伪仿射自共形测度的傅里叶衰减与非衰减
Fourier decay and non-decay for pseudo-affine self-conformal measures
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中文总结 AI 辅助
研究直线上自共形测度多项式傅里叶衰减充分条件的尖锐性，通过伪仿射迭代函数系统框架，构造非\(C^1\)共轭的迭代函数系统及特定微分同胚，得到相关测度的傅里叶衰减情况。
中文摘要 AI 辅助
我们研究了近期关于直线上自共形测度多项式傅里叶衰减充分条件的尖锐性。首先，构造了一个\(C^\infty\)迭代函数系统，它与自相似系统不是\(C^1\)共轭的，但其平稳测度不是拉伊赫曼测度。其次，对于每个强分离的伯努利卷积\(\mu\)以及每个\(1\leq r<\infty\)，构造了一个\(C^r\)微分同胚\(h\)，使得\(h'\)在\(\mu\)的支撑集上为常数，而像测度\(h\mu\)具有多项式傅里叶衰减。所有构造都在作者之前引入的伪仿射迭代函数系统框架内。
英文摘要
We study the sharpness of recent sufficient conditions for polynomial Fourier decay of self conformal measures on the line. First, we construct a $C^\infty$ iterated function system which is not $C^1$-conjugate to self-similar, but which nevertheless admits a stationary measure that is not Rajchman. Second, for every strongly separated Bernoulli convolution $μ$ and every $1\leq r<\infty$, we construct a $C^r$-diffeomorphism $h$ such that $h'$ is constant on $\operatorname{supp}μ$, yet the image measure $hμ$ has polynomial Fourier decay. All constructions are within the framework of pseudo-affine iterated function systems, previously introduced by the authors.
