Chapter 6: Starting from Randomness

Section 6: Special Initial Conditions

Sarkovskii's theorem

For any iterated map based on a continuous function such as a polynomial it was shown in 1962 that if an initial condition exists that gives period 3, then other initial conditions must exist that give any other period. In general, if a period m is possible then so must all periods n for which p={m, n} satisfies

OrderedQ[Transpose[If[MemberQ[p/#, 1], Map[Reverse, {p/#, #} ], {#, p/#}]] &[2^IntegerExponent[p, 2]]]

Extensions of this to other types of systems seem difficult to find, but it is conceivable that when viewed as continuous mappings on a Cantor set (see page 869) at least some cellular automata might exhibit similar properties.

From Stephen Wolfram: A New Kind of Science [citation]