Consider the map defined on $\mathbb{R}$ by

$$F(x)= \begin{cases}3 x & x \leqslant \frac{1}{2} \\ 3(1-x) & x \geqslant \frac{1}{2}\end{cases}$$

and let $I$ be the open interval $(0,1)$. Explain what it means for $F$ to have a horseshoe on $I$ by identifying the relevant intervals in the definition.

Let $\Lambda=\left\{x: F^{n}(x) \in I, \forall n \geqslant 0\right\}$. Show that $F(\Lambda)=\Lambda$.

Find the sets $\Lambda_{1}=\{x: F(x) \in I\}$ and $\Lambda_{2}=\left\{x: F^{2}(x) \in I\right\}$.

Consider the ternary (base-3) representation $x=0 \cdot x_{1} x_{2} x_{3} \ldots$ of numbers in $I$. Show that

$$F\left(0 \cdot x_{1} x_{2} x_{3} \ldots\right)=\left\{\begin{array}{ll} x_{1} \cdot x_{2} x_{3} x_{4} \ldots & x \leqslant \frac{1}{2} \\ \sigma\left(x_{1}\right) \cdot \sigma\left(x_{2}\right) \sigma\left(x_{3}\right) \sigma\left(x_{4}\right) \ldots & x \geqslant \frac{1}{2} \end{array},\right.$$

where the function $\sigma\left(x_{i}\right)$ of the ternary digits should be identified. What is the ternary representation of the non-zero fixed point? What do the ternary representations of elements of $\Lambda$ have in common?

Show that $F$ has sensitive dependence on initial conditions on $\Lambda$, that $F$ is topologically transitive on $\Lambda$, and that periodic points are dense in $\Lambda$. [Hint: You may assume that $F^{n}\left(0 \cdot x_{1} \ldots x_{n-1} 0 x_{n+1} x_{n+2} \ldots\right)=0 \cdot x_{n+1} x_{n+2} \ldots$ for $x \in \Lambda$.]

Briefly state the relevance of this example to the relationship between Glendinning's and Devaney's definitions of chaos.