A Classification of Postcritically Finite Newton Maps

Russell Lodge , Yauhen Mikulich , Dierk Schleicher (1)

1 I2M - Institut de Mathématiques de Marseille

The 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.

Mots clés: Newton map; Rational map; Parameter space; Renormalization; Hubbard tree; Combinatorial classification; Extended Newton graph; Thurston’s theorem

Domaines: Mathématiques [math]

Russell Lodge, Yauhen Mikulich, Dierk Schleicher. A Classification of Postcritically Finite Newton Maps. Springer Verlag, 2022, In the Tradition of Thurston II, pp.421-448. ⟨10.1007/978-3-030-97560-9_13⟩. ⟨hal-04086088⟩