{"title":"Renormalization of Bicritical Circle Maps","authors":"Gabriela Estevez, Pablo Guarino","doi":"10.1007/s40598-022-00199-x","DOIUrl":null,"url":null,"abstract":"<div><p>A general <i>ansatz</i> in Renormalization Theory, already established in many important situations, states that exponential convergence of renormalization orbits implies that topological conjugacies are actually smooth (when restricted to the attractors of the original systems). In this paper, we establish this principle for a large class of <i>bicritical circle maps</i>, which are <span>\\(C^3\\)</span> circle homeomorphisms with irrational rotation number and exactly two (non-flat) critical points. The proof presented here is an adaptation, to the bicritical setting, of the one given by de Faria and de Melo in (J Eur Math Soc 1:339–392, 1999) for the case of a single critical point. When combined with the recent papers (Estevez et al. in Complex bounds for multicritical circle maps with bounded type rotation number, arXiv:2005.02377, 2020; Yampolsky in C R Math Rep Acad Sci Can 41:57–83, 2019), our main theorem implies <span>\\(C^{1+\\alpha }\\)</span> rigidity for real-analytic bicritical circle maps with rotation number of <i>bounded type</i> (Corollary 1.1).</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Arnold Mathematical Journal","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s40598-022-00199-x","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 3
Abstract
A general ansatz in Renormalization Theory, already established in many important situations, states that exponential convergence of renormalization orbits implies that topological conjugacies are actually smooth (when restricted to the attractors of the original systems). In this paper, we establish this principle for a large class of bicritical circle maps, which are \(C^3\) circle homeomorphisms with irrational rotation number and exactly two (non-flat) critical points. The proof presented here is an adaptation, to the bicritical setting, of the one given by de Faria and de Melo in (J Eur Math Soc 1:339–392, 1999) for the case of a single critical point. When combined with the recent papers (Estevez et al. in Complex bounds for multicritical circle maps with bounded type rotation number, arXiv:2005.02377, 2020; Yampolsky in C R Math Rep Acad Sci Can 41:57–83, 2019), our main theorem implies \(C^{1+\alpha }\) rigidity for real-analytic bicritical circle maps with rotation number of bounded type (Corollary 1.1).
重整化理论中已经在许多重要情况下建立了一个一般的假设，即重整化轨道的指数收敛意味着拓扑共轭实际上是光滑的（当限制于原始系统的吸引子时）。本文建立了一大类双临界圆映射的这一原理，这些映射是具有无理旋转数和恰好两个（非平坦）临界点的（C^3）圆同胚。这里给出的证明是对de Faria和de Melo在（《欧洲数学会杂志》1999年第1卷第139-392页）中给出的单临界点情况下的双临界设置的改编。当与最近的论文相结合时（Estevez等人在《具有有界型旋转数的多临界圆映射的复界》中，arXiv:2005.023772020；Yampolsky在《C R Math Rep Acad Sci Can 41:57–832019》中），我们的主要定理暗示了具有有界类型旋转数的实解析双临界圆映射（推论1.1）的\（C^｛1+\alpha｝）刚性。
期刊介绍：
The Arnold Mathematical Journal publishes interesting and understandable results in all areas of mathematics. The name of the journal is not only a dedication to the memory of Vladimir Arnold (1937 – 2010), one of the most influential mathematicians of the 20th century, but also a declaration that the journal should serve to maintain and promote the scientific style characteristic for Arnold''s best mathematical works. Features of AMJ publications include: Popularity. The journal articles should be accessible to a very wide community of mathematicians. Not only formal definitions necessary for the understanding must be provided but also informal motivations even if the latter are well-known to the experts in the field. Interdisciplinary and multidisciplinary mathematics. AMJ publishes research expositions that connect different mathematical subjects. Connections that are useful in both ways are of particular importance. Multidisciplinary research (even if the disciplines all belong to pure mathematics) is generally hard to evaluate, for this reason, this kind of research is often under-represented in specialized mathematical journals. AMJ will try to compensate for this.Problems, objectives, work in progress. Most scholarly publications present results of a research project in their “final'' form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned, but the very process of mathematical discovery remains hidden. Following Arnold, publications in AMJ will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. AMJ publishes well-motivated research problems on a regular basis. Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold''s principle, a general formulation is less desirable than the simplest partial case that is still unknown.Being interesting. The most important requirement is that the article be interesting. It does not have to be limited by original research contributions of the author; however, the author''s responsibility is to carefully acknowledge the authorship of all results. Neither does the article need to consist entirely of formal and rigorous arguments. It can contain parts, in which an informal author''s understanding of the overall picture is presented; however, these parts must be clearly indicated.
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