Recall the highest point of the tent map has height s/2. | |||

Consequently, if | |||

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 |

Return to Deterministic Chaos.