树和多项式的动力学
Trees and the dynamics of polynomials
摘要：
多项式映射$f : {\bf C} \arrow {\bf C}$的无穷盆带有自然的叶理和带有奇点的平坦度量，使其成为度量化的黎曼曲面$X(f)$。当$f$在多项式的模空间中发散时，表面$X(f)$沿着其叶理折叠以产生具有极限动力学$F : T \arrow T$的度量化单纯树$(T,η)$。在本文中，我们的特点，出现的极限树，并显示他们提供了一个自然的边界$\PT_d$紧化的模空间的多项式的次数$d$。我们发现，$(T,η,F)$记录的限制行为的乘数在周期点，任何发散亚纯族的多项式$\{f_t(z) : t \mem Δ^* \}$可以完成一个独特的树在其中心纤维。最后证明了在立方模空间$\PT_3$的边界是一棵树。度量化树$(T,η,F)$在迭代有理映射的设置中提供了作为双曲流形极限出现的${\bf R}$ -树的对应物。
英文摘要：
The basin of infinity of a polynomial map $f : {\bf C} \arrow {\bf C}$ carries a natural foliation and a flat metric with singularities, making it into a metrized Riemann surface $X(f)$. As $f$ diverges in the moduli space of polynomials, the surface $X(f)$ collapses along its foliation to yield a metrized simplicial tree $(T,η)$, with limiting dynamics $F : T \arrow T$. In this paper we characterize the trees that arise as limits, and show they provide a natural boundary $\PT_d$ compactifying the moduli space of polynomials of degree $d$. We show that $(T,η,F)$ records the limiting behavior of multipliers at periodic points, and that any divergent meromorphic family of polynomials $\{f_t(z) : t \mem Δ^* \}$ can be completed by a unique tree at its central fiber. Finally we show that in the cubic case, the boundary of moduli space $\PT_3$ is itself a tree. The metrized trees $(T,η,F)$ provide a counterpart, in the setting of iterated rational maps, to the ${\bf R}$-trees that arise as limits of hyperbolic manifolds.
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