Chaos in the sense of Li-Yorke and the order of the inverse limit space

Let I = [0,1] and omega(0) be the first limit ordinal number. Assume that f :I --> I is continuous, piece-wise monotone and the set of periods of f is {2' : i is an element of {0} boolean OR N}. It is known that the order of ( I, f) is omega(0) or omega(0) + 1. It is shown that the order of the invers e limit space (I, f) is omega(0) (resp. omega(0) + 1) if and only if f is n ot (resp. is) chaotic in the sense of Li-Yorke.