A Classification of Postcritically Finite Newton Maps

Abstract

AbstractThe dynamical classification of rational maps is a central concern of holomorphic dynamics. Much progress has been made, especially on the classification of polynomials and some approachable one-parameter families of rational maps; the goal of finding a classification of general rational maps is so far elusive. Newton maps (rational maps that arise when applying Newton’s method to a polynomial) form a most natural family to be studied from the dynamical perspective. Using Thurston’s characterization and rigidity theorem, a complete combinatorial classification of postcritically finite Newton maps is given in terms of a finite connected graph satisfying certain explicit conditions.KeywordsNewton mapRational mapParameter spaceRenormalizationHubbard treeCombinatorial classificationExtended Newton graphThurston’s theorem2010 Mathematics Subject ClassificationPrimary 30D0537F1037F20