On Postcritically Minimal Newton Maps

  • The Newton map of an entire map f is defined by N_f(z):=z-f(z)/f'(z). For f(z)=p(z)e^q(z), where p and q are polynomials, its Newton map N_pe^q=z-p(z)/(p'(z)+p(z) q'(z)}) is a rational function. For a Newton map N_pe^q the finite fixed points are (super)attracting and are roots of p. The point at infinity is a parabolic fixed point with deg(q) petals for a Newton map N_pe^q. For fixed integers d>=3 and 1<=n<=d, let deg(p)=d-n with p having only simple roots and deg(q)=n, then we have deg(N_pe^q)=d. The parameter plane (parametrized by coefficients of p and q) of Newton maps satisfying properties above is of complex dimension d-2. Due to existence of the parabolic fixed point for these Newton maps, we can not have post-critically finiteness condition in this family. But there exists an analogous notion, that we call it ``post-critically minimal", to the notion of post-critically finite. The properties of post-critically minimal Newton maps are also similar to those of post-critically finite Newton maps of polynomials. Using surgery tools developed by P. Haissinsky and G. Cui we give a full classification of post-critically minimal Newton maps in terms of post-critically finite Newton maps of polynomials. The latter was recently classified by Y. Mikulich, R. Lodge and D. Schleicher.

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Meta data
Publishing Institution:IRC-Library, Information Resource Center der Jacobs University Bremen
Granting Institution:Jacobs Univ.
Author:Khudoyor Mamayusupov
Referee:Dierk Schleicher, John Hubbard, Marcel Oliver
Advisor:Dierk Schleicher
Persistent Identifier (URN):urn:nbn:de:gbv:579-opus-1000025
Document Type:PhD Thesis
Date of Successful Oral Defense:2015/08/12
Year of Completion:2015
Date of First Publication:2015/12/10
Academic Department:Mathematics & Logistics
PhD Degree:Mathematics
Library of Congress Classification:Q Science / QA Mathematics (incl. computer science)
Focus Area:Mobility
Call No:Thesis 2015/40

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