Absolute continuity in families of parametrised non-homogeneous self-similar measures

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2018-12-12 DOI:10.4171/jfg/127

A. Käenmäki, Tuomas Orponen

引用次数: 5

Abstract

In 2016, Shmerkin and Solomyak showed that if $U \subset \mathbb{R}$ is an interval, and $\{\mu_{u}\}_{u \in U}$ is an analytic family of homogeneous self-similar measures on $\mathbb{R}$ with similitude dimensions exceeding one, then, under a mild transversality assumption, $\mu_{u} \ll \mathcal{L}^{1}$ for all parameters $u \in U \setminus E$, where $\dim_{\mathrm{H}} E = 0$. The purpose of this paper is to generalise the result of Shmerkin and Solomyak to non-homogeneous self-similar measures. As a corollary, we obtain new information about the absolute continuity of projections of non-homogeneous planar self-similar measures.

查看原文本刊更多论文

参数化非齐次自相似测度族中的绝对连续性

2016年，Shmerkin和Solomyak证明，如果$U \子集\mathbb{R}$是一个区间，$\ \mu_{U} \}_{U \ \ U \ \ U \是$\mathbb{R}$上齐次自相似测度的解析族，相似维数超过1，则在温和的横向性假设下，$U \ \ set- E$中所有参数$U \ \ \ math_ {U} \ll \mathcal{L}^{1}$，其中$\dim_{\ mathm {H}} E = 0$。本文的目的是将Shmerkin和Solomyak的结果推广到非齐次自相似测度。作为一个推论，我们得到了关于非齐次平面自相似测度投影的绝对连续性的新信息。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.