Local uniform convergence and convergence of Julia sets

Abstract

Let fn and f be entire functions and suppose that fn converges to f locally uniformly on C. Then if the Fatou set of f consists only of basins of attracting cycles or is empty, the Julia set of fn converges to that of f in the Hausdorff metric. We also show that expanding f implies the above assumption. Next we show that for each singular value c of f there exists a singular value c(n) of fn (for each sufficiently large n) converging to c. As an application, we propose a criterion which determines by the approximating sequence (fn)n=0infinity whether the Julia set of S is the whole sphere C for a certain class of entire functions.