{"title":"Dynamics on ℙ1: preperiodic points and pairwise stability","authors":"Laura DeMarco, Niki Myrto Mavraki","doi":"10.1112/s0010437x23007546","DOIUrl":null,"url":null,"abstract":"<p>DeMarco, Krieger, and Ye conjectured that there is a uniform bound <span>B</span>, depending only on the degree <span>d</span>, so that any pair of holomorphic maps <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180226992-0669:S0010437X23007546:S0010437X23007546_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$f, g :{\\mathbb {P}}^1\\to {\\mathbb {P}}^1$</span></span></img></span></span> with degree <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180226992-0669:S0010437X23007546:S0010437X23007546_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$d$</span></span></img></span></span> will either share all of their preperiodic points or have at most <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180226992-0669:S0010437X23007546:S0010437X23007546_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$B$</span></span></img></span></span> in common. Here we show that this uniform bound holds for a Zariski open and dense set in the space of all pairs, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180226992-0669:S0010437X23007546:S0010437X23007546_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathrm {Rat}_d \\times \\mathrm {Rat}_d$</span></span></img></span></span>, for each degree <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180226992-0669:S0010437X23007546:S0010437X23007546_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$d\\geq 2$</span></span></img></span></span>. The proof involves a combination of arithmetic intersection theory and complex-dynamical results, especially as developed recently by Gauthier and Vigny, Yuan and Zhang, and Mavraki and Schmidt. In addition, we present alternate proofs of the main results of DeMarco, Krieger, and Ye [<span>Uniform Manin-Mumford for a family of genus 2 curves</span>, Ann. of Math. (2) <span>191</span> (2020), 949–1001; <span>Common preperiodic points for quadratic polynomials</span>, J. Mod. Dyn. <span>18</span> (2022), 363–413] and of Poineau [<span>Dynamique analytique sur</span> <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180226992-0669:S0010437X23007546:S0010437X23007546_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbb {Z}$</span></span></img></span></span> <span>II : Écart uniforme entre Lattès et conjecture de Bogomolov-Fu-Tschinkel</span>, Preprint (2022), arXiv:2207.01574 [math.NT]]. In fact, we prove a generalization of a conjecture of Bogomolov, Fu, and Tschinkel in a mixed setting of dynamical systems and elliptic curves.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"2 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Compositio Mathematica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1112/s0010437x23007546","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
DeMarco, Krieger, and Ye conjectured that there is a uniform bound B, depending only on the degree d, so that any pair of holomorphic maps $f, g :{\mathbb {P}}^1\to {\mathbb {P}}^1$ with degree $d$ will either share all of their preperiodic points or have at most $B$ in common. Here we show that this uniform bound holds for a Zariski open and dense set in the space of all pairs, $\mathrm {Rat}_d \times \mathrm {Rat}_d$, for each degree $d\geq 2$. The proof involves a combination of arithmetic intersection theory and complex-dynamical results, especially as developed recently by Gauthier and Vigny, Yuan and Zhang, and Mavraki and Schmidt. In addition, we present alternate proofs of the main results of DeMarco, Krieger, and Ye [Uniform Manin-Mumford for a family of genus 2 curves, Ann. of Math. (2) 191 (2020), 949–1001; Common preperiodic points for quadratic polynomials, J. Mod. Dyn. 18 (2022), 363–413] and of Poineau [Dynamique analytique sur$\mathbb {Z}$II : Écart uniforme entre Lattès et conjecture de Bogomolov-Fu-Tschinkel, Preprint (2022), arXiv:2207.01574 [math.NT]]. In fact, we prove a generalization of a conjecture of Bogomolov, Fu, and Tschinkel in a mixed setting of dynamical systems and elliptic curves.
DeMarco、Krieger 和 Ye 猜想存在一个均匀约束 B,它只取决于度数 d,因此任何一对度数为 $d$ 的全形映射 $f, g :{\mathbb {P}}^1\to {\mathbb {P}}^1$ 要么共享它们所有的前周期点,要么最多有 $B$ 的共同点。在这里,我们将证明,对于所有线对空间中的一个扎里斯基开放致密集合,即 $\mathrm {Rat}_d \times \mathrm {Rat}_d$,在每个度为 $d\geq 2$的情况下,这个统一约束成立。证明涉及算术交集理论和复动态结果的结合,特别是高特和维尼、袁和张以及马夫拉基和施密特最近发展的结果。此外,我们还提出了德马科、克里格和叶 [Uniform Manin-Mumford for a family of genus 2 curves, Ann. of Math. (2) 191 (2020), 949-1001; Common preperiodic points for quadratic polynomials, J. Mod.Dyn.18 (2022), 363-413] 和 Poineau [Dynamique analytique sur $\mathbb {Z}$ II : Écart uniforme entre Lattès et conjecture de Bogomolov-Fu-Tschinkel, Preprint (2022), arXiv:2207.01574 [math.NT]].事实上,我们证明了博戈莫洛夫、傅氏和钦克尔在动力系统和椭圆曲线混合背景下的猜想的一般化。
期刊介绍:
Compositio Mathematica is a prestigious, well-established journal publishing first-class research papers that traditionally focus on the mainstream of pure mathematics. Compositio Mathematica has a broad scope which includes the fields of algebra, number theory, topology, algebraic and differential geometry and global analysis. Papers on other topics are welcome if they are of broad interest. All contributions are required to meet high standards of quality and originality. The Journal has an international editorial board reflected in the journal content.