Renormalization in Lorenz maps -- completely invariant sets and periodic orbits
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866910579583614976 |
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| author | Cholewa, Łukasz Oprocha, Piotr |
| author_facet | Cholewa, Łukasz Oprocha, Piotr |
| contents | The paper deals with dynamics of expanding Lorenz maps, which appear in a natural way as Poincarè maps in geometric models of well-known Lorenz attractor. Using both analytical and symbolic approaches, we study connections between periodic points, completely invariant sets and renormalizations. We show that some renormalizations may be connected with completely invariant sets while some others don't. We provide an algorithm to detect the renormalizations that can be recovered from completely invariant sets. Furthermore, we discuss the importance of distinguish one-side and double-side preimage. This way we provide a better insight into the structure of renormalizations in Lorenz maps. These relations remained unnoticed in the literature, therefore we are correcting some gaps existing in the literature, improving and completing to some extent the description of possible dynamics in this important field of study. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_00110 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Renormalization in Lorenz maps -- completely invariant sets and periodic orbits Cholewa, Łukasz Oprocha, Piotr Dynamical Systems 37E05, 37B05, 37E20 The paper deals with dynamics of expanding Lorenz maps, which appear in a natural way as Poincarè maps in geometric models of well-known Lorenz attractor. Using both analytical and symbolic approaches, we study connections between periodic points, completely invariant sets and renormalizations. We show that some renormalizations may be connected with completely invariant sets while some others don't. We provide an algorithm to detect the renormalizations that can be recovered from completely invariant sets. Furthermore, we discuss the importance of distinguish one-side and double-side preimage. This way we provide a better insight into the structure of renormalizations in Lorenz maps. These relations remained unnoticed in the literature, therefore we are correcting some gaps existing in the literature, improving and completing to some extent the description of possible dynamics in this important field of study. |
| title | Renormalization in Lorenz maps -- completely invariant sets and periodic orbits |
| topic | Dynamical Systems 37E05, 37B05, 37E20 |
| url | https://arxiv.org/abs/2104.00110 |