Strange Cantor repellerFig.1 shows, how the classic Cantor set emerges in dynamics of the tent map f_{c}(x) for c = 3 . Really, as since x_{o} = 0 is an unstable fixed point, therefore iterations for x outside the [0, 1] interval diverge to infinity. f(x) maps open interval (1/3,2/3) beyond [0,1], thus we can throw away these points too. As since intervals [0,1/3] and [2/3,1] are linear mapped onto [0,1], therefore we can continue this process ad infinitum. In every iteration we cut the central one third of an interval. The limit Cantor set is nowhere dense as since it has holes in any small interval. |
In the ternary notation after the first iterations all numbers 0.1... are thrown away and so on. Therefore the Cantor set consists of all ternary numbers without digit 1. After substitution 2 → 1 one can see that it is continuum. Rational numbers correspond to unstable periodic orbits. For a random sequence made of digits 0 and 2 one gets chaotic orbit.
Map dynamics shown to the left is very easy. In the ternary notation multiplication by 3 is equivalent to the left shift. Then the highest 0 and 1 digits are thrown off. If the highest digit is 1 then the point gets into the central (1/3,2/3) interval and leaves [0,1]. If we start at a point which have the first 1 digit at the n position, then during n-1 iterations the orbit wanders chaotically in [0,1]. Further it gets in the central interval and go to infinity demonstrating temporarily chaotic transient. |