Combinatorics, Dynamics, and Geometry of Postcritically Finite Rational Maps

后临界有限有理图的组合学、动力学和几何

• 批准号: 0400852
• 负责人: Kevin Pilgrim
• 金额: --
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-05-01 至 2008-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400852&HistoricalAwards=false
• 关键词: Combinatorics; Dynamics; Geometry; Postcritically; Finite

DMS 0400852K PilgrimIndiana UniversityIn many instances, a dynamical system $f: X \to X$ having good metric expanding properties can be shown to be determined by purely combinatorial data. That is, there is a computable model $f^{comb}: X^{comb} \to X^{comb}$ and a homeomorphism $h: X^{comb} \to X$ conjugating $f^{comb}$ to $f$. A recent new construction shows that for a very wide class of examples, one may take $X^{comb}$ to be the Boundary at infinity of a certain associated infinite Gromov hyperbolic one-complex and $f^{comb}$ to be a simplicial covering map from this complex to itself. The construction is analogous to describing the action of a Kleinian group on its limit set via its action on theboundary of its Cayley graph. Techniques from geometric group theory imply the existence of a natural class of metrics on $X^{comb}$ such that the system becomes ``uniformly quasiregular'', i.e. iterates of $f^{comb}$distort the roundness of balls by at most a constant factor independent of scale and of iterate. Succintly: this process uniformizes $f: X \to X$ by providing a metric on $X$ for which $f$ is nearly conformal. The PI proposes to apply this new point of view to the study of rational maps of the Riemann sphere to itself. The PI will develop a unified framework with which to describe the dynamics of rational maps, Kleinian groups, and their natural generalizations. The PI will thereby (i)reinterpret Thurston's characterization of rational functions from thispoint of view; (ii) bring new methods into the study of dynamical systems; (iii) further the development of the ``dictionary'' between rational maps and Kleinian groups, focusing on those aspects related to Cannon's conjecture and the regularity of the new metric; (iv) suggestnew directions for the study of dynamical systems by formulating and developing a self-contained theory of uniformly quasiregular expanding dynamical systems, and (v) connect several areas of mathematics of current active interest, namely conformal dynamics, geometric grouptheory, geometric topology, and analysis on metric spaces.

DMS 0400852 K PilgrimIndiana大学在许多情况下，具有良好度量扩展性质的动力系统$f：X \to X$可以被证明是由纯粹的组合数据确定的。 也就是说，有一个可计算模型$f^{comb}：X^{comb} \to X^{comb}$和一个同胚$h：X^{comb} \to X$将$f^{comb}$共轭到$f$。 最近的一个新的构造表明，对于一类非常广泛的例子，可以取$X^{comb}$作为某个相关联的无限Gromov双曲一复形在无穷远处的边界，而$f^{comb}$作为从该复形到自身的单纯覆盖映射。 这种构造类似于描述Kleinian群在其极限集上的作用通过它在其Cayley图的边界上的作用。 几何群论的技巧暗示了在X^{comb}$上存在一个自然的度量类，使得系统成为“一致拟正则的”，也就是说，f^{comb}$的迭代使球的圆度最多扭曲一个与尺度和曲率无关的常数因子。 成功地：这个过程通过在$X$上提供一个度量使$f $近似共形，从而使$f：X \到X$一致化。 PI建议将这一新的观点应用于研究黎曼球面到自身的有理映射。 PI将开发一个统一的框架来描述有理映射，Kleinian群及其自然推广的动力学。 因此，PI将（i）从这个角度重新解释Thurston的有理函数的特征;（ii）将新的方法引入动力系统的研究;（iii）进一步发展有理映射和Kleinian群之间的“词典”，重点关注与Cannon猜想和新度量的规律性有关的方面;（iv）通过制定和发展一致拟正则扩展动力系统的自包含理论，为动力系统的研究提供了最新的方向，以及（v）连接当前积极感兴趣的几个数学领域，即共形动力学，几何群论，几何拓扑，和度量空间的分析。