## Section2.1Basic 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.

###### 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 $f\text{.}$

Some orbits don't move; they are fixed.

###### Definition2.1.2
A point $x_0 \in \mathbb R$ is a fixed point of $f$ if $f(x_0)=x_0\text{.}$

###### Definition2.1.3
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.

###### Definition2.1.4
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.
###### Example2.1.5

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