Chapter 4 - Interval maps

Summary

In this chapter we shall concentrate on the special case of continuous maps on the closed interval I = [0, 1]. This level of specialization allows us to prove some particularly striking results on periodic points and topological entropy.

Fixed points and periodic points

Let T : I → I be a continuous map of the interval I = [0, 1] to itself. Recall that a fixed point x ∈ I satisfies Tx = x and that a periodic point (of period n) satisfies Tⁿx = x. We say that x has prime period n if n is the smallest positive integer with this property (i.e. T^kx ≠ x for k = 1,…, n – 1).

For interval maps a very simple visualization of fixed points exists. We can draw the graph G_T of T : I → I and the diagonal D = {(x, x) : x ∈ I}.

Lemma 4.1. The fixed points Tx = x occur at the intersection points (x, x) ∈ G_T ∩ D (see figure 4.1).

Similarly, if for n ≥ 2 we look for intersections of the graph (of n-compositions Tⁿ : I → I) with the diagonal D then the intersection points (x, x) ∈ G_T ∩ D are periodic points of period n.

Lemma 4.2. Assume that we have an interval J ⊂ I with T(J) ⊃ J; then there exists a fixed point Tx = x ∈ J.