几类不具有捏制轨道系列完整性的单峰函数族

SOME FAMILIES OF UNIMODAL FUNCTIONS WITHOUT THE INTEGRITY OF KNEADING SEQUENCES

本文证明，存在着由单峰函数构成的Ｃｏ－函数族｛ｆλ），｛ｇλ｝及具有下列性质：（ｉ）各ｆλ均是分段线性的单峰平顶函数，各ｇλ及均是Ｃ∞－单峰平顶函数．但族｛ｆλ｝，｛ｇλ｝及均非一致平顶；（ｉｉ）族｛ｆλ｝及｛ｇλ｝均满足一致的Ｌｉｐｓｃｈｉｔｚ条件，但在峰顶处均非一致地可微；（ｉｉｉ）族在峰顶处一致地可微，但不满足一致的Ｌｉｐｓｃｈｉｔｚ条件；（ｉｖ）当０≤λ≤７／８时捏制序列Ｋ（ｆλ），Ｋ（ｇλ）及以均不大于ＲＬＣ，当７／８＜λ－≤１时Ｋ（ｆλ），Ｋ（ｇλ）及Ｋ均不小于ＲＬＬ（ＲＬＲ）∞，因而族｛ｆλ｝，｛ｇλ｝及均不具有捏制轨道系列的完整性．本文的结果解答了文［５］中提出的两个猜测．

It is proved that there exist C0-families {fλ},{gλ} and {λ} of unimodal functions having the following properties: (i) Each fλ in the family {fλ} is a piecewise linear unimodal flat-top function, each gλ in the family {gλ} and each λ in the family {λ} are C∞ unimodal flat-top functions, but none of the families {fλ}, {gλ} and {λ} is uniformly flat-top; (ii)Families {fλ} and {gλ} satisfy the uniform Lipschitz condition, but they are not uniformly differentiable at tops; (iii)The family {λ} is uniformly differentiable at tops, but it does not satisfy the uniform Lipschitz condition; (iv) The kneading sequences K(fλ), K(fλ,) and K(λ,) are not greater than RLC for k e [0, 7/8] and not less than .RLL(RLR)-∞ for λ (7/8,1], and hence none of the families {fλ}, {gλ} and {λ} has the integrity of the series of the kneading orbits.The results of this paper answer the two conjectures posed in Ref. [5].