Abstract
Motivated by a problem in complex dynamics, we examine the block structure of the natural action of iterated monodromy groups on the tree of preimages of a generic point. We show that in many cases, including when the polynomial has prime power degree, there are no large blocks other than those arising naturally from the tree structure. However, using a method of construction based on real graphs of polynomials, we exhibit a non-trivial example of a degree 6 polynomial failing to have this property. This example settles a problem raised in a recent paper of the second author regarding constant weighted sums of polynomials in the complex plane. We also show that degree 6 is exceptional in another regard, as it is the lowest degree for which the monodromy group of a polynomial is not determined by the combinatorics of the post-critical set. These results give new applications of iterated monodromy groups to complex dynamics.
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Acknowledgments
The authors wish to thank Laurent Bartholdi for reading a preliminary version of the article and for several helpful comments.
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Rafe Jones’ work was partially supported by NSF grant DMS-0852826 and Han Peters’ work was partially supported by NSF grant DMS-0757856.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Jones, R., Peters, H. Blocks of monodromy groups in complex dynamics. Geom Dedicata 150, 137–150 (2011). https://doi.org/10.1007/s10711-010-9499-2
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DOI: https://doi.org/10.1007/s10711-010-9499-2
Keywords
- Iterated monodromy groups
- Complex dynamics
- Polynomial iteration
- Post-critically finite polynomials
- Conservative polynomials
- Constant weighted sum of iterates