Verification of mixing properties in two-dimensional shifts of finite type
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2011
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| _version_ | 1866909226050256896 |
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| author | Ban, Jung-Chao Hu, Wen-Guei Lin, Song-Sun Lin, Yin-Heng |
| author_facet | Ban, Jung-Chao Hu, Wen-Guei Lin, Song-Sun Lin, Yin-Heng |
| contents | The degree of mixing is a fundamental property of a dynamical system. General multi-dimensional shifts cannot be systematically determined. This work introduces constructive and systematic methods for verifying the degree of mixing, from topological mixing to strong specification (or strong irreducibility) for two-dimensional shifts of finite type. First, transition matrices on infinite strips of width $n$ are introduced for all $n\geq 2$. To determine the primitivity of the transition matrices, connecting operators are introduced to reduce the order of high-order transition matrices to yield lower-order transition matrices. Two sufficient conditions for primitivity are provided; they are invariant diagonal cycles and primitive commutative cycles of connecting operators. After primitivity is established, the corner-extendability and crisscross-extendability are used to demonstrate topological mixing. In addition, the hole-filling condition yields the strong specification. All mentioned conditions can be verified to apply in a finite number of steps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1112_2471 |
| institution | arXiv |
| publishDate | 2011 |
| record_format | arxiv |
| spellingShingle | Verification of mixing properties in two-dimensional shifts of finite type Ban, Jung-Chao Hu, Wen-Guei Lin, Song-Sun Lin, Yin-Heng Dynamical Systems The degree of mixing is a fundamental property of a dynamical system. General multi-dimensional shifts cannot be systematically determined. This work introduces constructive and systematic methods for verifying the degree of mixing, from topological mixing to strong specification (or strong irreducibility) for two-dimensional shifts of finite type. First, transition matrices on infinite strips of width $n$ are introduced for all $n\geq 2$. To determine the primitivity of the transition matrices, connecting operators are introduced to reduce the order of high-order transition matrices to yield lower-order transition matrices. Two sufficient conditions for primitivity are provided; they are invariant diagonal cycles and primitive commutative cycles of connecting operators. After primitivity is established, the corner-extendability and crisscross-extendability are used to demonstrate topological mixing. In addition, the hole-filling condition yields the strong specification. All mentioned conditions can be verified to apply in a finite number of steps. |
| title | Verification of mixing properties in two-dimensional shifts of finite type |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/1112.2471 |