proceedings of the american mathematical society Volume 118, Number 1, May 1993 PIECEWISE LINEAR DISCONTINUOUS DOUBLE COVERINGSOF THE CIRCLE ROZA GALEEVA AND CHARLES TRESSER (Communicated by Charles Pugh) Abstract. In his study of a particular Lorenz-like semiflow, S. F. Kennedy introduced a two-parameter family of endomorphisms of the circle with two marked points. These are piecewise affine double coverings of the circle with a pair of discontinuities, which all have topological entropy log 2 . We answer the question Kennedy raised about when two such maps are topologically conjugate. 1. Introduction and statement of the results In the wake of Williams's work on the Lorentz equations [7], Kennedy has described the semiflows on the branched 2-manifold W represented in Figure 1 (on the next page), where the sketch of phase portrait illustrates the main features of these semiflows [3]. Following backward the branches of the stable manifold of O, one gets two first intersection points at P and Q with the circle C. The circle C with its two marked points is a natural section of this semiflow, so the topological dynamics of the semiflow can be captured by studying K-maps, i.e., double coverings of the circle with two marked points where the map can be discontinuous, and such that each arc between the marked points is sent to the full circle less one point. In this note, we shall restrict ourselves to the case of expanding A"-maps with a constant factor 2 on the two arcs of continuity. Opening the circle at one marked point to form the interval [0, 1) with a marked point at \ means that we shall examine the two-parameter family f(a,bf- [0> U ~* [0> 1)» where f(a,b)\[oAI2)(x) = (2x + a) modi and •f(«.*)l[i/2, i)(-x) = (2x + b-\) mod 1. For simplicity, we shall also denote by {fia,b)} the corresponding family of K-maos. Following a question raised by Kennedy in [3, 4] we shall more pre- cisely describe conditions on the parameters, which are necessary and sufficient for two A^-maps in this family to be K-topologically conjugate, i.e., be the same up to a continuous change of coordinates, which preserves the set of marked points. Received by the editors August 19, 1991. 1991 Mathematics Subject Classification. Primary 58F20, 58F14. Key words and phrases. Kneading theory, topological conjugacy, branched manifolds. ©1993 American Mathematical Society 0002-9939/93 $1.00+ $.25 per page 285 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use