Complex dynamics (i.e., the theory of dynamical systems on the complex plane) is a rich area in which powerful techniques from complex analysis are available. In our presentation, we try to avoid the more involved techniques as much as possible. Therefore, we work in settings that allow simplified definitions and theorems yet still allow for challenging results. For example, our definition of Julia sets (Section 14.1) is only valid for polynomials, not for arbitrary analytic functions on ℂ. Comprehensive introductions can be found in [9, 50, 53, 64, 118, 121].

We only discuss quadratic polynomials on ℂ because they are the straightforward, and the most frequently studied, complexifications of unimodal maps. Our goal is to introduce and study symbolic dynamics of them. Just as in the real case, symbolic approaches give a lot of information on the various dynamical behaviors that the system can exhibit. In principle, itineraries can be defined for complex quadratic maps just the same as for real unimodal maps. The difficulty is that the Julia set, the set on which interesting dynamics take place, is much more complicated than the interval. Instead of just “left of the critical point” and “right of critical point,” we need to be much more inventive to decide on which side of the critical point points lie. To make this decision, we will introduce the notion of external rays and external angles. At that point, we will want to determine the external angle of a given point in the Julia set. This is the contents of Section 14.1.

Given a compact metric space (X, d) and a map ƒ : X →X (not necessarily continuous) one asks: How many orbits does the dynamical system ƒ : X → X have? Of course, there are as many orbits as there are points, because every point has its own orbit. However, in many cases, these orbits behave in the same way. For example, take ƒ : S1 – S1 a rigid circle rotation (recall Definition 8.1.2) and x, y ∈ S1. If x and y are ε apart, then ƒ(x) and ƒ(y) are also ε apart, and for every So the orbits of x and y behave in essentially the same way.

How many essentially different orbits does a dynamical system have? This depends on what we think is “essentially different”; however, it is reasonable to say that orb(a:) and orb(y) are at least ε apart if there exists i such that If “essentially different” means “at least ε apart,” then a rigid circle rotation has “1/ε essentially different orbits.” A dynamical system with sensitive dependence, on the other hand, has infinitely many essentially different orbits: In every neighborhood of every x, there is a y that eventually gets e apart from x, so x and y have essentially different orbits.

If x and y are very close together, it could take many iterations before (or if) we find. It is therefore more useful to compute how many essentially different orbits there are up to the nth iteration.

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.

In this chapter we present results of [48] proving that, given any p ∈ [0,1]\ℚ there exists a unimodal map ƒ such that (S1, Rρ) 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 {Sk}. In the event that limk→∞Q(k) = ∞, we have that (ω(c, ƒ),ƒ) is a factor of (Ω, P) (Theorem 11.1.15). The condition limk→∞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 limk→∞Q(k) = ∞.

If asked what is the measure of the interval (a, b) ℝ, we reply b-a without thinking. Indeed, the Lebesgue measure of any interval in ℝ is its length; sets of the form (∞, a] or (b, ∞) have infinite measure, while bounded intervals have finite measure. But what about a set that is much more irregular, such as the Middle Third Cantor set defined in the previous chapter? How do we compute its size? This is the question that Henri Lebesgue addressed in his Ph.D. thesis in 1902. His goal was to come up with a tool for integrating functions that were horribly discontinuous, unbounded, or both, functions not covered by the classical Riemann integration theory. In order to reach that goal, he designed a simple means for determining the “length” or measure of a set of real numbers that is not necessarily the union of intervals. We present the basic ideas here of Lebesgue measure on IR and refer to measure theory texts for generalizing Lebesgue's ideas to other spaces. We will return to the ideas of integration in the chapter on measurable dynamics (Chapter 4). See [55, 70] for further discussion. Material from this chapter is used in Chapter 4.

Basics of Lebesgue Measure on ℝ

There are many ways one could generalize length, and perhaps this is why measure theory seems like a complicated subject to many. In addition, there is no reasonable way to preserve the desirable properties of length and have it work on every subset of ℝ, as the example in the next section reveals.

Following Lebesgue's simple approach to the subject, our starting point is that the measure of an interval should be its length.

Given a unimodal map ƒ: [0, 1] → [0, 1], we are interested in closed invariant subsets E ⊂ [0, 1] such that the restriction of ƒ to E (denoted f|E) is an onto homeomorphism (recall Definition 1.1.37). If such an E were a finite set, it would consist of a finite number of periodic orbits. Hence, we are interested in the case when E is not finite. Chapter 3 and Sections 6.1, 6.2, and 11.1 contain background material for this chapter.

Our interest in this dynamical behavior is motivated by the following example and question. In Chapter 5 we investigated the 2∞ map, g*, from the unimodal logistic family ga{x) - ax{1 – x). We found that ω(c, g*) was minimal and Cantor, and that g*|ω(c,g*) is an onto homeomorphism (recall Exercises 3.2.5, 5.1.9, and 5.4.7). There are many other maps in the logistic family such that:

1. ω(c, ga) is minimal.

2. ω(c, ga) is a Cantor set.

3. ga|ω(c,ga) is an onto homeomorphism.

In fact, any infinitely renormalizable map in this family has these three properties; moreover, the Lebesgue measure of such an ω(c,ga) is zero [78, 115]. One then asks, are there nonrenormalizable maps with these three properties? If some ga has an attracting periodic orbit, then w(c,ga) is precisely that orbit. Hence, we are interested in maps from the logistic family that are not renormalizable and which do not have an attracting periodic orbit. Any such map is topologically conjugate to a symmetric tent map (recall Theorem 3.4.27 and see [115, 39]).

We frequently consider the long-term behavior of a dynamical system on a large set of orbits without knowing the behavior of every orbit. In particular, Poincaré proved a type of recurrence for all orbits except for those lying in a set of measure 0 for dynamical systems. This result is presented below and contrasts with the recurrence definitions and results of Section 3.5. This is referred to as the qualitative theory of dynamics since frequently, even though one can predict what the orbits will do on a set of full measure, it is not known precisely what will happen at even a single point! We introduce the ideas behind measurable dynamics and ergodic transformations in this chapter but refer to texts on ergodic theory such as [80, 137, 168].

The reader should be familiar with the material in Chapter 2 before beginning this chapter. The notion of a measurable isomorphism, introduced in Section 4.2, is used in Chapter 12; otherwise, this chapter is not used elsewhere in the text.

Preliminaries

We restrict our setting to the examples of interest in this book. As before, f: I → I denotes a map of a compact interval, that is, I = [a, b]. We assume in addition that I is endowed with the Lebesgue measure structure restricted to I, so we denote the measurable sets as and normalized Lebesgue measure as mI; that is, if I = [a,b], then m(I) = b a and mI(A) = m(A)/(b a) for all.