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Section 2.1 Basic notions

We begin with some of the most fundamental definitions and examples. While these definitions are stated for real functions, many of them extend quite easily to other contexts.

Definition 2.1.

Let \(x_0\in\mathbb R\) be an initial point and define a sequence \((x_n)\) recursively by \(x_{n+1} = f(x_n)\text{.}\) This sequence is called the orbit of \(x_0\) under iteration of \(f\text{.}\)

Some orbits don't move; they are fixed.

Definition 2.2.

A point \(x_0 \in \mathbb R\) is a fixed point of \(f\) if \(f(x_0)=x_0\text{.}\)

Sometimes an orbit might return to the original starting point.

Definition 2.3.

Suppose that the orbit \((x_n)\) satisfies

\begin{equation*} x_0 \to x_1 \to x_2 \cdots \to x_{n-1} \to x_0 \end{equation*}

and \(x_n=x_0\text{.}\) Such an orbit is called a periodic orbit and the points themselves are called periodic points. If \(x_k \neq x_0\) for \(k=1,2,\ldots,n-1\text{,}\) then \(n\) is called the period of the orbit.

Note that a fixed point is a periodic point with period one.

Sometimes, the orbit of a non-periodic point might land on a periodic orbit.

Definition 2.4.

If the zeroth term \(x_0\) of an orbit \((x_n)\) is not periodic but \(x_n\) is periodic for some \(n\text{,}\) then \(x_0\) and its orbit are called pre-periodic.

Example 2.5.

Let \(f(x) = 2x+1\text{.}\) We could compute a few terms of the orbit of \(x_0=1\) by direct comutation:

\begin{equation*} 1\to3\to7\to14\to31\to\cdots \end{equation*}

It's not too hard to guess that a general formula for \(n^{\text{th}}\) term might be \(x_n = 2^{n+1}-1\text{.}\) You should check this for the first few values of \(n\text{.}\)

It's easy to check that \(x_0=-1\) is a fixed point. It seems unlikely that there are other fixed points or periodic orbits of any period.

Example 2.6.

Let \(f(x) = x^2-1\text{.}\) Then zero is a periodic point and one is a pre-periodic point, as the reader may easily verify.

To find a fixed point, we can simply set \(f(x)=x\) and solve the resulting equation. In this case, we get

\begin{equation*} x^2-1=x \: \text{ or } \: x^2-x-1 = 0. \end{equation*}

We can then apply the quadratic formula to find that

\begin{equation*} x=\frac{1\pm\sqrt{5}}{2} \end{equation*}

are both fixed.

Often, it helps to express these ideas in terms of composition of functions. We denote the \(n\) fold composition of a function with itself by \(f^n\text{.}\) That is, \(f^2 = f\circ f\) and \(f^n = f\circ f^{n-1}\text{.}\) (Be careful note to confuse this with raising a function to a power.) A more complete understanding of periodicity arises from the study of the functions \(f^n\text{.}\) For example, a point \(x_0\) has period \(n\) iff \(f^n(x_0)=x_0\) but \(f^k(x_0)\neq x_0\) for \(k=1,2,\ldots,n-1\text{.}\)

Let \(f(x) = 2 x^2-3 x-6\text{.}\)

  1. Find the first three terms of the orbit of \(x_0=1\text{.}\)

  2. Find all fixed points of \(f.\)

  3. Write down an equation that any point of period 2 should satisfy.