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Blocks of monodromy groups in complex dynamics
- Source :
- Geometriae dedicata, 150(1), 137-150. Springer Netherlands
- Publication Year :
- 2011
- Publisher :
- Springer Netherlands, 2011.
-
Abstract
- Motivated by a problem in complex dynamics, we examine the block structure of the natural action of 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.<br />15 pages, 4 figures
- Subjects :
- 37F10, 20B25, 30D05
Polynomial
Group (mathematics)
010102 general mathematics
Monodromy theorem
Dynamical Systems (math.DS)
Group Theory (math.GR)
010103 numerical & computational mathematics
01 natural sciences
Generic point
Combinatorics
Monodromy
Iterated function
ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION
FOS: Mathematics
Degree of a polynomial
Geometry and Topology
Mathematics - Dynamical Systems
0101 mathematics
Mathematics - Group Theory
Prime power
Mathematics
Subjects
Details
- Language :
- English
- ISSN :
- 15729168 and 00465755
- Volume :
- 150
- Issue :
- 1
- Database :
- OpenAIRE
- Journal :
- Geometriae dedicata
- Accession number :
- edsair.doi.dedup.....d8020b049b82013ae49b0a57c6652ddb