Jason P. Bell, Jeffrey Diller, Mattias Jonsson, Holly Krieger
{"title":"Birational maps with transcendental dynamical degree","authors":"Jason P. Bell, Jeffrey Diller, Mattias Jonsson, Holly Krieger","doi":"10.1112/plms.12573","DOIUrl":null,"url":null,"abstract":"We give examples of birational selfmaps of <mjx-container aria-label=\"double struck upper P Superscript d Baseline comma d greater than or slanted equals 3\" ctxtmenu_counter=\"0\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/plms12573-math-0001.png\"><mjx-semantics><mjx-mrow data-semantic-children=\"2,3,7\" data-semantic-content=\"3\" data-semantic- data-semantic-role=\"sequence\" data-semantic-speech=\"double struck upper P Superscript d Baseline comma d greater than or slanted equals 3\" data-semantic-type=\"punctuated\"><mjx-msup data-semantic-children=\"0,1\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"numbersetletter\" data-semantic-type=\"superscript\"><mjx-mi data-semantic-font=\"double-struck\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"numbersetletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: 0.363em;\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\" size=\"s\"><mjx-c></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo data-semantic- data-semantic-operator=\"punctuated\" data-semantic-parent=\"8\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\" rspace=\"3\" style=\"margin-left: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"4,6\" data-semantic-content=\"5\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"inequality\" data-semantic-type=\"relseq\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"relseq,⩾\" data-semantic-parent=\"7\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:00246115:media:plms12573:plms12573-math-0001\" display=\"inline\" location=\"graphic/plms12573-math-0001.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"2,3,7\" data-semantic-content=\"3\" data-semantic-role=\"sequence\" data-semantic-speech=\"double struck upper P Superscript d Baseline comma d greater than or slanted equals 3\" data-semantic-type=\"punctuated\"><msup data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-parent=\"8\" data-semantic-role=\"numbersetletter\" data-semantic-type=\"superscript\"><mi data-semantic-=\"\" data-semantic-font=\"double-struck\" data-semantic-parent=\"2\" data-semantic-role=\"numbersetletter\" data-semantic-type=\"identifier\" mathvariant=\"double-struck\">P</mi><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">d</mi></msup><mo data-semantic-=\"\" data-semantic-operator=\"punctuated\" data-semantic-parent=\"8\" data-semantic-role=\"comma\" data-semantic-type=\"punctuation\">,</mo><mrow data-semantic-=\"\" data-semantic-children=\"4,6\" data-semantic-content=\"5\" data-semantic-parent=\"8\" data-semantic-role=\"inequality\" data-semantic-type=\"relseq\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">d</mi><mo data-semantic-=\"\" data-semantic-operator=\"relseq,⩾\" data-semantic-parent=\"7\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\">⩾</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"number\">3</mn></mrow></mrow>$\\mathbb {P}^d, d \\geqslant 3$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, whose dynamical degree is a transcendental number. This contradicts a conjecture by Bellon and Viallet. The proof uses a combination of techniques from algebraic dynamics and diophantine approximation.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"16 1","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1112/plms.12573","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We give examples of birational selfmaps of , whose dynamical degree is a transcendental number. This contradicts a conjecture by Bellon and Viallet. The proof uses a combination of techniques from algebraic dynamics and diophantine approximation.
期刊介绍：
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