11 - Unimodal Maps and Rigid Rotations

Summary

In this chapter we present results of [48] proving that, given any p ∈ [0,1]\ℚ there exists a unimodal map ƒ such that (S^1, R_ρ) is a factor of (ω(c, ƒ),ƒ) (recall Remark 3.3.2 for the definition of factor). One might ask whether one can obtain the stronger result of conjugacy? As S¹ is not homeomorphic to a Cantor set, and in this setting ω(c, ƒ) is indeed a Cantor set, a conjugacy is not possible. See [48, 47] for further details and results.

Chapter 3 and Sections 6.1, 7.2, and 8.3 contain background material for this chapter.

Adding Machines in Unimodal Maps

Given a unimodal map ƒ with turning point c and kneading map Q(k), we construct an adding machine (ω,P) from the sequence of cutting times {S_k}. In the event that lim_k→∞Q(k) = ∞, we have that (ω(c, ƒ),ƒ) is a factor of (Ω, P) (Theorem 11.1.15). The condition lim_k→∞Q(k) = ∞ is not required to define the adding machine (ω, P), but rather comes into play for the continuity of the map P. In this section we provide only the information on the adding machine (ω, P) needed to obtain Theorem 11.1.15; see Section 13.3 for a more detailed discussion of (ω,P).

Let S₀ < S₁ < S₂ < S₃,… be the sequence of cutting times for some unimodal map ƒ (recall it is always the case that S₀ = 1 and S₁ = 2). We do not assume that lim_k→∞Q(k) = ∞.