setting of ideal-constrained dynamics. Additionally, we explore the relationship between shadowing modulo $\mathcal I$ and the conventional shadowing property. |
| title | Topological Stability and Shadowing in Uniform Transformation Semigroups Modulo an Ideal |
| topic | Dynamical Systems Geometric Topology 54H15, 37B52 |
| url | https://arxiv.org/abs/2508.17257 |