Subsection 2.10.1 Tent maps
The tent map \(T_a:[0,1]\to\mathbb R\) is defined by
\begin{equation*}
T_a(x) = \begin{cases}
ax & \text{if } 0\leq x \leq 1/2 \\
a - ax & \text{if } 1/2 \leq x \leq 1.
\end{cases}
\end{equation*}
Note that \(T_a(1/2) = a/2\) using either definition; thus, the map is well defined and continuous at \(x=1/2\text{.}\) Figure 2.10.1 shows the graph of \(T_a\) for several choices of \(a\text{.}\)
Note that \(T_a:\mathbb [0,1] \to \mathbb [0,1]\) for \(0\leq a \leq 2\text{.}\) We say that the the unit interval is invariant under the action of \(T_a\) for these parameters.