Combinatorics, Dynamics, and Geometry of Postcritically Finite Rational Maps

DMS 0400852K朝圣者印第安纳大学在许多情况下，具有良好度规扩展性质的动力系统$f：X\to X$可以被证明是由纯组合数据确定的。也就是说，存在一个可计算模型$f^{comb}：X^{comb}\to X^{comb}$和一个同胚$h：X^{comb}\to X$共轭$f^{comb}$到$f$。最近的一个新构造表明，对于一类非常广泛的例子，可以用$X^{comb}$作为某个相关的无限Gromov双曲型1-复形的无穷远边界，而$f^{comb}$则是从这个复形到它自己的单纯覆盖映射。这种构造类似于通过一个Kleian群在其Cayley图的边界上的作用来描述它在其极限集上的作用。几何群论的技巧表明，在$X^{comb}$上存在一类自然的度量，使得系统变得“一致准正则”，即$f^{comb}$的迭代至多使球的圆度扭曲一个与尺度和迭代无关的恒定因子。简明扼要地说：这个过程通过提供$X$上的度量来将$f：X统一为X$，其中$f$几乎是共形的。PI建议将这一新观点应用于Riemann球面到自身的有理映射的研究。PI将开发一个统一的框架来描述有理映射、Klein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瑟斯顿对有理函数的刻画；(Ii)将新的方法引入到动力系统的研究中；(Iii)进一步发展有理映射与Kleinan群之间的“词典”，重点关注与Cannon猜想和新度量的正则性有关的那些方面；(Iv)通过建立和发展一致准正则扩展动力系统的自含式理论，为动力系统的研究提出新的方向；以及(V)将当前活跃的几个数学领域联系起来，即共形动力学、几何群论、几何拓扑学和度量空间分析。

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.