In this chapter we present two combinatoric tools, Hofbauer towers and kneading maps, developed by Hofbauer and Keller [87]. These tools allow combinatoric characterizations (Section 6.2) for certain dynamical behaviors of unimodal maps that will prove useful in the remaining chapters. We next investigate shadowing for symmetric tent maps and identify, using these tools, a combinatoric characterization for shadowing in this family of maps. Lastly, we use these tools to construct examples of unimodal maps where ω(c, f) = [c2, c1] or ω(c, f) is a Cantor set.

The reader should be familiar with the material from Sections 3.1 through 3.5 before working in this chapter. Section 6.1 is needed for Section 9.3 and Chapter 11. Both Sections 6.1 and 6.2 are required for Chapters 10 and 13. Sections 6.3 and 6.4 are not used elsewhere in the text.

Hofbauer Towers and Kneading Maps

Recall that a continuous map f : [0,1] → [0,1] is called unimodal if there exists a unique turning or critical point, c, such that is increasing, is decreasing, and f(0) = f(1) = 0. As before, ci = fi(c) for i ≥ 0.

We assume c2 < c < c1 and c2 ≤ c3; otherwise, the asymptotic dynamics are uninteresting. Note that the interval [c2, c1] is invariant, that is, f maps [c2)c1] onto itself. Hence, to study the asymptotic dynamics of the system, it suffices to restrict our attention to [c2)c1]. We call [c2)c1] the core of the map f.