On the accumulation sets of exponential rays

Abstract

We show that there exist non-landing exponential rays with bounded accumulation sets. By introducing folding models of certain rays, we prove that each of the corresponding accumulation sets is an indecomposable continuum containing part of the ray, an indecomposable continuum disjoint from the ray or a Jordan arc.

A new type of non-landing exponential rays

“…Proof. By Lemma 3.1 and a similar argument as in the proof of Lemma 5.3 in [6], we get that there is a universal constant…”

“…The organization of the paper is as follows. In §2, we recall some basic knowledge on the escaping rays of exponential maps, hyperbolic expansion lemma and Bounded-wiggling lemma appeared in [6]. In §3, we show how to determine the itineraries of the non-landing rays whose accumulation sets will be homeomorphic to the closed topologist's sine curve and define the folding points of the rays.…”

“…Proof. According to Lemma 5.1 of [6], there is a universal constant D 2 > 0 such that for all γ 0 n (s) ⊂ T 1 and all 0…”

“…For example, the authors in [3,4,5,14] constructed some non-landing rays, each of whose accumulation sets is unbounded and an indecomposable continuum containing some ray in the extended complex plane. Recently, the first author and G. Zhang constructed some non-landing rays, each of whose accumulation sets is bounded and can be an indecomposable continuum containing part of the ray, an indecomposable continuum disjoint from the ray or a Jordan arc [6].…”

“…For later use, we sometimes denote G ± k by G k for simplicity. Denote by I ± k the corresponding itinerary block (1, ±1, 0, 1 k ) of G ± k , where 1 k is the block of 1 of length k. For our purpose, we will be only concerned with the itineraries of the following form [6]). There is a δ 0 ∈ (0, 1) such that for all k ≥ 1 and z ∈ X, we have…”