Dynamics of the tall tent map

mark
Mon, 23 Oct 2017 14:25:44 +0000
Problems

Let [imath]C[/imath] denote Cantor's ternary set and let

[dmath]
f(x) =
\begin{cases}
3x & \text{if } x \leq 1/2 \\
3(1-x) & \text{if } x > 1/2.
\end{cases}
[/dmath]

a. Show that Cantor's ternary set is invariant under [imath]f[/imath], i.e. [imath]f:C\to C[/imath].
b. Describe they dynamics of [imath]f[/imath] on [imath]C[/imath] in terms of ternary expansions.
c. Explain why [imath]f[/imath] is chaotic on [imath]C[/imath].