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2018 The Existence and Structure of Rotational Systems in the Circle
Jayakumar Ramanathan
Abstr. Appl. Anal. 2018: 1-11 (2018). DOI: 10.1155/2018/8752012

Abstract

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 F θ = d · θ mod 1 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.

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Jayakumar Ramanathan. "The Existence and Structure of Rotational Systems in the Circle." Abstr. Appl. Anal. 2018 1 - 11, 2018. https://doi.org/10.1155/2018/8752012

Information

Received: 15 December 2017; Revised: 8 March 2018; Accepted: 20 March 2018; Published: 2018
First available in Project Euclid: 11 July 2018

zbMATH: 06929597
MathSciNet: MR3816084
Digital Object Identifier: 10.1155/2018/8752012

Rights: Copyright © 2018 Hindawi

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