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Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups


Authors: Rich Stankewitz and Hiroki Sumi
Journal: Trans. Amer. Math. Soc. 363 (2011), 5293-5319
MSC (2010): Primary 37F10, 37F50, 30D05
DOI: https://doi.org/10.1090/S0002-9947-2011-05199-8
Published electronically: May 20, 2011
MathSciNet review: 2813416
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Abstract: We discuss the dynamic and structural properties of polynomial semigroups, a natural extension of iteration theory to random (walk) dynamics, where the semigroup $ G$ of complex polynomials (under the operation of composition of functions) is such that there exists a bounded set in the plane which contains any finite critical value of any map $ g \in G$. In general, the Julia set of such a semigroup $ G$ may be disconnected, and each Fatou component of such $ G$ is either simply connected or doubly connected. In this paper, we show that for any two distinct Fatou components of certain types (e.g., two doubly connected components of the Fatou set), the boundaries are separated by a Cantor set of quasicircles (with uniform dilatation) inside the Julia set of $ G.$ Important in this theory is the understanding of various situations which can and cannot occur with respect to how the Julia sets of the maps $ g \in G$ are distributed within the Julia set of the entire semigroup $ G$. We give several results in this direction and show how such results are used to generate (semi) hyperbolic semigroups possessing this postcritically boundedness condition.


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Additional Information

Rich Stankewitz
Affiliation: Department of Mathematical Sciences, Ball State University, Muncie, Indiana 47306
Email: rstankewitz@bsu.edu

Hiroki Sumi
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, 1-1, Machikaneyama, Toyonaka, Osaka, 560-0043, Japan
Email: sumi@math.sci.osaka-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-2011-05199-8
Keywords: Complex dynamics, Julia sets, polynomial semigroups, random iteration, random complex dynamics
Received by editor(s): May 14, 2007
Received by editor(s) in revised form: August 13, 2009
Published electronically: May 20, 2011
Additional Notes: The first author was partially supported by the BSU Lilly V grant. He would also like to thank Osaka University for their hospitality during his stay there while this work was begun.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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