A characterization of smooth Cantor bouquets

Abstract

Abstract.We prove that all smooth fans having a dense set of endpoints are topologically equivalent.Let I bea smooth fan whose set of endpoints is dense in X. Such fans have been constructed, e.g., by J. H. Roberts [6], who proved that the space of rational sequences of the Hubert cube can be embedded in the Cantor fan, and by A. Lelek [3], who showed the existence of a fan whose (one-dimensional) set of endpoints can be connectified by adding the vertex. Lately, spaces similar to X \{v}, where v is the vertex of X, were discovered to be Julia sets of some nice analytic functions (see R. L. Devaney and M. Krych [2]; see also J. C. Mayer [4]). We are going to prove that all such examples are homeomorphic.Theorem. All smooth fans having dense set of endpoints are topologically equivalent.Let us recall that a continuum X is said to be hereditarily unicoherent if K n L is connected for every pair K , L of subcontinua of X . A continuum X is called a dendroid if it is arcwise connected and hereditarily unicoherent. By a fan we will mean a dendroid having exactly one ramification point; we will call this point the vertex of X . A fan X is said to be smooth if the sequence of arcs [w,xn] converges to the arc [v,x] for every sequence xn converging to x , where x, xn e X and v is the vertex of X. If X is a fan, then E(X) will denote the set of endpoints ofX.Ifx.yGÍHxíH, then by \x -y\ we will denote the Euclidean distance between points x and y and by [x, v] we will mean the linear segment with endpoints x and y . Mappings between inverse systemsThe following lemma is similar to [5, Theorem 2 ]. The proof is a standard inductive argument and is left to the reader.

Complete Erdős space is unstable

“…for some m. By hypothesis (5) we have that this implies that Proof. Let m be such that x ∈ Y m and let ε > 0 be such that F ε (x) Y ⊂ C. Select a k ∈ N such that k > m, 1/k < ε/2, and x − ξ k (x) < ε/2 and let i be such that ξ k (x) = t i and k = n i .…”

“…The Lelek fan is a space that can be obtained by identifying the base arc of the hairy arc to a point. A similar argument as we used for the hairy arc shows that the homeomorphism group of the Lelek fan is not homeomorphic to E c (use [7] or [5] instead of [1]).…”

Iteration of meromorphic functions

“…Besides the papers already mentioned we refer to [1,41,106,120] for a further discussion of Cantor bouquets. 6.…”

The Lelek fan and the Poulsen simplex as Fraïssé limits

“…From now on, L will denote the limit of a Fraïssé sequence in ‡F. Fraïssé theory combined with our geometric theory of finite fans provides a simple proof of the uniqueness result, originally proved by Charatonik [5] and independently by Bula & Oversteegen [3]. Corollary 3.9 (Uniqueness).…”