In this article we discuss combinatorial structure of the parameter plane of the family  The parameter space contains components where the dynamics are conjugate on their Julia sets. The complement of each of these components is the bifurcation locus. These are the hyperbolic components where the post-singular set is disjoint from the Julia set. We prove that all hyperbolic components are bounded except the four components of period one which are all simply connected.

capture component, shell component, quasi-conformal surgery, multiplier map, virtual center, bud component, period doubling.

Received: August 12, 2020; Accepted: November 14, 2020; Published: January 13, 2021

How to cite this article: Santanu Nandi, Combinatorial Structure of the Parameter Plane of the Family λ tan z², Far East Journal of Dynamical Systems 33(1) (2021), 1-38. DOI: 10.17654/DS033010001

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] B. Branner and A. Douady, Surgery on complex polynomials, Holomorphic Dynamics, Springer, 1988, pp 11-72.
[2] B. Branner, N. Fagella and X. Buff, Quasiconformal surgery in holomorphic dynamics, Vol. 141, Cambridge University Press, 2014.
[3] N. Fagella and L. Keen, Dynamics of purely meromorphic functions of bounded type, arXiv preprint arXiv:1702.06563, 2017.
[4] L. Keen and J. Kotus, Dynamics of the family λ tan z, Conformal Geometry and Dynamics of the American Mathematical Society 1(4) (1997), 28-57.
[5] W. H. Jiang, The parameter space of λ tan z, CUNY, Unpublished Ph.D. Thesis, 1991.