One-Dimensional Dynamics

This book is about real one-dimensional discrete dynamical systems. We consider continuous maps f: N → N where N is an interval or a circle and — if this leads to better results — we shall often assume that f is smooth.

This chapter is devoted to the study of invertible one-dimensional dynamical systems. In the later chapters we shall see that, although the non-invertible case is quite different, many techniques and theorems for analyzing these non-invertible systems find their roots in the invertible case. One of our aims in this book is to emphasize these similarities.

In this chapter we will discuss endomorphisms of the circle and of the interval from a combinatorial point of view. The aim is to develop an analogue to the topological description of circle homeomorphisms given in Section I.1. As in that section, the main ingredient here is symbolic dynamics and the structure to be considered is the order structure of the interval or of the circle.

In the last chapter we described the structure of the unimodal maps with negative Schwarzian derivative. The ingredients we used were: i) the combinatorial theory of Section II.3; ii) the finiteness of attractors; iii) the non-existence of wandering intervals. For ii) and iii) we used the assumption on the Schwarzian derivative.

In this chapter we want to study the typical behaviour of orbits. Up till now we have seen that the topological behaviour of interval maps is quite well understood: for example if f is a unimodal interval map with negative Schwarzian derivative and such that the fixed point on the boundary is repelling, then by Guckenheimer’s theorem, see Theorem III.4.1, there are three possibilities:

In this chapter we will discuss the renormalization techniques which were introduced in one-dimensional dynamics independently by Feigenbaum (1978), (1979) and Coullet and Tresser (1978) to explain some quantitative and universal phenomena appearing in bifurcations of one parameter families of unimodal maps. More precisely, let f _t be a full one parameter family of unimodal maps of the interval I = [−1, 1]. For instance, f _t may be the quadratic family. As we saw in Section II.5, because f _t is a full family, there exists an interval [a_l, b₁] in the parameter space such that for every t in this interval, f _t has a restrictive interval I_l,t = [p′(t),p(t)]of period 2 where p(t) is a fixed point of f _t and f _t(p′(t)) = p(t). Furthermore, f _t ² is a unimodal map from I_l,t into itself and the family [a _l, b ₁] ∋t↦f _t ² | I _1,t is again full. In particular, \(f_{b1}^2|{I_{1,b1}}\) is a surjective unimodal map and there is a parameter value \({\tilde b_1}\) ∈(a ₁, b₁) such that the critical point of \(f_{{{\tilde b}_1}}^2|{I_{1,{{\tilde b}_1}}}\) is a fixed point. Since this family of first return maps is again full, we can repeat the argument and we get, by induction, a decreasing sequence of intervals [a _n , b _n ] in the parameter space, and, for each t∈[a _n , b _n ] an interval I _n,t ⊂ I such that the first return map of f _t to I _{n, t} is a unimodal map which coincides with the restriction of \(f_t^{{2^n}}\) to this interval. Furthermore, \(t \mapsto f_t^{{2^n}}|{I_{n,t}}\) is a full family of unimodal maps. In particular, there exists \({\tilde b_n} \in [{a_n},{b_n}]\) such that the critical point of \({f_{\tilde b}}_n\) is periodic of period 2 ⁿ and f _t has zero topological entropy for \(t \leqslant {\tilde b_n}\). Let a _∞, be the limit of a _n when n → ∞. As we have seen before \({f_{a\infty }}\) has an attracting Cantor set and the dynamics of the restriction of \({f_{a\infty }}\) to this Cantor set is conjugate to the adding machine, see Section III.4.

The purpose of this appendix is to present some basic definitions, theorems and background material for this book. We assume the reader to be familiar with manifolds.