Equicontinuity of maps on dendrites

Abstract

Given a dendrite X and a continuous map f : X → X, we show the following are equivalent: (i) ω f is continuous and Per(f ) = n∈N f n (X); (ii) ω(x, f ) = Ω(x, f ) for each x ∈ X; and (iii) f is equicontinuous. Furthermore, we present some examples illustrating our results.2010 Mathematics Subject Classification. Primary: 54H20. Secondary: 37B45. 1 Definitions and preliminariesLet Z be a metric space, then given A ⊆ Z and ǫ > 0, the open ball about A of radius ǫ is denoted by V(A, ǫ). The interior, clousure, boundary and cardinality of A are denoted by A • , A, Bd(A) and |A|, respectively. A map is a continuous function. Given a compact metric space X, we denote by 2 X the set of all nonempty closed subsets of X, topologized by the Hausdorff metric which is defined as follows: for A, B ∈ 2 X , we setA continuum is a nonempty, compact, connected metric space. We say that a continuum X is a simple closed curve if X is homeomorphic to S 1 = {z ∈ C : |z| = 1}. A continuum X is a dendrite provided that X is locally connected and does not contain a simple closed curve. It is clear that dendrites are uniquely arcwise connected continua. So, if X is a dendrite and x, y ∈ X, we denote by [x, y] the unique arc joining x and y; also (x, y) will denote [x, y] \ {x, y}.Let X be a continuum. A point p ∈ X is called an end point of X provided that whenever U is open and p ∈ U, there exists an open set V ⊆ U such that x ∈ V and |Bd(V )| = 1. We denote by End(X) the collection of end points of X. Furthermore, p is a cut point of X, if X \ {p} is disconnected.

Equicontinuous mappings on finite trees

“…A few words regarding the equivalence between (a) and (d) are in order. By [5,Theorem 4.12] together with [14,Lemma 2.6], the implication (a)⇒(d) still holds if X is merely a dendrite with finitely many branching points. The reverse implication holds for an arbitrary dendrite and a surjective map by [10,Theorem 5.2].…”

“…The dendrite X together with the map f appears in [5,Example 5.1]. We reproduce their description here for three reasons: for the reader's convenience, to point out a few observations about f that are not made in [5], and in order to be able to also describe the map g. We build X by taking infinitely many disjoint arcs indexed by Z, {I n | n ∈ Z}, with each I n of length 1/2 |n| , and identifying in a single point v (the vertex) one end of each I n . The result X = n∈Z I n is a dendrite with a single infinite-order branching point v, as in Figure 1.…”

“…, I 2 n ) in such a way that f 2 n−1 2 n i=2 n−1 +1 I i is the identity map (and g fixes every point in each I m with m ≤ 2). As a result, Per(g) = X, and so g is equicontinuous by [5,Theorem 4.14]; at the same time, although f is pointwise-periodic, X contains points of arbitrarily high period (if x ∈ I m for 2 n−1 +1 ≤ m ≤ 2 n , n ∈ N \ {1}, then the period of x is 2 n−1 ) and therefore, for every n ∈ N, we have Fix(g n ) = ∞ m=1 g m [X] and g n ∞ m=1 g m [X] is not the identity map. 4.2.…”

“…Further interesting results regarding equicontinuity of a map f : X → X have been obtained by Mai [10] when X is a finite graph, and by Camargo, Rincón and Uzcátegui [5] when X is a dendrite. May [10,Theorem 5.2] shows that if X is a finite graph, then f is equicontinuous if and only if ∞ m=1 f m [X] = Rec(f ) (here Rec(f ) denotes the set of recurrent points of f , to be defined later in Definition 2.3; for the moment just note that Per(f ) ⊆ Rec(f )); and Camargo et al [5,Theorem 4.12] prove that if X is a dendrite, then f is equicontinuous if and only if cl X (Per(f )) = ∞ m=1 f m [X] plus an extra condition on the ω-limit sets of f . Obtaining a simultaneous strengthening of these two results at the expense of considering a less general class of spaces, we prove that when X is a finite tree, f is equicontinuous if and only if Per(f ) = ∞ m=1 f m [X].…”

“…(4) Further conditions equivalent to equicontinuity of a map f on a space X have been established in [15,Theorem 2,p. 62] for X a finite tree, in [10,Theorem 5.2] when X is a finite graph, and in [5,Theorem 4.12] in the case of X being an arbitrary dendrite.…”