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A combinatorial classification of postcritically fixed Newton maps

Published online by Cambridge University Press:  13 March 2018

KOSTIANTYN DRACH ,
YAUHEN MIKULICH ,
JOHANNES RÜCKERT  and
DIERK SCHLEICHER
KOSTIANTYN DRACH
Affiliation:
Jacobs University Bremen, Research I, Campus Ring 1, 28759 Bremen, Germany email k.drach@jacobs-university.de, y.mikulich@gmail.com, jrueckert@jacobs-alumni.de, dierk@jacobs-university.de
YAUHEN MIKULICH
Affiliation:
Jacobs University Bremen, Research I, Campus Ring 1, 28759 Bremen, Germany email k.drach@jacobs-university.de, y.mikulich@gmail.com, jrueckert@jacobs-alumni.de, dierk@jacobs-university.de
JOHANNES RÜCKERT
Affiliation:
Jacobs University Bremen, Research I, Campus Ring 1, 28759 Bremen, Germany email k.drach@jacobs-university.de, y.mikulich@gmail.com, jrueckert@jacobs-alumni.de, dierk@jacobs-university.de
DIERK SCHLEICHER
Affiliation:
Jacobs University Bremen, Research I, Campus Ring 1, 28759 Bremen, Germany email k.drach@jacobs-university.de, y.mikulich@gmail.com, jrueckert@jacobs-alumni.de, dierk@jacobs-university.de
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Abstract

We give a combinatorial classification for the class of postcritically fixed Newton maps of polynomials as dynamical systems. This lays the foundation for classification results of more general classes of Newton maps. A fundamental ingredient is the proof that for every Newton map (postcritically finite or not) every connected component of the basin of an attracting fixed point can be connected to $\infty$ through a finite chain of such components.

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 11 , November 2019 , pp. 2983 - 3014
Copyright
© Cambridge University Press, 2018 

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References

Branner, B. and Fagella, N.. Quasiconformal Surgery in Holomorphic Dynamics. Cambridge University Press, Cambridge, 2014.Google Scholar
Bielefeld, B., Fisher, Y. and Hubbard, J.. The classification of critically preperiodic polynomials as dynamical systems. J. Amer. Math. Soc. 5(4) (1992), 721–762.Google Scholar
Douady, A. and Hubbard, J.. A proof of Thurston’s topological characterization of rational functions. Acta Math. 171 (1993), 263–297.Google Scholar
Dudko, D., Hlushchanka, M. and Schleicher, D.. A decomposition theorem for rational maps. In preparation.Google Scholar
Head, J.. The combinatorics of Newton’s method for cubic polynomials. Thesis, Cornell University, 1987.Google Scholar
Hubbard, J.. Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. Volume 2: Surface Homeomorphisms and Rational Functions. Matrix Editions, Ithaca, NY, 2016.Google Scholar
Hubbard, J., Schleicher, D. and Sutherland, S.. How to find all roots of complex polynomials by Newton’s method. Invent. Math. 146 (2001), 1–33.Google Scholar
Lodge, R., Mikulich, Y. and Schleicher, D.. Combinatorial properties of Newton maps. Preprint, 2015,arXiv:1510.02761.Google Scholar
Lodge, R., Mikulich, Y. and Schleicher, D.. A classification of postcritically finite Newton maps. Preprint, 2015, arXiv:1510.02771.Google Scholar
Luo, J.. Newton’s method for polynomials with one inflection value, Preprint, 1993, Cornell University.Google Scholar
Mamayusupov, K.. On postcritically minimal Newton maps. PhD Thesis, Jacobs University, 2015.Google Scholar
Mamayusupov, K.. Newton maps of complex exponential functions and parabolic surgery. Fund. Math., doi:10.4064/fm345-9-2017. Published online January 2018.Google Scholar
Mamayusupov, K.. A characterization of postcritically minimal Newton maps of complex exponential functions. Ergod. Th. & Dynam. Sys. doi:10.1017/etds.2017.137. Published online 25 January 2018.Google Scholar
Milnor, J.. Dynamics in One Complex Variable, 3rd edn. Princeton University Press, Princeton, NJ, 2006.Google Scholar
Przytycki, F.. Remarks on the simple connectedness of basins of sinks for iterations of rational maps. Collection: Dynamical Systems and Ergodic Theory, Warsaw, 1986 (Banach Center Publications, 23) . PWN, Warsaw, 1989, pp. 229–235.Google Scholar
Pilgrim, K. and Lei, Tan. Combining rational maps and controlling obstructions. Ergod. Th. & Dynam. Sys. 18 (1998), 221–245.Google Scholar
Roesch, P.. Topologie locale des méthodes de Newton cubiques: plan dynamique. C. R. Acad. Sci. Paris Série I 326 (1998), 1221–1226.Google Scholar
Rückert, J. and Schleicher, D.. On Newton’s method for entire functions. J. Lond. Math. Soc. (2) 75(3) (2007), 659–676.Google Scholar
Randig, M., Schleicher, D. and Stoll, R.. Newton’s method in practice II: The iterated refinement Newton method and near-optimal complexity for finding all roots of some polynomials of very large degrees. Preprint, 2017, arXiv:1703.05847v2, submitted.Google Scholar
Shishikura, M.. The connectivity of the Julia set and fixed points. Complex Dynamics: Families and Friends. Ed. Schleicher, D.. A. K. Peters, Welleseley, MA, 2009.Google Scholar
Schleicher, D.. On the efficient global dynamics of Newton’s method for complex polynomials. Preprint, 2011, arXiv:1108.5773.Google Scholar
Schleicher, D. and Stoll, R.. Newton’s method in practice: Finding all roots of polynomials of degree one million efficiently. J. Theoret. Comput. Sci. 681 (2017), 146–166.Google Scholar
Lei, Tan. Branched coverings and cubic Newton maps. Fund. Math. 154 (1997), 207–260.Google Scholar