Equivalent Conditions for Domination of $\mathrm{M}(2,\mathbb{C})$-sequences
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909466707886080 |
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| author | Sun, Chang Zhang, Zhenghe |
| author_facet | Sun, Chang Zhang, Zhenghe |
| contents | It is well known that a $\mathrm{SL}(2,\mathbb{C})$-sequence is uniformly hyperbolic if and only it satisfies a uniform exponential growth condition. Similarly, for $\mathrm{GL}(2,\mathbb{C})$-sequences whose determinants are uniformly bounded away from zero, it has dominated splitting if and only if it satisfies a uniform exponential gap condition between the two singular values. Inspired by [QTZ], we provide a similar equivalent description in terms of singular values for $\mathrm{M}(2,\mathbb{C})$-sequences that admit dominated splitting. We also prove a version of the Avalanche Principle for such sequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_15940 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equivalent Conditions for Domination of $\mathrm{M}(2,\mathbb{C})$-sequences Sun, Chang Zhang, Zhenghe Dynamical Systems 37D30 It is well known that a $\mathrm{SL}(2,\mathbb{C})$-sequence is uniformly hyperbolic if and only it satisfies a uniform exponential growth condition. Similarly, for $\mathrm{GL}(2,\mathbb{C})$-sequences whose determinants are uniformly bounded away from zero, it has dominated splitting if and only if it satisfies a uniform exponential gap condition between the two singular values. Inspired by [QTZ], we provide a similar equivalent description in terms of singular values for $\mathrm{M}(2,\mathbb{C})$-sequences that admit dominated splitting. We also prove a version of the Avalanche Principle for such sequences. |
| title | Equivalent Conditions for Domination of $\mathrm{M}(2,\mathbb{C})$-sequences |
| topic | Dynamical Systems 37D30 |
| url | https://arxiv.org/abs/2501.15940 |