A note on the topology of escaping endpoints

Abstract

We study topological properties of the escaping endpoints and fast escaping endpoints of the Julia set of complex exponential $\exp (z)+a$ when $a\in (-\infty ,-1)$ . We show neither space is homeomorphic to the whole set of endpoints. This follows from a general result stating that for every transcendental entire function $f$ , the escaping Julia set $I(f)\cap J(f)$ is first category.

The topological dimension of radial Julia sets

“…For Ė(f a ) this is a consequence of the proof of [3,Theorem 3.6] and the characterization in [12]. It is unknown whether Ė(f a ) and E are in fact topologically equivalent; see [13,Question 1]. Based on Corollary 11 and the fact that E is not a G δσ -space, we note the following.…”

“…The escaping endpoint set Ė(f a ) shares many of its topological properties with Erdős space E := {x ∈ ℓ 2 : x n ∈ Q for all n < ω}. For example, Ė(f a ) and E are both almost zero-dimensional first category F σδ -spaces (see [13]), and Ė(f a ) ∪ {∞} and E ∪ {∞} are connected (see [3] and [7]). Also, each point in either space is contained in a closed copy of E c .…”

On cohesive almost zero-dimensional spaces

“…Erdős space E = {x ∈ 2 : x i ∈ Q for each i < ω} and complete Erdős space E c = {x ∈ 2 : x i ∈ {0} ∪ {1/n : n = 1, 2, 3, ...} for each i < ω}, where 2 stands for the Hilbert space of square summable sequences of real numbers. Other examples include the homeomorphism groups of the Sierpiński carpet and Menger universal curve [33,8], and various endpoint sets in complex dynamics [2,31]. Almost zero-dimensionality of E and E c follows from the fact that each closed ε-ball in either space is closed in the zero-dimensional topology inherited from Q ω , which is weaker than the 2 -norm topology.…”

On cohesive almost zero-dimensional spaces

“…Erdős space E = {x ∈ 2 : x i ∈ Q for each i < ω} and complete Erdős space E c = {x ∈ 2 : x i ∈ {0} ∪ {1/n : n = 1, 2, 3, ...} for each i < ω}, where 2 stands for the Hilbert space of square summable sequences of real numbers. Other examples include the homeomorphism groups of the Sierpiński carpet and Menger universal curve [33,8], and various endpoint sets in complex dynamics [2,31].…”