The Existence and Structure of Rotational Systems in the Circle
By a rotational system, we mean a closed subset X of the circle, T=R/Z, together with a continuous transformation f:X→X with the requirements that the dynamical system (X,f) be minimal and that f respect the standard orientation of T. We show that infinite rotational systems (X,f), with the property...
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| Format: | Article |
| Language: | English |
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Wiley
2018-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2018/8752012 |
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| _version_ | 1832559066070646784 |
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| author | Jayakumar Ramanathan |
| author_facet | Jayakumar Ramanathan |
| author_sort | Jayakumar Ramanathan |
| collection | DOAJ |
| description | By a rotational system, we mean a closed subset X of the circle, T=R/Z, together with a continuous transformation f:X→X with the requirements that the dynamical system (X,f) be minimal and that f respect the standard orientation of T. We show that infinite rotational systems (X,f), with the property that map f has finite preimages, are extensions of irrational rotations of the circle. Such systems have been studied when they arise as invariant subsets of certain specific mappings, F:T→T. Because our main result makes no explicit mention of a global transformation on T, we show that such a structure theorem holds for rotational systems that arise as invariant sets of any continuous transformation F:T→T with finite preimages. In particular, there are no explicit conditions on the degree of F. We then give a development of known results in the case where Fθ=d·θmod1 for an integer d>1. The paper concludes with a construction of infinite rotational sets for mappings of the unit circle of degree larger than one whose lift to the universal cover is monotonic. |
| format | Article |
| id | doaj-art-30f0a22867014ce8891efa315d3d60b1 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-30f0a22867014ce8891efa315d3d60b12025-02-03T01:30:56ZengWileyAbstract and Applied Analysis1085-33751687-04092018-01-01201810.1155/2018/87520128752012The Existence and Structure of Rotational Systems in the CircleJayakumar Ramanathan0Department of Mathematics, Eastern Michigan University, Ypsilanti, MI 48197, USABy a rotational system, we mean a closed subset X of the circle, T=R/Z, together with a continuous transformation f:X→X with the requirements that the dynamical system (X,f) be minimal and that f respect the standard orientation of T. We show that infinite rotational systems (X,f), with the property that map f has finite preimages, are extensions of irrational rotations of the circle. Such systems have been studied when they arise as invariant subsets of certain specific mappings, F:T→T. Because our main result makes no explicit mention of a global transformation on T, we show that such a structure theorem holds for rotational systems that arise as invariant sets of any continuous transformation F:T→T with finite preimages. In particular, there are no explicit conditions on the degree of F. We then give a development of known results in the case where Fθ=d·θmod1 for an integer d>1. The paper concludes with a construction of infinite rotational sets for mappings of the unit circle of degree larger than one whose lift to the universal cover is monotonic.http://dx.doi.org/10.1155/2018/8752012 |
| spellingShingle | Jayakumar Ramanathan The Existence and Structure of Rotational Systems in the Circle Abstract and Applied Analysis |
| title | The Existence and Structure of Rotational Systems in the Circle |
| title_full | The Existence and Structure of Rotational Systems in the Circle |
| title_fullStr | The Existence and Structure of Rotational Systems in the Circle |
| title_full_unstemmed | The Existence and Structure of Rotational Systems in the Circle |
| title_short | The Existence and Structure of Rotational Systems in the Circle |
| title_sort | existence and structure of rotational systems in the circle |
| url | http://dx.doi.org/10.1155/2018/8752012 |
| work_keys_str_mv | AT jayakumarramanathan theexistenceandstructureofrotationalsystemsinthecircle AT jayakumarramanathan existenceandstructureofrotationalsystemsinthecircle |