## 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.