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首页> 外文期刊>Memoirs of the American Mathematical Society >Newton's methond applied to two quadratic equations in C-2 viewed as a global dynamical system
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Newton's methond applied to two quadratic equations in C-2 viewed as a global dynamical system

机译：牛顿方法应用于C-2中的两个二次方程，被视为一个全局动力系统

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In this article we study the Newton map N : C-2 -> C-2 associated to two equations in two unknowns, as a dynamical system. We focus on the first non-trivial case: two simultaneous quadratics, to intersect two conics. In the first two chapters we prove among other things - The Russakovski-Shiffman measure does not charge the points of indeterminacy; - The lines joining pairs of roots are invariant, and the Julia set of the restriction of N to such a line has under appropriate circumstances an invariant manifold, which shares features of a stable manifold and a center manifold. The main part of the article concerns the behavior of N at infinity. To compactify C-2 in such a way that N extends to the compactification, we must take the projective limit of an infinite sequence of blow-ups. The simultaneous presence of points of indeterminacy and of critical curves forces us to define a new kind of blow-up: the Furey blow-up. This construction is studies in its own right in Chapter 4, where we show among others that the real oriented blow-up of the Farey blow-up has a topological structure reminiscent of the invariant tori of the KAM theorem. We also show that the cohomology, completed under the intersection inner product, is naturally isomorphic to the classical Sobolev space of functions with square-integrable derivatives. In Chapter 5, we apply these results to the mapping N in a particular case, which we generalize in Chapter 6 to the intersection of any two conics.
机译：在本文中，我们将牛顿映射N：C-2-> C-2与两个未知数中的两个方程相关联，作为一个动力系统。我们关注第一个非平凡的情况：两个同时的二次方，以与两个圆锥相交。在前两章中，我们除其他外证明：Russakovski-Shiffman测度不包含不确定性; -连接成对的根的线是不变的，并且在适当的情况下，将N限制为Julia的集具有不变的流形，该流形具有稳定流形和中心流形的特征。本文的主要部分涉及N在无穷大处的行为。要以使N扩展到压缩的方式压缩C-2，我们必须采用无穷大爆破序列的投影极限。同时存在不确定点和临界曲线，这迫使我们定义一种新的爆炸：Fury爆炸。在第4章中对这种构造进行了单独研究，除其他外，我们证明了Farey炸弹的真正定向炸弹具有拓扑结构，让人联想到KAM定理的不变托里。我们还表明，在相交内积下完成的同调与具有平方可积导数的经典Sobolev函数空间自然同构。在第5章中，我们将这些结果应用于特定情况下的映射N，在第6章中将其推广到任意两个圆锥曲线的交点。

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