NON-TRIVIAL DEFORMATION OF AN ENTIRE FUNCTION abz +ebz + c

Abstract

In this note, we consider the dynamics of the family of transcendental entire functions f(z; a, b, c) = abz + e bz + c. In particular, while studying the case when they have a Baker domains, we investigate the area of Julia sets and the Hausdorff convergence of Julia sets. Consider the family f (z; a, b, c) = abz + e bz + c. These f (z; a, b, c) with a, b = 0 are structurally infinite [7], for they have infinitely many critical values. For the sake of simplicity, we assume that a, b and c are real. Theorem 1. If ab ≥ 1, a > 0, c ≤ −1 and a log a − a + c < 0, then f = f (z; a, b, c) has a Baker domain B = B a,b,c , which is completely invariant.In particular, the Julia set is the boundary of B.Proof. We set D 0 = {z|Re z < 0}. Elementary calculation shows that D 0 is mapped by f to a proper subdomain in D 0 , and hence D 0 is invariant. The Fatou component B containing D 0 is a Baker domain, for if z ∈ D 0 , then Re f (z) < abRe z.

Singular values of two-parameter families λ((bz − 1)/z)μ and λ(z/(bz − 1))η

“…This exponential family is simpler than other families which have more than one or infinitely many singular values. Some dynamical properties of such types of families of functions including exponential map were studied in [7][8][9][10][11]. Often these investigations are very applicable for studying of iterative methods associate to entire and meromorphic transcendental functions which have either critical values or asymptotic values, or both.…”

“…Often these investigations are very applicable for studying of iterative methods associate to entire and meromorphic transcendental functions which have either critical values or asymptotic values, or both. Enormously, singular values are useful in the dynamics of entire and meromorphic transcendental functions for describing the Julia sets and the Fatou sets [8][9][10][11][12].…”

On the Stability of Julia Sets of Functions having Baker Domains

“…Our main tool will be the fact that each f c is a logarithmic lift of an entire transcendental function, an easy calculation shows that if g λ (z) = λze z and λ = e −c , then exp •f c = g λ • exp. The family {g λ (z) = λze z , λ ∈ C * } has been intensively studied, for example in [20], [21] and [27], while the case f c , c ∈ R + , is examined in [43]. Before we state a theorem, recall the definition of the Brjuno set:…”

“…Morosawa and Taniguchi [43] proved this result for c = 0 directly in the c−plane, but it seems to be difficult to carry over their method to all c ∈ R .…”

“…0 is a parabolic fixed point, using that there is only one singular value which is contained in the Fatou set we see that the Fatou set consists of one cycle of parabolic domains (and possible preimages). The Fatou set of f 0 has already been described in [43], [44] or [49], it consists of infinitely many invariant Baker domains contained in horizontal strips of height 2π, the horizontal lines at 0 and 2πik, k ∈ N , are contained in the Julia set. For each point in such a Baker domain, the real part tends to −∞ under iteration.…”