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

LetI= 0, 1] and ω₀ be the first limit ordinal number. Assume thatf: 1→- 1 is continuous, piece-wise monotone and the set of periods off is |2′: iε |0|U|. It is known that the order of (1, 1) is ω₀ or w₀ + 1. It is shown that the order of the inverse limit space (1, f) is ω₀ (resp. ω₀ + 1) if and only iff is not (resp. is) chaotic in the sense of Li-Yorke.