Deterministic Chaos

6.I. Dust in the Tent Map

Recall the highest point of the tent map has height s/2.
Consequently, if s > 2 the top of the tent map extends above the top of the unit square.
Graphical iteration implies the points near 1/2 iterate out of the unit square, and then on to -∞ (top figure).
Then points that iterate to the middle will escape to -∞ (bottom picture).
Continuing, the points that do not escape to -infinity form a Cantor set. Can you find the dimension of this Cantor set as a function of s?
On the Cantor set, the tent map is chaotic. One way to prove this is to coarse-grain orbits (L for left side of 1/2, R for the right side), and investigate the sequences that can be produced. This is a reasonably subtle problem.

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