复杂动力学中的同斜轨道、乘子谱和刚性定理

IF 2.8 1区 数学 Q1 MATHEMATICS Forum of Mathematics Pi Pub Date : 2022-05-26 DOI:10.1017/fmp.2023.12
Zhuchao Ji, Junyi Xie
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引用次数: 10

摘要

摘要本文的目的是结合复杂和非阿基米德动力学的工具,回答关于有理映射乘谱的几个猜想和问题,并给出复杂动力学中几个刚性定理的新证明。McMullen的一个显著定理断言,除了灵活的Lattès族之外,周期点的乘谱决定了有理映射的共轭类,可以有有限多个选择。证明依赖于后临界有限映射的Thurston刚性定理,其中Teichmüller理论是一个重要的工具。在不使用拟共形映射或Teichmüller理论的情况下,我们将给出McMullen定理的一个新证明(因此也是Thurston定理的一种新证明)。我们证明,除了灵活的Lattès族之外,周期点的长度谱决定了有理映射的共轭类,多达有限多个选择。这推广了前面提到的McMullen定理。我们还将证明标记长度谱的一个刚度定理。类似的想法也产生了Zdunik刚性定理的简单证明。我们证明了有理映射是例外的,当且仅当以下其中一个成立:(i)周期点的乘子包含在虚二次域的整数环中,以及(ii)除有限多个周期点外的所有周期点都具有相同的李雅普诺夫指数。这解决了米尔诺的两个猜想。
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Homoclinic orbits, multiplier spectrum and rigidity theorems in complex dynamics
Abstract The aims of this paper are to answer several conjectures and questions about the multiplier spectrum of rational maps and giving new proofs of several rigidity theorems in complex dynamics by combining tools from complex and non-Archimedean dynamics. A remarkable theorem due to McMullen asserts that, aside from the flexible Lattès family, the multiplier spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices. The proof relies on Thurston’s rigidity theorem for post-critically finite maps, in which Teichmüller theory is an essential tool. We will give a new proof of McMullen’s theorem (and therefore a new proof of Thurston’s theorem) without using quasiconformal maps or Teichmüller theory. We show that, aside from the flexible Lattès family, the length spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices. This generalizes the aforementioned McMullen’s theorem. We will also prove a rigidity theorem for marked length spectrum. Similar ideas also yield a simple proof of a rigidity theorem due to Zdunik. We show that a rational map is exceptional if and only if one of the following holds: (i) the multipliers of periodic points are contained in the integer ring of an imaginary quadratic field, and (ii) all but finitely many periodic points have the same Lyapunov exponent. This solves two conjectures of Milnor.
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来源期刊
Forum of Mathematics Pi
Forum of Mathematics Pi Mathematics-Statistics and Probability
CiteScore
3.50
自引率
0.00%
发文量
21
审稿时长
19 weeks
期刊介绍: Forum of Mathematics, Pi is the open access alternative to the leading generalist mathematics journals and are of real interest to a broad cross-section of all mathematicians. Papers published are of the highest quality. Forum of Mathematics, Pi and Forum of Mathematics, Sigma are an exciting new development in journal publishing. Together they offer fully open access publication combined with peer-review standards set by an international editorial board of the highest calibre, and all backed by Cambridge University Press and our commitment to quality. Strong research papers from all parts of pure mathematics and related areas are welcomed. All published papers are free online to readers in perpetuity.
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