Dynamical approximation and kernels of nonescaping-hyperbolic components

Abstract

Let F_n be families of entire functions, holomorphically parametrized by a complex manifold M. We consider those parameters in M that correspond to nonescaping-hyperbolic functions, i.e., those maps f in F_n for which the postsingular set P(f) is a compact subset of the Fatou set F(f) of f. We prove that if F_n converge to a family F in the sense of a certain dynamically sensible metric, then every nonescaping-hyperbolic component in the parameter space of F is a kernel of a sequence of nonescaping-hyperbolic components in the parameter spaces of F_n. Parameters belonging to such a kernel do not always correspond to hyperbolic functions in F. Nevertheless, we show that these functions must be J-stable. Using quasiconformal equivalences, we are able to construct many examples of families to which our results can be applied.