TY - JOUR
AU - Chen, Tao
AU - Keen, Linda
PY - 2021/09/19
Y2 - 2021/12/08
TI - Dynamics of the meromorphic families $f_\lambda=\lambda \tan^pz^q$
JF - New Zealand Journal of Mathematics
JA - NZ J Math
VL - 52
IS -
SE - Vaughan Jones Memorial Special Issue
DO - 10.53733/135
UR - https://nzjmath.org/index.php/NZJMATH/article/view/135
SP - 469--510
AB - <p>This paper continues our investigation of the dynamics of families of transcendental meromorphic functions with finitely many singular values all of which are finite. <span class="Apple-converted-space"> </span>Here we<span class="Apple-converted-space"> </span>look at a generalization of the family of polynomials $P_a(z)=z^{d-1}(z- \frac{da}{(d-1)})$, the family $f_{\lambda}=\lambda \tan^p z^q$.<span class="Apple-converted-space"> </span>These functions have a super-attractive fixed point, and, depending on $p$, one or two asymptotic values. <span class="Apple-converted-space"> </span>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. <span class="Apple-converted-space"> </span>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. <span class="Apple-converted-space"> </span>We also give a description of the parameter plane of the family $f_{\lambda}$.<span class="Apple-converted-space"> </span>Again there are similarities to and differences from<span class="Apple-converted-space"> </span>the parameter plane of the family $P_a$ and again<span class="Apple-converted-space"> </span>there are new techniques. <span class="Apple-converted-space"> </span>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<span class="Apple-converted-space"> </span>points.</p>
ER -