###### Definition 2.1.1

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.

Some orbits don't move; they are fixed.

Sometimes an orbit might return to the original starting point.

\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 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.

Let \(f(x) = x^2-1\text{.}\) Then zero is a periodic point, since

\begin{equation*}
0\to-1\to0\to-1\to0\cdots.
\end{equation*}

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

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