与康托级数展开相对应的非自治动力系统中的运动递归问题

IF 0.6 3区 数学 Q3 MATHEMATICS
Z. Shen
{"title":"与康托级数展开相对应的非自治动力系统中的运动递归问题","authors":"Z. Shen","doi":"10.1007/s10474-025-01514-6","DOIUrl":null,"url":null,"abstract":"<div><p>\nWe investigate a moving recurrent problem for the nonautonomous dynamical system induced by the Cantor series expansion.\nTo be precise, let <span>\\(Q=\\{q_{k}\\}_{k\\geq1}\\)</span> be a sequence of positive integers with <span>\\(q_{k}\\geq2\\)</span> for all <span>\\(k\\geq1\\)</span>. Put\n<span>\\(T_{Q}^{n}(x)=q_{1}\\cdots q_{n}x-\\lfloor q_{1}\\cdots q_{n}x\\rfloor\\)</span> for each <span>\\(n\\geq1\\)</span>, which gives the <span>\\(Q\\)</span>-Cantor series expansion.\nWe focus on the following <span>\\(\\{n_{k},r_{k}\\}\\)</span>-moving recurrent points proposed by Boshernitzan and Glasner:\n</p><div><div><span>$$\\inf_{k\\geq1}|T^{n_{k}}_{Q}(x)-T^{n_{k}+r_{k}}_{Q}(x)|=0,$$</span></div></div><p>\nwhere <span>\\(\\{n_{k}\\}_{k\\geq1}\\)</span> and <span>\\(\\{r_{k}\\}_{k\\geq1}\\)</span> are two given sequences of integers. It is proved that when <span>\\(\\{n_{k}\\}_{k\\geq1}\\)</span>\nand <span>\\(\\{r_{k}\\}_{k\\geq1}\\)</span> tend to infinity, the set of <span>\\(\\{n_{k},r_{k}\\}\\)</span>-moving recurrent points is of full Lebesgue measure. In addition,\nwe study the size of the following quantitative version of <span>\\(\\{n_{k},r_{k}\\}\\)</span>-moving recurrent set:\n</p><div><div><span>$$ R(\\{n_{k},r_{k}\\}):=\\big\\{x\\in [0,1] : |T^{n_{k}}_{Q}(x)-T^{n_{k}+r_{k}}_{Q}(x)|&lt;\\varphi(k)~\\text{for i.m.}~k\\in \\mathbb{N}\\big\\},$$</span></div></div><p>\nwhere <span>\\(\\varphi \\colon \\mathbb{N}\\rightarrow\\mathbb{R}^{+}\\)</span> is a positive function and ``i.m.'' stands for ``infinitely many''. It is proved that when <span>\\(\\{n_{k}\\}_{k\\geq1}\\)</span> and <span>\\(\\{r_{k}\\}_{k\\geq1}\\)</span> tend to infinity, the\nLebesgue measure and Hausdorff measure of <span>\\(R(\\{n_{k},r_{k}\\})\\)</span> respectively fulfill a dichotomy law according to the convergence or divergence of certain series.\n</p></div>","PeriodicalId":50894,"journal":{"name":"Acta Mathematica Hungarica","volume":"175 2","pages":"433 - 451"},"PeriodicalIF":0.6000,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Moving recurrent problems in the nonautonomous dynamical systems corresponding to Cantor series expansions\",\"authors\":\"Z. Shen\",\"doi\":\"10.1007/s10474-025-01514-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>\\nWe investigate a moving recurrent problem for the nonautonomous dynamical system induced by the Cantor series expansion.\\nTo be precise, let <span>\\\\(Q=\\\\{q_{k}\\\\}_{k\\\\geq1}\\\\)</span> be a sequence of positive integers with <span>\\\\(q_{k}\\\\geq2\\\\)</span> for all <span>\\\\(k\\\\geq1\\\\)</span>. Put\\n<span>\\\\(T_{Q}^{n}(x)=q_{1}\\\\cdots q_{n}x-\\\\lfloor q_{1}\\\\cdots q_{n}x\\\\rfloor\\\\)</span> for each <span>\\\\(n\\\\geq1\\\\)</span>, which gives the <span>\\\\(Q\\\\)</span>-Cantor series expansion.\\nWe focus on the following <span>\\\\(\\\\{n_{k},r_{k}\\\\}\\\\)</span>-moving recurrent points proposed by Boshernitzan and Glasner:\\n</p><div><div><span>$$\\\\inf_{k\\\\geq1}|T^{n_{k}}_{Q}(x)-T^{n_{k}+r_{k}}_{Q}(x)|=0,$$</span></div></div><p>\\nwhere <span>\\\\(\\\\{n_{k}\\\\}_{k\\\\geq1}\\\\)</span> and <span>\\\\(\\\\{r_{k}\\\\}_{k\\\\geq1}\\\\)</span> are two given sequences of integers. It is proved that when <span>\\\\(\\\\{n_{k}\\\\}_{k\\\\geq1}\\\\)</span>\\nand <span>\\\\(\\\\{r_{k}\\\\}_{k\\\\geq1}\\\\)</span> tend to infinity, the set of <span>\\\\(\\\\{n_{k},r_{k}\\\\}\\\\)</span>-moving recurrent points is of full Lebesgue measure. In addition,\\nwe study the size of the following quantitative version of <span>\\\\(\\\\{n_{k},r_{k}\\\\}\\\\)</span>-moving recurrent set:\\n</p><div><div><span>$$ R(\\\\{n_{k},r_{k}\\\\}):=\\\\big\\\\{x\\\\in [0,1] : |T^{n_{k}}_{Q}(x)-T^{n_{k}+r_{k}}_{Q}(x)|&lt;\\\\varphi(k)~\\\\text{for i.m.}~k\\\\in \\\\mathbb{N}\\\\big\\\\},$$</span></div></div><p>\\nwhere <span>\\\\(\\\\varphi \\\\colon \\\\mathbb{N}\\\\rightarrow\\\\mathbb{R}^{+}\\\\)</span> is a positive function and ``i.m.'' stands for ``infinitely many''. It is proved that when <span>\\\\(\\\\{n_{k}\\\\}_{k\\\\geq1}\\\\)</span> and <span>\\\\(\\\\{r_{k}\\\\}_{k\\\\geq1}\\\\)</span> tend to infinity, the\\nLebesgue measure and Hausdorff measure of <span>\\\\(R(\\\\{n_{k},r_{k}\\\\})\\\\)</span> respectively fulfill a dichotomy law according to the convergence or divergence of certain series.\\n</p></div>\",\"PeriodicalId\":50894,\"journal\":{\"name\":\"Acta Mathematica Hungarica\",\"volume\":\"175 2\",\"pages\":\"433 - 451\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2025-04-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematica Hungarica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10474-025-01514-6\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Hungarica","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10474-025-01514-6","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

研究了一类由康托级数展开引起的非自治动力系统的运动递推问题。更精确地说,设\(Q=\{q_{k}\}_{k\geq1}\)是一个正整数序列,所有的\(k\geq1\)都是\(q_{k}\geq2\)。将\(T_{Q}^{n}(x)=q_{1}\cdots q_{n}x-\lfloor q_{1}\cdots q_{n}x\rfloor\)代入\(n\geq1\),得到\(Q\) -Cantor级数展开式。我们关注以下由Boshernitzan和Glasner提出的\(\{n_{k},r_{k}\}\) -移动循环点:$$\inf_{k\geq1}|T^{n_{k}}_{Q}(x)-T^{n_{k}+r_{k}}_{Q}(x)|=0,$$其中\(\{n_{k}\}_{k\geq1}\)和\(\{r_{k}\}_{k\geq1}\)是两个给定的整数序列。证明了当\(\{n_{k}\}_{k\geq1}\)和\(\{r_{k}\}_{k\geq1}\)趋于无穷时,\(\{n_{k},r_{k}\}\)移动的循环点集是完全勒贝格测度。此外,我们研究了以下\(\{n_{k},r_{k}\}\) -移动循环集的定量版本的大小:$$ R(\{n_{k},r_{k}\}):=\big\{x\in [0,1] : |T^{n_{k}}_{Q}(x)-T^{n_{k}+r_{k}}_{Q}(x)|<\varphi(k)~\text{for i.m.}~k\in \mathbb{N}\big\},$$,其中\(\varphi \colon \mathbb{N}\rightarrow\mathbb{R}^{+}\)是一个正函数,‘ ’ i.m。代表“无限多”。证明了当\(\{n_{k}\}_{k\geq1}\)和\(\{r_{k}\}_{k\geq1}\)趋于无穷时,\(R(\{n_{k},r_{k}\})\)的elebesgue测度和Hausdorff测度分别根据一定级数的收敛性或发散性满足二分律。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Moving recurrent problems in the nonautonomous dynamical systems corresponding to Cantor series expansions

We investigate a moving recurrent problem for the nonautonomous dynamical system induced by the Cantor series expansion. To be precise, let \(Q=\{q_{k}\}_{k\geq1}\) be a sequence of positive integers with \(q_{k}\geq2\) for all \(k\geq1\). Put \(T_{Q}^{n}(x)=q_{1}\cdots q_{n}x-\lfloor q_{1}\cdots q_{n}x\rfloor\) for each \(n\geq1\), which gives the \(Q\)-Cantor series expansion. We focus on the following \(\{n_{k},r_{k}\}\)-moving recurrent points proposed by Boshernitzan and Glasner:

$$\inf_{k\geq1}|T^{n_{k}}_{Q}(x)-T^{n_{k}+r_{k}}_{Q}(x)|=0,$$

where \(\{n_{k}\}_{k\geq1}\) and \(\{r_{k}\}_{k\geq1}\) are two given sequences of integers. It is proved that when \(\{n_{k}\}_{k\geq1}\) and \(\{r_{k}\}_{k\geq1}\) tend to infinity, the set of \(\{n_{k},r_{k}\}\)-moving recurrent points is of full Lebesgue measure. In addition, we study the size of the following quantitative version of \(\{n_{k},r_{k}\}\)-moving recurrent set:

$$ R(\{n_{k},r_{k}\}):=\big\{x\in [0,1] : |T^{n_{k}}_{Q}(x)-T^{n_{k}+r_{k}}_{Q}(x)|<\varphi(k)~\text{for i.m.}~k\in \mathbb{N}\big\},$$

where \(\varphi \colon \mathbb{N}\rightarrow\mathbb{R}^{+}\) is a positive function and ``i.m.'' stands for ``infinitely many''. It is proved that when \(\{n_{k}\}_{k\geq1}\) and \(\{r_{k}\}_{k\geq1}\) tend to infinity, the Lebesgue measure and Hausdorff measure of \(R(\{n_{k},r_{k}\})\) respectively fulfill a dichotomy law according to the convergence or divergence of certain series.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
1.50
自引率
11.10%
发文量
77
审稿时长
4-8 weeks
期刊介绍: Acta Mathematica Hungarica is devoted to publishing research articles of top quality in all areas of pure and applied mathematics as well as in theoretical computer science. The journal is published yearly in three volumes (two issues per volume, in total 6 issues) in both print and electronic formats. Acta Mathematica Hungarica (formerly Acta Mathematica Academiae Scientiarum Hungaricae) was founded in 1950 by the Hungarian Academy of Sciences.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:604180095
Book学术官方微信