Non-landing hairs in Sierpiński curve Julia sets of transcendental entire maps

Abstract

We consider the family of transcendental entire maps given by f a (z) = a(z − (1 − a)) exp(z + a) where a is a complex parameter. Every map has a superattracting fixed point at z = −a and an asymptotic value at z = 0. For a > 1 the Julia set of f a is known to be homeomorphic to the Sierpiński universal curve [19], thus containing embedded copies of any one-dimensional plane continuum. In this paper we study subcontinua of the Julia set that can be defined in a combinatorial manner. In particular, we show the existence of nonlanding hairs with prescribed combinatorics embedded in the Julia set for all parameters a ≥ 3. We also study the relation between non-landing hairs and the immediate basin of attraction of z = −a. Even as each non-landing hair accumulates onto the boundary of the immediate basin at a single point, its closure, nonetheless, becomes an indecomposable subcontinuum of the Julia set.

Singular perturbations of complex polynomials

“…Presumably, one would get a combination of entire and rational dynamics in certain cases. Such an interesting mixture of entire and polynomial dynamics has been seen in some families recently [34]. In addition, some of the features we observed earlier, such as Sierpiński curve Julia sets, do arise in other families, such as the meromorphic maps known as the Weierstrass elliptic p-functions [38].…”

Convergence of rays with rational argument in hyperbolic components—an illustration in transcendental dynamics

“…In the literature, the family (1) appears first in [26], where it was proved that the Julia set is locally connected for real parameter values a > 1, in particular, a Sierpinski carpet. In [17], existence of nonlanding hairs with prescribed combinatorics was shown for all real parameters ⩾ a 3, and also the topological structure of some nonlanding hairs were studied.…”

Spiders’ webs and locally connected Julia sets of transcendental entire functions

“…It is a characteristic of the Sierpiński curve that it contains a homeomorphic copy of every one-dimensional plane continuum. This was explored by Garijo, Jarque and Moreno Rocha [18], who have made a detailed study of the function g a , and demonstrated the existence of indecomposable continua in its Julia set.…”