91Ó°ÊÓ

Show that the middle-thirds Cantor set contains no intervals. But also show that no point in the set is isolated.

The tent map on the interval \([0,1]\) is defined by \(x_{n+1}=f\left(x_{n}\right)\), where $$ f(x)= \begin{cases}r x, & 0 \leq x \leq \frac{1}{2} \\ r(1-x), & \frac{1}{2} \leq x \leq 1\end{cases} $$ and \(r>0\). In this exercise we assume \(r>2\). Then some points get mapped outside the interval \([0,1]\). If \(f\left(x_{0}\right)>1\) then we say that \(x_{0}\) has "escaped" after one iteration. Similarly, if \(f^{\prime \prime}\left(x_{0}\right)>1\) for some finite \(n\), but \(f^{k}\left(x_{0}\right) \in[0,1]\) for all \(k

A fat fractal is a fractal with a nonzero measure. Here's a simple example: start with the unit interval \([0,1]\) and delete the open middle \(1 / 2\), \(1 / 4,1 / 8\), etc., of each remaining sub-interval. (Thus a smaller and smaller fraction is removed at each stage, in contrast to the middle-thirds Cantor set, where we always remove \(1 / 3\) of what's left.) a) Show that the limiting set is a topological Cantor set. b) Show that the measure of the limiting set is greater than zero. Find its exact value if you can, or else just find a lower bound for it. Fat fractals answer a fascinating question about the logistic map. Farmer ( 1985\()\) has shown numerically that the set of parameter values for which chaos occurs is a fat fractal. In particular, if \(r\) is chosen at random between \(r_{\infty}\) and \(r=4\), there is about an \(89 \%\) chance that the map will be chaotic. Farmer's analysis also suggests that the odds of making a mistake (calling an orbit chaotic when it's actually periodic) are about one in a million, if we use double precision arithmetic!