Rigidity of Newton dynamics
Aix-Marseille Université - Site St Charles
3, place Victor Hugo - case 39
13331 MARSEILLE Cedex 03
Kostiantyn DRACH (I2M et V. N. Karazin Kharkiv National University)
The Newton map of a complex polynomial is a rational map coming from Newton’s root-finding method. This is "a map that wants to be iterated", as the numerical method suggests. From the point of view of complex dynamics, the Newton maps of degree d>2 form an important family of rational maps, arguably the largest family (beyond polynomials) that gained our substantial level of understanding in recent years. We contribute to this development by studying rigidity properties of Newton maps of arbitrary degree. One of our key results is that any two combinatorially equivalent Newton maps are quasi-conformally conjugate provided that they are either non-renormalizable, or renormalizable in the same way. In other words, Newton maps can be distinguished among each other in purely combinatorial terms modulo "embedded" polynomial dynamics. A similar statement in the dynamical plane ("dynamical" rigidity of Newton maps) also holds true. In the talk, we will outline the proofs of the aforementioned results with an emphasis on the construction of puzzle pieces for Newton maps. This is an important step because it allows us to employ the methods of symbolic dynamics in the spirit of celebrated Yoccoz puzzles, and the results on rigidity of polynomials (by Yoccoz, Kahn-Lyubich, Kozlovski-Shen-van Strien and others). Based on joint work with Dierk Schleicher.
- Kostiantyn DRACH