Virtual immediate basins of Newton maps and asymptotic values
Virtual immediate basins of Newton maps and asymptotic values
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DOI：
10.1155/imrn/2006/65498
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发表时间：
2006-01
影响因子：
1
通讯作者：
Xavier Buff;Johannes Rueckert
中科院分区：
文献类型：
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作者：
Xavier Buff;Johannes Rueckert
应用于(超越的)整函数f：C-gt；C的牛顿求根法是亚纯函数N的迭代。众所周知，如果对于某个初值z，牛顿法收敛于C中的点x，则f在x处有根。我们证明了在许多情况下，如果牛顿法的轨道收敛到无穷远，则f在无穷远处有一个虚根。更确切地说，我们证明了如果N有一个满足一些温和假设的不变的Baker域，则0是f的一个渐近值。相反，如果f在0处有一个对数型的渐近值，则0上的奇点包含在N的一个不变的Baker域中，我们称之为虚拟直接盆。我们通过反例证明，对于更一般类型的奇点，情况并非如此。
Newton's root finding method applied to a (transcendental) entire function f:C->C is the iteration of a meromorphic function N. It is well known that if for some starting value z, Newton's method converges to a point x in C, then f has a root at x. We show that in many cases, if an orbit converges to infinity for Newton's method, then f has a `virtual root' at infinity. More precisely, we show that if N has an invariant Baker domain that satisfies some mild assumptions, then 0 is an asymptotic value for f. Conversely, we show that if f has an asymptotic value of logarithmic type at 0, then the singularity over 0 is contained in an invariant Baker domain of N, which we call a virtual immediate basin. We show by way of counterexamples that this is not true for more general types of singularities.