Symbolic dynamics for Hénon maps near the boundary of the horseshoe locus

Pub Date : 2024-05-23 DOI:10.1017/etds.2024.34
Yuki Hironaka, Yutaka Ishii
{"title":"Symbolic dynamics for Hénon maps near the boundary of the horseshoe locus","authors":"Yuki Hironaka, Yutaka Ishii","doi":"10.1017/etds.2024.34","DOIUrl":null,"url":null,"abstract":"\n\t <jats:p>Bedford and Smillie [A symbolic characterization of the horseshoe locus in the Hénon family. <jats:italic>Ergod. Th. & Dynam. Sys.</jats:italic><jats:bold>37</jats:bold>(5) (2017), 1389–1412] classified the dynamics of the Hénon map <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000348_inline1.png\"/>\n\t\t<jats:tex-math>\n$f_{a, b} : (x, y)\\mapsto (x^2-a-by, x)$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> defined on <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000348_inline2.png\"/>\n\t\t<jats:tex-math>\n$\\mathbb {R}^2$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> in terms of a symbolic dynamics when <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000348_inline3.png\"/>\n\t\t<jats:tex-math>\n$(a, b)$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> is close to the boundary of the horseshoe locus. The purpose of the current article is to generalize their results for all <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000348_inline4.png\"/>\n\t\t<jats:tex-math>\n$b\\ne 0$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> (including the case <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000348_inline5.png\"/>\n\t\t<jats:tex-math>\n$b < 0$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> as well). The method of the proof is first to regard <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000348_inline6.png\"/>\n\t\t<jats:tex-math>\n$f_{a, b}$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> as a complex dynamical system in <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000348_inline7.png\"/>\n\t\t<jats:tex-math>\n$\\mathbb {C}^2$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> and second to introduce the new Markov-like partition in <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000348_inline8.png\"/>\n\t\t<jats:tex-math>\n$\\mathbb {R}^2$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> constructed by us [On parameter loci of the Hénon family. <jats:italic>Comm. Math. Phys.</jats:italic><jats:bold>361</jats:bold>(2) (2018), 343–414].</jats:p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/etds.2024.34","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

Abstract

Bedford and Smillie [A symbolic characterization of the horseshoe locus in the Hénon family. Ergod. Th. & Dynam. Sys.37(5) (2017), 1389–1412] classified the dynamics of the Hénon map $f_{a, b} : (x, y)\mapsto (x^2-a-by, x)$ defined on $\mathbb {R}^2$ in terms of a symbolic dynamics when $(a, b)$ is close to the boundary of the horseshoe locus. The purpose of the current article is to generalize their results for all $b\ne 0$ (including the case $b < 0$ as well). The method of the proof is first to regard $f_{a, b}$ as a complex dynamical system in $\mathbb {C}^2$ and second to introduce the new Markov-like partition in $\mathbb {R}^2$ constructed by us [On parameter loci of the Hénon family. Comm. Math. Phys.361(2) (2018), 343–414].
分享
查看原文
马蹄形位置边界附近赫农图的符号动力学
贝德福德和斯米利 [赫农家族马蹄形基因座的符号特征。Ergod.Th. & Dynam.Sys.37(5) (2017), 1389-1412]将定义在 $\mathbb {R}^2$ 上的 Hénon 映射 $f_{a, b} : (x, y)\mapsto (x^2-a-by, x)$ 的动力学分类为符号动力学,当 $(a, b)$ 接近马蹄形位置的边界时。本文的目的是将他们的结果推广到所有 $b\ne 0$(也包括 $b < 0$ 的情况)。证明的方法首先是把 $f_{a, b}$ 视为 $\mathbb {C}^2$ 中的复杂动力系统,其次是引入我们在 $\mathbb {R}^2$ 中构建的新的类似马尔可夫的分区[On parameter loci of the Hénon family.Comm.Math.Phys.361(2) (2018), 343-414].
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
×
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:481959085
Book学术官方微信