标题： 谱间隙和重叠自共形集的傅里叶维数 显示英文标题

标题： Spectral gaps and Fourier dimension for self-conformal sets with overlaps

摘要： 我们证明了在满足均匀非可积性（UNI）条件的实线上重叠的$C^2$迭代函数系统相关的临界线附近复转移算子的均匀谱间隙。 我们的工作扩展了Naud（2005）关于非线性康托尔集的谱间隙的研究，以允许重叠。 证明构建了一种新方法，将缺乏马尔可夫结构的问题转化为随机Dolgopyat算子乘积的平均收缩。 这种方法受到Algom、第一作者和Shmerkin在研究正常数时开发的分解技术的启发。 作为第二作者和Stevens方法的结果，我们的谱间隙结果表明，任何满足UNI的$C^{2}$迭代函数系统的非原子自共形测度的傅里叶变换以多项式速率趋于零。 这一结果导致了具有任意重叠的分形不确定性原理。 显示英文摘要

摘要： We prove a uniform spectral gap for complex transfer operators near the critical line associated to overlapping $C^2$ iterated function systems on the real line satisfying a Uniform Non-Integrability (UNI) condition. Our work extends that of Naud (2005) on spectral gaps for nonlinear Cantor sets to allow overlaps. The proof builds a new method to reduce the problem of the lack of Markov structure to average contraction of products of random Dolgopyat operators. This approach is inspired by a disintegration technique developed by Algom, the first author and Shmerkin in the study of normal numbers. As a consequence of the method of the second author and Stevens, our spectral gap result implies that the Fourier transform of any non-atomic self-conformal measure decays to zero at a polynomial rate for any $C^{2}$ iterated function system satisfying UNI. This latter result leads to Fractal Uncertainty Principles with arbitrary overlaps.