Indecomposable Continua and Misiurewicz Points in Exponential Dynamics

Abstract

In this paper we describe several new types of invariant sets that appear in the Julia sets of the complex exponential functions Eλ(z) = λez where λ ∈ ℂ in the special case when λ is a Misiurewicz parameter, so that the Julia set of these maps is the entire complex plane. These invariant sets consist of points that share the same itinerary under iteration of Eλ. Previously, the only known types of such invariant sets were either simple hairs that extend from a definite endpoint to ∞ in the right half plane or else indecomposable continua for which a single hair accumulates everywhere upon itself. One new type of invariant set that we construct in this paper is an indecomposable continuum in which a pair of hairs accumulate upon each other, rather than a single hair having this property. The second type consists of an indecomposable continuum together with a completely separate hair that accumulates on this continuum.

Joining polynomial and exponential combinatorics for some entire maps

“…The union of a tail with its endpoint is known as a hair associated to s and in this case, we say that the hair lands at z s . For examples of non-landing hairs that instead accumulate on an indecomposable continuum, see [10].…”

Escaping Endpoints Explode

“…Nonetheless, there is still a natural notion of "rays" or "hairs", generalising the curves mentioned above. While some of these may no longer land at finite endpoints [11,31], Schleicher and Zimmer [42] have shown that there is always a setẼ( f a ) of escaping endpoints, which moreover has a certain universal combinatorial structure that is independent of the parameter a, making it a natural object for dynamical considerations. For the purpose of this introduction, we define the relevant concepts as follows; see Sect.…”

Singular Values Of One Parameter Family \lambda\frac{b^2-1}{z}

“…The dynamics of one parameter family z e  , that has only one singular value, is studied in detail [1,2,3]. This exponential family is simpler than other families of functions which involving exponential maps [4,5,6,7] and having more than one or infinitely many singular values.…”