# Families of Rational Maps and Iterative Root-Finding Algorithms

 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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