A Monotonicity Theorem for the Family f a (x) = a - x 2

Abstract

Let fa(x) = ax2 1x E [-i I 1+4aI + IVFT4] and a E [0, 2]. It iB proved that if fa has a periodic orbit of odd period n and if b > a, then fb has a periodic orbit of period n. This is equivalent to the corresponding result for the function family gx(x) = Xx(1 x), x E [0, 1], X E [0,4]. Among one-dimensional discrete dynamical systems those which are simplest and best understood are the unimodal functions of an interval into itself. These functions can be analysed with the help of Milnor and Thurston's kneading theory [7, 3] which associates to the function f a formal power series v(f) E Z[[t]] called variously the kneading invariant or (in [7]) the kneading determinant. The kneading invariant almost completely specifies the periodic orbits and completely specifies the topological entropy of f [7, 3, 4, 5]. If in addition the function f has negative Schwarzian derivative, then the family of unimodal functions with the same kneading invariant either constitutes a conjugacy class of such functions or else is the union of two conjugacy classes [2]. Thus the kneading invariant almost characterizes f up to conjugacy. Milnor and Thurston showed that in a C' family of unimodal functions fa the topological entropy is a continuous function of the parameter a (see [7]). The bifurcations in such a family are completely understood, and occur in a versal pattern that is related to Sarkovskii's ordering of the natural numbers [6]. However, there is no function family fa for which it is known that V(fa) is a monotone function of a or that the topological entropy h(fa) is monotone in the parameter. Milnor and Thurston conjecture that the family of functions gx(x) =Xx(l -x), xe [0,1], has this property. Indeed computer calculations seem to indicate that h(gx) is monotone increasing. In this paper we present a modest result in this direction. It is known that the topological entropy increases monotonically with the appearance of periodic orbits of new periods. We prove the following. THEOREM. If g> has a periodic orbit of odd period n and if q > X, then g9 has a periodic orbit of period n.