3 research outputs found
A classification of postcritically finite Newton maps
The dynamical classification of rational maps is a central concern of
holomorphic dynamics. Much progress has been made, especially on the
classification of polynomials and some approachable one-parameter families of
rational maps; the goal of finding a classification of general rational maps is
so far elusive. Newton maps (rational maps that arise when applying Newton's
method to a polynomial) form a most natural family to be studied from the
dynamical perspective. Using Thurston's characterization and rigidity theorem,
a complete combinatorial classification of postcritically finite Newton maps is
given in terms of a finite connected graph satisfying certain explicit
conditions
A Combinatorial classification of postcritically fixed Newton maps
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 through a finite chain of
such components.Comment: 37 pages, 5 figures, published in Ergodic Theory and Dynamical
Systems (2018). This is the final author file before publication. Text
overlap with earlier arxiv file observed by arxiv system relates to an
earlier version that was erroneously uploaded separately. arXiv admin note:
text overlap with arXiv:math/070117