论文标题
复杂动力学中的同型轨道，乘数光谱和刚性定理
Homoclinic orbits, multiplier spectrum and rigidity theorems in complex dynamics
论文作者
论文摘要
本文的目的是回答有关理性地图的乘数频谱的几个猜想和问题，并通过结合复杂和非架构的动力学工具，提供了复杂动力学中几种刚性定理的新证明。 由于麦克穆伦（McMullen）的一个显着定理断言，除了灵活的lattès家族外，周期性点的乘数频谱还决定了合理地图的共轭类别，从而有限地选择了许多选择。证据依赖于瑟斯顿的刚性定理来进行后有限的地图，其中teichmüller理论是必不可少的工具。我们将在不使用列表图或teichmüller理论的情况下提供麦克穆伦定理的新证明。 我们表明，除了灵活的Lattès家族外，周期性点的长度频谱决定了合理地图的共轭类别，以有限多种选择。这概括了上述麦克穆伦定理。我们还将证明具有标记长度光谱的刚性定理。类似的想法也产生了由于Zdunik引起的刚性定理的简单证明。 我们表明，当且仅当以下一个保存之一（i）周期点的乘数之一中包含在假想二次段落的整数中时，有理图是例外的； （ii）几乎有限的许多周期点都具有相同的lyapunov指数。这解决了米尔诺的两个猜想。
The aims of this paper are answering several conjectures and questions about 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 where Teichmüller theory is an essential tool. We will give a new proof of McMullen'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 generalize 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; (ii) all but finitely many periodic points have the same Lyapunov exponent. This solves two conjectures of Milnor.
