Local uniform convergence and convergence of Julia sets

Abstract

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

Dynamical convergence of a certain polynomial family to f_a(z) = z + e^z + a

“…Proof. In the case that a belongs to an a-component of M, the claim is shown by the similar argument in [14]. If a ∈ B, then Re a < 0 by definition.…”

“…It is clear that R a,d converges uniformly on compact sets to f a as d → ∞. If |a| < 1, that is, F (f a ) only consists of attracting basins, then J(R a,d ) converges to J(f a ) in the Hausdorff metric by the similar argument of Kisaka [14]. However, it was shown in [23] that if a = −1, that is, f −1 has a Baker domain, then J(R −1,2d ) does not converge to J(f −1 ) in the Hausdorff metric.…”

“…Kisaka [14] extended this result as follows: Assume a sequence of polynomials P n converges uniformly on compact sets to a transcendental entire function f as n → ∞. If F (f ) contains all the singular values and consists only of basins of attracting cycles, then J(P n ) converges to J(f ) in the Hausdorff metric (see also [16] as remark).…”

Kernel convergence of hyperbolic components

“…The following lemma is well-known for rational maps on the Riemann sphere, but is more complicated to show in the current situation; see [14,16,17]. LEMMA 2.…”

“…In the dynamical plane the convergence of Julia sets with respect to the Hausdorff metric is shown in [15] for the above families for suitable values of the parameter λ. The convergence of Julia sets was obtained in [7] for polynomials of constant degree, in [19] for rational functions, and has now been generalized to larger classes of functions; see [14,16,17,20].…”