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Conformal Geometry and Dynamics

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Attractor sets and Julia sets in low dimensions


Author: A. Fletcher
Journal: Conform. Geom. Dyn. 23 (2019), 117-129
MSC (2010): Primary 30D05; Secondary 30C62, 30C65
DOI: https://doi.org/10.1090/ecgd/334
Published electronically: June 25, 2019
MathSciNet review: 3973918
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Abstract: If $ X$ is the attractor set of a conformal IFS (iterated function system) in dimension two or three, we prove that there exists a quasiregular semigroup $ G$ with a Julia set equal to $ X$. We also show that in dimension two, with a further assumption similar to the open set condition, the same result can be achieved with a semigroup generated by one element. Consequently, in this case the attractor set is quasiconformally equivalent to the Julia set of a rational map.


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Additional Information

A. Fletcher
Affiliation: Department of Mathematical Sciences, Northern Illinois University, Dekalb, Illinois 60115
Email: fletcher@math.niu.edu

DOI: https://doi.org/10.1090/ecgd/334
Received by editor(s): October 23, 2018
Received by editor(s) in revised form: March 6, 2019, and May 6, 2019
Published electronically: June 25, 2019
Additional Notes: This work was supported by a grant from the Simons Foundation (#352034, Alastair Fletcher).
Article copyright: © Copyright 2019 American Mathematical Society