Abstract and Applied Analysis

Volume 2018, Article ID 8752012, 11 pages

https://doi.org/10.1155/2018/8752012

## The Existence and Structure of Rotational Systems in the Circle

Department of Mathematics, Eastern Michigan University, Ypsilanti, MI 48197, USA

Correspondence should be addressed to Jayakumar Ramanathan; ude.hcime@htanamarj

Received 15 December 2017; Revised 8 March 2018; Accepted 20 March 2018; Published 3 June 2018

Academic Editor: Simeon Reich

Copyright © 2018 Jayakumar Ramanathan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

By a rotational system, we mean a closed subset of the circle, , together with a continuous transformation with the requirements that the dynamical system be minimal and that respect the standard orientation of . We show that infinite rotational systems , with the property that map 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, . Because our main result makes no explicit mention of a global transformation on , we show that such a structure theorem holds for rotational systems that arise as invariant sets of any continuous transformation with finite preimages. In particular, there are no explicit conditions on the degree of . We then give a development of known results in the case where for an integer . 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.

#### 1. Introduction

In what follows, denotes the unit circle with the standard orientation.

Suppose is a compact metric space and is a continuous transformation. The dynamical system is* minimal* if, for every , the orbitis dense in .

*Definition 1. *Let . A continuous transformation * preserves cyclic order* if, for any with distinct images, the arcs and have the same orientation.

*Definition 2. *A* rotational system* is a subset and a continuous transformation , with the properties that(i)the dynamical system is minimal,(ii)the transformation preserves cyclic order. In this situation, we will simply say that is rotational.

We need to recall one more definition, before stating the main theorem.

*Definition 3. *Let be continuous transformations of compact metric spaces for . The dynamical system is an* extension* of if there is a continuous, surjective function such that .

Our main result is as follows.

Theorem 4. *Let be a rotational system such that is an infinite, proper subset of . In addition, suppose that the continuous mapping has finite preimages, that is, for each . Then,*(i)*the dynamical system is an extension of an irrational rotation of the circle via a map that is compatible with the cyclic ordering on both and ;*(ii)*the function has preimages of cardinality one except at countably many points of . The preimages of these exceptional points have cardinality two and are the endpoints of gaps of the set in ;*(iii)* has a unique ergodic measure and is the standard Lebesgue measure on .*

*The angle of the rotation of in the preceding theorem is called the rotation number of .*

*Such systems are of particular interest when they arise as invariant subsets of a continuous mapping on the whole circle, . Recall that a closed subset of is invariant with respect to if . In this situation, we may set and consider the dynamical system . Such a system, , is a subsystem of .*

*Our main theorem has the following obvious corollary.*

*Corollary 5. Let be a continuous mapping with finite preimages. Suppose, moreover, that is a closed, infinite, proper subset of that is invariant with respect to . If is rotational, then all three conclusions of Theorem 4 hold.*

*In the case where is given by for an integer , Corollary 5 has an extensive history. Ideas related to the case were studied by Morse and Hedlund [1] in their work on Sturmian trajectories. The problem was taken up later by several authors, including Gambaudo et al. [2], Veerman [3, 4], Goldberg [5], Goldberg and Milnor [6], and Bullett and Sentenac [7]. The case was studied by Goldberg and Tresser [8], Blokh et al. [9], and Bowman et al. [10]. In sum, these works provide a complete characterization of the rotational subsystems, with rational and irrational rotation number, for the uniform cover of with positive degree.*

*In this paper, we point out that parts of the analysis of rotational systems with irrational rotation number can be done without explicit reference to an ambient transformation on the unit circle. This leads to a structure result for rotational subsystems of a wide class of continuous transformations , those with finite preimages.*

*The proof of Theorem 4 will be accomplished over the next two sections. An important step is to solve the functional equation found in Proposition 16. This equation is mentioned in the appendix in [8] where an analytical approach to the uniform cover case is sketched. (In particular, for the direction we are interested in, a solution to the functional equation is claimed but not given.) We provide a solution to this problem in our more general setting using the existence of an invariant measure together with the Mean Ergodic Theorem. Sections 4 and 5 then revisit the known -fold cover case (see [3, 4, 8, 9]) from our point of view.*

*For a given continuous , Theorem 4 and its corollary shed no light on how to determine which irrational numbers can be realized by rotational subsystems of . When , it is well known that there can be at most one such rotation number. In the uniform cover case with , every irrational number in can be achieved. In the last section, we consider degree mappings of that are monotonic with respect to the usual orientation. Mappings of this type can have intervals of constancy or an arbitrary number of fixed points (many of which may be attractive). In particular, they are not conjugate to the uniform cover case via a homeomorphism. We show that, with at most countable exceptions, every irrational rotation number can be realized when the degree is two. When the degree is larger than , there are examples for every irrational rotation number.*

*2. Structure of Rotational Subsets*

*For this section and the next, we will work under the assumptions of Theorem 4:(i) is a rotational system.(ii) is an infinite, proper subset of .(iii) has finite preimages.*

*Proposition 6. The set is a Cantor set; that is, it is a compact set that is perfect and has empty interior.*

*Proof. *Suppose is an isolated point of such a dynamical system. Minimality implies that the orbit, , is dense in . As a consequence, we have that for some positive integer . The set must then be finite as well as dense. Therefore, . This contradiction implies that cannot have any isolated points. Thus, is a perfect subset of ; that is, it is closed and has no isolated points.

Now, consider the possibility that contains a closed interval of positive length. Since is closed and a proper subset of , we may choose a closed interval of maximal length. The set cannot be left invariant by any power of . If were such a power, then would have a fixed point. The orbit of this fixed point would be a finite invariant subset of , which is impossible. Next, fix . Since the orbit of this point is dense and has a nonempty interior, there must be a positive integer such that . On the other hand, there must a point with . Let , , be the linear path in that connects to . Then connects with . But the range of is contained in . An argument using the intermediate value theorem shows that this contradicts the maximality of . We have shown that has empty interior.

*Remark 7. *The properties of the dynamical system used in the above result are that is minimal and that is an infinite, closed, proper subset of .

*Since is a proper subset of , we may select a point with . Parameterize by the map defined by This continuous bijection is orientation preserving, provided we orient in the standard way. is also a homeomorphism from to the open set . Set and (see Figure 1). Thus, is homeomorphic to and and are isomorphic dynamical systems. In particular, is an infinite, perfect compact subset of the open interval and is a minimal dynamical system.*