Twist like behavior in non-twist patterns of triods
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916541967106048 |
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| author | Bhattacharya, Sourav Yadav, Ashish |
| author_facet | Bhattacharya, Sourav Yadav, Ashish |
| contents | We prove a sufficient condition for a \emph{pattern} $π$ on a \emph{triod} $T$ to have \emph{rotation number} $ρ_π$ coincide with an end-point of its \emph{forced rotation interval} $I_π$. Then, we demonstrate the existence of peculiar \emph{patterns} on \emph{triods} that are neither \emph{triod twists} nor possess a \emph{block structure} over a \emph{triod twist pattern}, but their \emph{rotation numbers} are an end point of their respective \emph{forced rotation intervals}, mimicking the behavior of \emph{triod twist patterns}. These \emph{patterns}, absent in circle maps (see \cite{almBB}), highlight a key difference between the rotation theories for \emph{triods} (introduced in \cite{BMR}) and that of circle maps. We name these \emph{patterns}: ``\emph{strangely ordered}" and show that they are semi-conjugate to circle rotations via a piece-wise monotone map. We conclude by providing an algorithm to construct unimodal \emph{strangely ordered patterns} with arbitrary \emph{rotation pairs}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18648 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Twist like behavior in non-twist patterns of triods Bhattacharya, Sourav Yadav, Ashish Dynamical Systems Primary 37E05, 37E15, 37E25, 37E40, Secondary 37E45 We prove a sufficient condition for a \emph{pattern} $π$ on a \emph{triod} $T$ to have \emph{rotation number} $ρ_π$ coincide with an end-point of its \emph{forced rotation interval} $I_π$. Then, we demonstrate the existence of peculiar \emph{patterns} on \emph{triods} that are neither \emph{triod twists} nor possess a \emph{block structure} over a \emph{triod twist pattern}, but their \emph{rotation numbers} are an end point of their respective \emph{forced rotation intervals}, mimicking the behavior of \emph{triod twist patterns}. These \emph{patterns}, absent in circle maps (see \cite{almBB}), highlight a key difference between the rotation theories for \emph{triods} (introduced in \cite{BMR}) and that of circle maps. We name these \emph{patterns}: ``\emph{strangely ordered}" and show that they are semi-conjugate to circle rotations via a piece-wise monotone map. We conclude by providing an algorithm to construct unimodal \emph{strangely ordered patterns} with arbitrary \emph{rotation pairs}. |
| title | Twist like behavior in non-twist patterns of triods |
| topic | Dynamical Systems Primary 37E05, 37E15, 37E25, 37E40, Secondary 37E45 |
| url | https://arxiv.org/abs/2412.18648 |