In this talk we will explore parameter spaces of meromorphic maps with finitely many singular values. We will list various types of parameters for which the orbits of one or more singular values have peculiar characteristics, for example, are preperiodic, converge to a parabolic cycle, or are truncated after a certain number of iterates. We will look at the relations between such parameters and show that several of these types of parameters are dense in the bifurcation locus. To do so we will need to implement new strategies with respect to the rational case. This is joint work with N. Fagella and M. Astorg.
Bifurcations arise when there is a drastic change in the solutions of some equation depending on a parameter, as the parameter varies. In this talk we study bifurcations in holomorphic families of meromorphic maps with finitely many singular values. The equation(s) that we will study are the equations defining periodic points of period n. Such equations are crucial in complex dynamics because the Julia set (the set on which the dynamics is chaotic) is the closure of repelling periodic points. The celebrated results by Mane-Sad-Sullivan for families of rational maps (and independently by Lyubich, and by Levin for polynomials) show that in a set of parameters where no bifurcations of periodic points occur, the Julia set stays almost the same and so does the dynamics; precisely speaking, all maps are topologically conjugate in such set. Moreover, they establish a precise correlation between bifurcations of periodic points and a change of behaviour in the orbits of singular values. The key new feature that appears for families of meromorphic maps is that periodic points can disappear at infinity at specific parameters, creating a new type of bifurcations. Our work connects this new type of bifurcations with change of behaviour in singular orbits, to establish Mane-Sad-Sullivan's Theorem for meromorphic maps. This is joint work with Matthieu Astorg and Nùria Fagella.