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Abstract
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In this paper, we investigate the dynamics of iterating the Weierstrass elliptic
functions on vertical real rhombic lattices. The main result of this paper is to show
that these functions can have at most one real attracting or parabolic cycle. If there
is no real attracting or parabolic cycle, we prove that the real and imaginary axes, as
well as translations of these lines by the lattice, lie in the Julia set. Further, we prove
that if there exists a real attracting fixed point, then the intersection of the Julia set
with the real axis is a Cantor set. Finally, we apply the theorem to find parameters in
every real rhombic shape equivalence class for which the Julia set is the entire
sphere.
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Keywords
complex dynamics, meromorphic functions, Julia sets
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Mathematical Subject Classification 2010
Primary: 54H20, 37F10, 37F20
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Milestones
Received: 15 May 2015
Revised: 22 April 2016
Accepted: 2 May 2016
Published: 14 December 2016
Communicated by Michael E. Zieve
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