As mentioned at the end of section one a periodic point for \(f\) of period \(n\) is a fixed point of \(f^n\text{.}\) Treating the points of a periodic orbit this way allows us to extend the classification as fixed points to periodic orbits.

Definition2.5.1

Let \(f:\mathbb R \to \mathbb R\) be continuously differentiable and suppose that \(x_0\in\mathbb R\) is a periodic point of \(f\) with period \(n\text{.}\) Let \(F=f^n\text{.}\) We classify \(x_0\) and its orbit as

attractive, if \(|F'(x_0)| < 1\text{,}\)

super-attractive, if \(F'(x_0) =0 \text{,}\)

repulsive or repelling, if \(|F'(x_0)| > 1\text{,}\) or

neutral, if \(|F'(x_0)| = 1\text{,}\)

The number \(F'(x_0)\) is called the multiplier of the orbit. If, in the attractive case, the multiplier is zero, we say that the orbit is super-attractive.

There is a nice characterization of the multiplier of an orbit that allows us to compute it without explicitly computing a formula for \(f^n\text{.}\)

Note that the only way the product in Lemma 2.5.2 is zero, is if one of the terms is zero. This yields the following corollary.

Corollary2.5.3

A periodic orbit is super-attracting if and only if it contains a critical point.

Example2.5.4

Let \(f(x) = x^2-1\text{.}\) Note that \(f(0)=-1\) and \(f(-1)=0\) so that \(0\to1\to0\) forms an orbit of period 2. To see if this orbit is attractive, we examine

Note that \(F'(0)=0\) and \(F'(-1)=0\text{;}\) thus, the orbit is super-attractive.

The plots of \(f\) and \(f^2\text{,}\) together with \(y=x\text{,}\) are shown in figure Figure 2.5.5. Note that \(f\) has two fixed points shown in red. They can found by solving the equation \(x^2-1=x\) and they are both repulsive under iteration of \(f\text{.}\) The two super-attractive orbits of \(f^2\) are shown in green.