非齐次自相似测度的绝对连续性
动力系统
2024-09-24 v3 经典分析与常微分方程
概率论
摘要
我们证明，在超临界区域中，实线上的自相似测度对几乎所有参数都是绝对连续的，特别地，这证实了S-M. Ngai和Y. Wang的一个猜想。尽管近来在理解齐次自相似测度的绝对连续性方面已取得诸多进展，但本文是一般（非齐次）情形下对经典横向性方法的首个改进。在证明过程中，我们建立了一类随机自相似测度的维数与傅里叶衰减的新结果。
引用
@article{arxiv.1709.05092,
title = {Absolute continuity of non-homogeneous self-similar measures},
author = {Santiago Saglietti and Pablo Shmerkin and Boris Solomyak},
journal= {arXiv preprint arXiv:1709.05092},
year = {2024}
}
备注
v3: the statement of Theorem 1.3 was changed (the selection measure for the "model" of a random self-similar measure is assumed to be Bernoulli rather than an arbitrary ergodic shift-invariant measure; this was implicitly used in the proof. The original formulation is still correct; see the footnote on p.8 for details). The main result: Theorem 1.1 is unchanged