##### Definition2.1.1

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 \(f\text{.}\)

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.

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.

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.

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.

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*.

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 \(k=1,2,\ldots,n-1\text{.}\)