Rotation intervals for a family of degree one circle maps

Abstract

Abstract. Let / be a C° circle map of degree one with precisely one local minimum and one local maximum, and let [p_(/), p+(/)] be the interval of rotation numbers of/ We study the structure of the function p(A) = p+(.R A°/ ), where R k is the rotation through the angle A. IntroductionThe rotation interval for a degree one circle map was first denned by Newhouse, Palis and Takens in [9] and was subsequently shown by Ito ([6]) to be closed. Newhouse, Palis and Takens also showed that if / varies continuously in the C°t opology then p_(/) and p+(/) vary continuously also. In another paper ([7]), Ito showed that if p + (f) e R\Q, then p + (/? A °f) > p + (/) for all A > 0.In this paper we study the behaviour of the function p+(/? A <•/) near rational values. There are four main theorems, Theorems 2.5, 3.5, 3.7, and 3.8. The first deals with the persistence of p+(A) at rational values in any C°-continuous family of continuous circle maps of degree one. On the assumption that f k is never a homeomorphism it is shown that if p+(/ A )eQ then there is an e > 0 such that either P+(U)^P+(fx) for all fi€(X-e,\ + e) or else p+(/ M )=sp+(/ A ) for all p.e (A-e, A + e). This theorem is a slight generalization of results obtained by Bamon, Malta and Pacifico in [2], and it serves to set the stage for the other two main theorems.Theorems 3.5, 3.7, and 3.8 deal with a more specific family R x °f. In addition they require that we impose several differentiability conditions on / Specifically we assume that / is C 3 with precisely two critical points, a quadratic local minimum and a quadratic local maximum, and that / has a negative Schwarzian derivative. If b>iv all these conditions are satisfied, for example, by the function family x + b sin (2irx) (modi). One further condition, also required for these theorems, ensures that p_(/? A °f)<P+(R\ °f) for all A. The function x + b sin(2wx) (mod 1) satisfies this last condition at least when b>£.To state the theorems we need a little number theory. For any rational number piq with (p, q) = 1 there is an associated rational number defined as follows: 5 is the smallest positive integer such that sp +1 = rq