The absolute continuity of the conjugation of certain diffeomorphisms of the circle
The absolute continuity of the conjugation of certain diffeomorphisms of the circle
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圆的某些微分同胚的共轭的绝对连续性
DOI：
10.1017/s0143385700005289
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发表时间：
1989
影响因子：
0.9
通讯作者：
D. Ornstein
中科院分区：
文献类型：
--
作者：
Y. Katznelson;D. Ornstein
令 f 为圆的保持ℋ-微分同胚的方向。如果旋转数 α = ρ(f) 是无理数且 log Df 具有有界变化，则根据众所周知的 Denjoy 定理，f 与刚性旋转 Rα 共轭。共轭意味着圆存在本质上唯一的同胚 h，使得 f = h−lRαh。在 α 上合适的丢番图条件下将 h 的平滑度与 f 的平滑度关联起来的一般问题已被广泛研究（参见 [H1]、[KO]、[Y] 以及其中给出的参考文献）。在 f 的平滑度尺度的底部，有一个 M. Herman 定理 [H2]，该定理指出，如果 Df 绝对连续且 D log Df ∈ Lp, p > 1，α = ρ (f) 是“常数类型”，这意味着“α 的连分式展开式中的系数有界”，如果 f 是 Rα 的扰动，则 h 绝对连续。我们本文的目的是给出赫尔曼定理的不同证明和改进版本。结果的主要区别在于我们不需要假设 f 接近 Rα；这个证明与赫尔曼的证明非常不同，并且非常符合[KO]的精神。
Let f be an orientation preserving ℋ-diffeomorphism of the circle. If the rotation number α = ρ(f) is irrational and log Df is of bounded variation then, by a wellknown theorem of Denjoy, f is conjugate to the rigid rotation Rα. The conjugation means that there exists an essentially unique homeomorphism h of the circle such that f = h−lRαh. The general problem of relating the smoothness of h to that of f under suitable diophantine conditions on α has been studied extensively (cf. [H1], [KO], [Y] and the references given there). At the bottom of the scale of smoothness for f there is a theorem of M. Herman [H2] which states that if Df is absolutely continuous and D log Df ∈ Lp, p > 1, α = ρ (f) is of ‘constant type’ which means ‘the coefficients in the continued fraction expansion of α are bounded’, and if f is a perturbation of Rα, then h is absolutely continuous. Our purpose in this paper is to give a different proof and an improved version of Herman's theorem. The main difference in the result is that we do not need to assume that f is close to Rα; the proof is very different from Herman's and is very much in the spirit of [KO].