Abstract
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.
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Notes
- 1.
We denote the n-th iterate of a dynamical system f : X → X by fn : X → X.
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Acknowledgements
This research was partially supported by the Deutsche Forschungsgemeinschaft (DFG), as well as the Advanced Grant 695621 HOLOGRAM of the European Research Council (ERC), which is gratefully acknowledged. The authors are most grateful to the anonymous referee for very helpful comments that have led to marked improvements.
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Lodge, R., Mikulich, Y., Schleicher, D. (2022). A Classification of Postcritically Finite Newton Maps. In: Ohshika, K., Papadopoulos, A. (eds) In the Tradition of Thurston II. Springer, Cham. https://doi.org/10.1007/978-3-030-97560-9_13
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