[论文解读] Quasisymmetric rigidity in one-dimensional dynamics

Trevor Clark, Sebastian van Strien|arXiv (Cornell University)|May 23, 2018

Mathematical Dynamics and Fractals参考文献 47被引用 9

一句话总结

本文通过分析实界和复界、多项式类映射以及拟共形共轭，建立了在一维动力系统中的拟对称刚性。研究证明，在拟共形等价下，双曲吸引子和无限可重正化系统表现出刚性，关键结果包括持久盒映射的存在性以及在吸引子域中的外部拟共形共轭。

ABSTRACT

In the late 1980's Sullivan initiated a programme to prove quasisymmetric rigidity in one-dimensional dynamics: interval or circle maps that are topologically conjugate are quasisymmetrically conjugate (provided some obvious necessary assumptions are satisfied). The aim of this paper is to conclude this programme in a natural class of $C^3$ mappings. Examples of such rigidity were established previously, but not, for example, for real polynomials with non-real critical points. Our results are also new for analytic mappings. The main new ingredients of the proof in the real analytic case are (i) the existence of infinitely many (complex) domains associated to its complex analytic extension so that these domains and their ranges are compatible, (ii) a methodology for showing that combinatorially equivalent complex box mappings are qc conjugate, (iii) a methodology for constructing qc conjugacies in the presence of parabolic periodic points. For a $C^3$ mapping, the dilatation of a high iterate of any complex extension of the real map will in general be unbounded. To deal with this, we introduce dynamically defined $qc\backslash bg$ $partitions$, where the appropriate mapping has bounded quasiconformal dilatation, except on sets with "bounded geometry". To obtain such a partition we prove that we have very good geometric control for infinitely many dynamically defined domains. Some of these results are new even for real polynomials, and in fact an important sequence of domains turn out to be quasidiscs. This technology also gives a new method for dealing with the infinitely renormalizable case. We will briefly also discuss why quasisymmetric rigidity is such a useful property in one-dimensional dynamics.

研究动机与目标

• 建立具有双曲吸引子的一维动力系统的拟对称刚性。
• 利用复界和实界分析无限可重正化系统的刚性。
• 证明在双曲吸引子的吸引域中存在拟共形外部共轭。
• 将刚性结果推广至具有持久盒映射和中心级联的系统。
• 研究在拟共形等价下周期点和临界点处的局部动力学。

提出的方法

• 利用复界和实界控制重正化映射的几何结构。
• 应用多项式类映射和拟圆盘构造拟共形划分。
• 对中心部分使用qcg划分以分析持久动力学。
• 通过定理 LABEL:thm:external_conjugacy 建立拟共形外部共轭的存在性。
• 利用定理 LABEL:thm:touching_box_map 分析接触盒映射及其刚性。
• 应用渐近全纯映射和改进的delta-好划分以控制畸变。

实验结果

研究问题

• RQ1 在一维动力系统具有双曲吸引子时，拟对称刚性在何种条件下成立？
• RQ2 复界和实界如何促进可重正化系统中盒映射的持久性？
• RQ3 拟共形共轭在具有无限可重正化临界点的系统的刚性中起什么作用？
• RQ4 中心级联和持久盒映射如何影响动力系统的刚性？
• RQ5 在何种情形下，同一动力类中的映射之间存在外部共轭？

主要发现

• 定理 LABEL:thm:box_mapping_persistent 建立了在重正化下盒映射的持久性。
• 定理 LABEL:thm:external_conjugacy 证明了在双曲吸引子的吸引域中存在拟共形外部共轭。
• 命题 LABEL:prop:rigidity_attracting 通过qc共轭确认了在双曲吸引子处的刚性。
• 定理 LABEL:thm:central_cascades 展示了无限可重正化系统中中心级联的刚性。
• 命题 LABEL:prop:good_bounds 和 LABEL:prop:modified_delta_nice 提供了对拟共形划分中畸变的控制。
• 定理 LABEL:thm:box_mapping_reluctant 将刚性结果扩展至在复界下具有无限分支的系统。