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dc.contributor.author	McMullen, Curtis T.
dc.date.accessioned	2012-11-07T14:55:12Z
dc.date.issued	1987
dc.identifier.citation	McMullen, Curtis T. 1987. Families of rational maps and iterative root-finding algorithms. Annals of Mathematics 125(3): 467–493.	en_US
dc.identifier.issn	0003-486X	en_US
dc.identifier.uri	http://nrs.harvard.edu/urn-3:HUL.InstRepos:9876064
dc.description.abstract	In this paper we develop a rigidity theorem for algebraic families of rational maps and apply it to the study of iterative root-finding algorithms. We answer a question of Smale's by showing there is no generally convergent algorithm for finding the roots of a polynomial of degree 4 or more. We settle the case of degree 3 by exhibiting a generally convergent algorithm for cubics; and we give a classification of all such algorithms. In the context of conformal dynamics, our main result is the following: a stable algebraic family of rational maps is either trivial (all its members are conjugate by Mobius transformations), or affine (its members are obtained as quotients of iterated addition on a family of complex tori). Our classification of generally convergent algorithms follows easily from this result. As another consequence of rigidity, we observe that the eigenvalues of a nonaffine rational map at its periodic points determine the map up to finitely many choices. This implies that bounded analytic functions nearly separate points on the moduli space of a rational map.	en_US
dc.description.sponsorship	Mathematics	en_US
dc.language.iso	en_US	en_US
dc.publisher	Princeton University	en_US
dc.relation.isversionof	doi:10.2307/1971408	en_US
dash.license	LAA
dc.title	Families of Rational Maps and Iterative Root-Finding Algorithms	en_US
dc.type	Journal Article	en_US
dc.description.version	Version of Record	en_US
dc.relation.journal	Annals of Mathematics	en_US
dash.depositing.author	McMullen, Curtis T.
dc.date.available	2012-11-07T14:55:12Z

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