Kneading Theory for a Family of Circle Maps with One Discontinuity

(3) F (x+ 1) = F (x) + 1 for all x ∈ R. For a map F ∈ C and for each a ∈ Z we set F (a) = limx↓a F (x) and F (a) = limx↑a F (x). In view of (3) we have F (a ) = F (0) + a and F (a) = F (0) + a. Note that the exact value of F (0) is not specified. Then in what follows we consider that F (0) is either F (0) or F (0−), or both, as necessary. Since every map F ∈ C has a discontinuity in each integer, the class C can be considered as a family of liftings of circle maps with one discontinuity. The maps of class C appear in a natural way in the study of many branches of dynamics. The simplest example of such maps is the family x → βx + α, which plays an important role in ergodic theory (see [H]). The case α = 0 gives the famous β-transformations (see [R]). Also, the class C contains the class of the Lorenz-Like maps which has been studied by several authors (see [ALMT], [G], [GS], [Gu], [HS], [S]). The aim of this paper is to extend the kneading theory developed in [AM] for continuous maps of the circle of degree one to class C, to obtain a characterization of the rotation interval of a map in terms of its kneading sequences. From this characterization we shall obtain models with maximum and minimum entropy