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A CHARACTERIZATION OF THE UNIQUELY ERGODIC ENDOMORPHISMS OF THE CIRCLE

By Lluis Alseda, Francesc Manosas, Wieslaw Szlenk and Communicated Charles Pugh

Abstract

Abstract. We characterize the uniquely ergodic endomorphisms of the circle in terms of their periodic orbits. Let /: S1- • S1 be a continuous endomorphism of the circle S1. Denote by F: R — ► R its lifting to the universal covering space. Then, for all x £ R, F satisfies F(x + 1) = F(x) + k for some k £ Z. The number k is called the degree of / (when necessary this number will also be called the degree of F). Denote by Sf the class of all liftings of continuous endomorphisms of the circle and by SCX the class of maps from J? having degree 1. Let F £ S?x. For each x e R we define the rotation number of x, denoted by Pf(x) , as (see [NPT]) limsup,,^ ^ Fn(x)/n. From [I] it follows that the set LP = {pf(x) : x £ R} is a closed interval (or perhaps a point). This interval is called the rotation interval of F. In what follows, whenever Lf degenerates to a point it will be denoted by p(F). In such a case we shall talk about the rotation number of F. It is well known that when F is nondecreasing, it has degenerate rotation interval. A continuous endomorphism of the circle / is said to be uniquely ergodic if there exists a unique /-invariant probability measure on S1. From [H] it follows that every homeomorphism of the circle of degree one with irrational rotation number is uniquely ergodic. The aim of this paper is to extend this result to the endomorphisms of the circle. In fact, we shall give necessary and sufficient conditions for an endomorphism of the circle to be uniquely ergodic. Our main result is the following Theorem. A circle endomorphism f is uniquely ergodic if and only if it has at most one periodic orbit. To prove the above theorem we shall use the following results. For them we have to introduce some notation

Topics: 0002-9939/93 $1.00+ $.25 per page
Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.309.138
Provided by: CiteSeerX
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