Combinatorics and Dynamics of Iterated Rational Maps

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The goal of this research project is a combinatorial classification of all postcritically finite rational maps that are Newton maps of polynomials. This is a large class of all rational maps of given degree, and this will provide the first classification of a family of rational maps beyond polynomials or special one-parameter families. Along the way, we plan to answer a question of Steven Smale who asked for a classification of all those Newton maps that have additional attracting cycles (that is, attracting cycles that are not the roots of the given polynomials). This classification will be done in terms of graphs that we call "Newton graphs" and that naturally occur in the dynamical plane of the Newton maps. We also intend to investigate which postcritically finite Newton maps are "matings" of two polynomials: the latter is a known method to describe the dynamics of certain rational maps in terms of two polynomials of the same degree. It is known that all cubic Newton maps can be understood in this way (together with a related method called "capture"), but in higher degrees very little is currently known. Finally, we plan to extend our classification to all those rational Newton maps that arise as Newton maps of transcendental functions: these differ from polynomial Newton maps in the way that they have a parabolic, rather than repelling, fixed point at infinity.