{"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].