A Solution to the Monotonicity Problem for Unimodal Families

In this note we consider a collection C of one parameter families of unimodal maps of [0, 1]. Each family in the collection has the form {μf} where μ ∈ [0, 1]. Denoting the kneading sequence of μf by K(μf), we will prove that for each member of C, the map μ 7→ K(μf) is monotone. It then follows that for each member of C the map μ 7→ h(μf) is monotone, where h(μf) is the topological entropy of μf. For interest, μf(x) = 4μx(1 − x) and μf(x) = μ sin(πx) are shown to belong to C. introduction Metropolis, Stein and Stein were among the first, to my knowledge, to study what are now called finite kneading sequences. These were associated with super stable limit cycles of one parameter families of interval maps, which included μf(x) = 4μx(1 − x) and μf(x) = μ sin(πx) [2]. Computer studies strongly suggested a universal topological dynamics for a large class of such families, and many workers were quickly drawn to this fascinating field of study. In the 1980’s and 1990’s there was intense interest in the behavior of the logistic map (affinely modified) in the setting of one complex dimension. A central question was (essentially) under what circumstances would finite kneading sequences be monotone with the parameter, as this question was associated with the structure of the Mandelbrot set, among other things. This question was successfully addressed (in the special case of a real quadratic) in [1,5]. Here we address the question generally for a large class that includes the logistic map, and give a sufficient condition for the solution. All known proofs of this type apply to the case of a quadratic polynomial only and use complex analytic methods (holomorphic techniques) or depend on complex analysis (Compare [1], [5], [6], [7] [8]). These methods are not used here. Let I = [0, 1]. Consider the collection of parameterized maps {μf | μf : I → I} with μf at least C in x. Notice that since μf(x) is linear in μ, it is C in μ. Denote the single critical point c ∈ (0, 1) and scale the map so that f(c) = 1, requiring that f(0) = 0 = f(1). Then μf(c) = μ. Denote the n iterate of μf by f μ (x) = (μf) ◦ · · · ◦ (μf) ◦ (μf)(x), where the composition is n-fold. For any x ∈ I, the orbit of x is the set O(x) = {f μ (x)|n ≥ 0}. Associate with O(x) the word ω(x) = ω0ω1ω2 · · · with ωk ∈ {L,C,R} where words are formed as follows: Date: May 15, 2008. 2000 Mathematics Subject Classification. Primary 37E05, 54H20; Secondary 37B40.