Does Sarkovskii's Theorem hold for Darboux functions of Baire
Class-1?
by
Kenneth Kellum
(kellum@sjsumcs.sjsu.edu)
Sarkovskii's Theorem states that if f:R-->R is continuous
and f has a point of prime period k, then f also has points
of prime period n, for each n that follows k in the
Sarkovskii ordering given by 3,5,...,6,10,...,4,2,1.
A very readable proof can be found in Robert Devaney's book "An
Introduction to Chaotic Dynamical Systems." As this theorem is
important in dynamics, it would be useful to understand it better.
In "Iterates of almost continuous functions and Sarkovskii's
theorem", Real Anal. Exchange 14 (1988/89), no. 2, 420-422; I proved
that Sarkovskii's Theorem cannot be extended to almost continuous
functions by proving that there exists an almost continuous function
f:R-->R which has a point of prime period 3 but for each x,
either x=f(x), f is of prime period 3 at x, or
x is not equal to the n-th iterate f^n(x) for all n>1.
In that paper I posed the question stated above, which, to my knowledge,
remains unanswered.
Kenneth Kellum