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Dynamics of the meromorphic families $f_\lambda=\lambda \tan^pz^q$
Author(s) -
Tao Chen,
Linda Keen
Publication year - 2021
Publication title -
new zealand journal of mathematics
Language(s) - English
Resource type - Journals
eISSN - 1179-4984
pISSN - 1171-6096
DOI - 10.53733/135
Subject(s) - meromorphic function , julia set , mathematics , lambda , singularity , complex plane , pure mathematics , multiplicity (mathematics) , combinatorics , mathematical analysis , physics , optics
This paper continues our investigation of the dynamics of families of transcendental meromorphic functions with finitely many singular values all of which are finite.   Here we  look at a generalization of the family of polynomials $P_a(z)=z^{d-1}(z- \frac{da}{(d-1)})$, the family $f_{\lambda}=\lambda \tan^p z^q$.  These functions have a super-attractive fixed point, and, depending on $p$, one or two asymptotic values.   Although many of the dynamical properties generalize, the existence of an essential singularity and of poles of multiplicity greater than one implies that significantly different techniques are required here.   Adding transcendental methods to standard ones, we give a description of the dynamical properties; in particular we prove the Julia set of a hyperbolic map is either connected and locally connected or a Cantor set.   We also give a description of the parameter plane of the family $f_{\lambda}$.  Again there are similarities to and differences from  the parameter plane of the family $P_a$ and again  there are new techniques.   In particular, we prove there is dense set of points on the boundaries of the hyperbolic components that are accessible along curves and we characterize these  points.

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