Skip to main content

Section 2.6 Critical orbits

A critical point of \(f\) is simply a point where the derivative of \(f\) is zero and the orbit of a critical point is called a critical orbit. As it turns out, critical orbits have an outsized influence on the global behavior of the dynamics of \(f\) - a fact due largely to the following theorem:

Note that this is really a theorem of complex dynamics. There is an analogous statement for real dynamics but its statement is much more complicated and takes us a bit farther astray than we want or need. This is a great example of complex analysis being, in some ways, more elegant than real analysis.

using Theorem 2.6.1 makes it easy to find attractive orbits.

Let \(f_c(x) = x^2 - 1.625\text{.}\) Find all attractive or super-attractive orbits of \(f\text{.}\)

Since \(f'(x)=0\text{,}\) \(f\) has exactly one critical point, namely \(x=0\text{.}\) Thus we can find any attractive behavior by iterating from zero.

Sure looks like we've found an orbit of period 5. Furthermore, this is necessarily the only periodic oribit.

It's worth pointing out that it is essential that we consider all complex critical points. It's easy to see, for example, that zero is an attractive fixed point for \(f(x)=\frac{1}{2}x+x^3\text{.}\) The derivative of \(f\) is \(f'(x)=\frac{1}{2}+3x^2\) which has no real roots. There are two complex roots of \(f'\text{,}\) though, that are attracted to the origin.

Nontheless, Theorem 2.6.1 is a powerful tool for understanding real iteration when the critical points are real and we'll use this to powerful effect in the next section.