Bifurcation loci of families of finite type meromorphic maps

TL;DR

This paper proves that in families of finite type meromorphic maps, stability is prevalent and introduces new bifurcation phenomena related to poles and essential singularities, extending classical results in complex dynamics.

Contribution

It establishes the density of stability in finite type meromorphic maps and introduces novel bifurcation analysis for maps with poles and essential singularities.

Findings

Stability is open and dense in families of finite type meromorphic maps.

New bifurcation types are characterized by periodic orbits exiting the domain.

The results extend classical stability theorems to broader classes of maps.

Abstract

We show that $J-$ stability is open and dense in natural families of meromorphic maps of one complex variable with a finite number of singular values, and even more generally, to finite type maps. This extends the results of Ma\~{n}\'e-Sad-Sullivan for rational maps of the Riemann sphere and those of Eremenko and Lyubich for entire maps of finite type of the complex plane, and essentially closes the problem of density of structural stability for holomorphic dynamical systems in one complex variable with finitely many singular values. This result is obtained as a consequence of a detailed study of a new type of bifurcation that arises with the presence of both poles and essential singularities (namely periodic orbits exiting the domain of definition of the map along a parameter curve), and in particular its relation with the bifurcations in the dynamics of singular values. The presence of these new bifurcation parameters require essentially different methods to those used in previous work for rational or entire maps.