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Regular version of the site

Article

Antiholomorphic perturbations of Weierstrass Zeta functions and Green's function on tori

Nonlinearity. 2017. Vol. 30. No. 8. P. 3241-3254.
Mamayusupov Khudoyor, Bogdanov K., Mukherjee S., Schleicher D.

In Bergweiler and Eremenko (2016 Proc. Am. Math. Soc. 144 2911–22), Bergweiler and Eremenko computed the number of critical points of the Green’s function on a torus by investigating the dynamics of a certain family of antiholomorphic meromorphic functions on tori. In this paper, we study the parameter space of this family of meromorphic functions, which can be written as antiholomorphic perturbations of Weierstrass Zeta functions. We give a complete topological description of the hyperbolic components and their boundaries, and we show that these sets admit natural parametrizations by associated dynamical invariants. This settles a conjecture, made in Lin and Wang (2010 Ann. Math. 172 911–54), on the topology of the regions in the upper half plane H where the number of critical points of the Green’s function remains constant.