SOME FAMILIES OF UNIMODAL FUNCTIONS WITHOUT THE INTEGRITY OF KNEADING SEQUENCES
Fanxin Zeng · Xitong kexue yu shuxue · 1995
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].