PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 3, March 1998, Pages 881–885 S 0002-9939(98)04420-7 FIXED POINTS OF THE BUCKET HANDLE JAN M. AARTS AND ROBBERT J. FOKKINK (Communicated by James West) Abstract. If a homeomorphism on the bucket handle has an invariant com- posant, it has a fixed point in that composant. It follows that a homeomor- phism on the bucket handle has at least two fixed points. Our methods apply to general Knaster continua. 1. Introduction Mahavier has asked whether every homeomorphism of the standard Knaster continuum or bucket handle has at least two fixed points [3, p 384, Problem 120]. In this paper we give a positive answer to this question. Our proof makes use of the fact that the 2-solenoid is a branched covering space of the bucket handle. The structure of the isotopy class group of the solenoid is also involved in the proof. See Section 2 for details. The proof is presented in Section 3. In the remainder of this section we recall some notation. The solenoid and bucket handle will be represented as limits of inverse sequences. We consider inverse sequences ··· f −→ X f −→ X f −→ X. (1) The shift σ of the inverse limit is defined by σ ( 〈x n 〉 n ) = 〈x n+1 〉 n . The bucket handle is the inverse limit of the sequence (1) with X = I , the unit interval, and f = τ , the tent map which is defined by τ (x) = min{ 2x, 2 − 2x }. The bucket handle is denoted by K and the shift on K is denoted by σ K . Since the tent map has two fixed points, so does the shift σ K . The sequence 〈0〉 n in K is called the initial point and is denoted by 0 K . The initial point is topologically unique. Thus the initial point is a fixed point of every homeomorphism of the bucket handle. Mahavier’s question is: does there exist yet another fixed point? The circle R/Z is denoted by C. Via the quotient map C is identified with [0, 1). The circle C has the structure of an additive topological group; the sum of x and y is x + y (mod 1). The 2-solenoid, or just solenoid, is the inverse limit of the sequence (1) with X = C and f = δ, the doubling map which is defined by δ(x)=2x (mod 1). The solenoid is denoted by S and the shift on S is denoted by σ S . The solenoid is a compact abelian group. The group operation is coordinatewise addition. The zero element is 〈0〉 n . It is denoted by 0 S . Received by the editors December 15, 1995. 1991 Mathematics Subject Classification. Primary 54H25, 54F15; Secondary 54H11. Key words and phrases. Fixed point, bucket handle, solenoid. c 1998 American Mathematical Society 881 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use