Colloquium

In this talk, I will give an introduction about our program to understand the parameter spaces of dynamical systems generated by meromorphic functions with finitely many singular values. I will give a description of the bifurcation diagram of the tangent family and the parameter space of meromorphic functions with precisely two finite asymptotic values and one attracting fixed point. It represents a step beyond the previous work for the parameter space of quadratic polynomials and the parameter space of degree two rational functions. In particular, for the parameter space of degree two rational maps that have analogous constraints: two critical values and an attracting fixed point. What is interesting and promising for pushing the general program even further, is that, despite the presence of the essential singularity and infinite degree for the covering, our family of meromorphic functions exhibits a dynamic structure as similar as one could hope to the rational case, and that the philosophy of the techniques used in the rational case could be adapted. This talk is based on a recent joint work with Tao Chen and Linda Keen.