Invariant Sets for Monotone Local Dendrite Maps

Abstract

Let [Formula: see text] be a local dendrite and let [Formula: see text] be a monotone map. Denote by [Formula: see text], RR[Formula: see text], UR[Formula: see text], [Formula: see text] the set of periodic (resp., regularly recurrent, uniformly recurrent, recurrent) points and [Formula: see text] the union of all [Formula: see text]-limit sets of [Formula: see text]. We show that if [Formula: see text] is nonempty, then (i) [Formula: see text]. (ii) R[Formula: see text] if and only if every cut point is a periodic point. If [Formula: see text] is empty, then (iii) [Formula: see text]. (iv) R[Formula: see text] if and only if [Formula: see text] is a circle and [Formula: see text] is topologically conjugate to an irrational rotation of the unit circle [Formula: see text]. On the other hand, we prove that [Formula: see text] has no Li–Yorke pair. Moreover, we show that the family of all [Formula: see text]-limit sets of [Formula: see text] is closed with respect to the Hausdorff metric.

Equicontinuous local dendrite maps

“…Note that, by [1,Proposition 3.6], if f is a monotone onto local dendrite map, then f (Γ(X)) = Γ(X). Now we introduce some notations used in the proof of Lemma 4.3.…”

“…Recently, several authors have been interested in studying local dendrite maps (for example one can see [1,2,4]).…”

“…Then, by [17, Corollary 5.1], f | Γ(X) is equicontinuous. We further have X \ Γ(X) is the union of pairwise disjoint dendrites (C k ) such that C k ∩ Γ(X) = {z k } and as f n0 (z k ) = z k hence by [1,Lemma 3.8] one has f n0 (C k ) = C k and since R(f ) = R(f n0 ) = X, f n0 : C k → C k is pointwise recurrent. Thus, by Lemma 4.1 and Lemma 4.4, f | C k is equicontinuous for each k which implies f | X\Γ(X) is equicontinuous ((C k ) are pairwise disjoint dendrites).…”

Minimal sets and orbit spaces for group actions on local dendrites

“…On the other hand, for a group G acting on a local dendrite X different from a circle, we show (see Theorem 5.5) that the following properties are equivalent: (1) (G, X) is pointwise almost periodic. (2) The orbit closure relation R = {(x, y) ∈ X × X : y ∈ G(x)} is closed. (3) Every non-endpoint of X is periodic.…”

“…[2],Lemma 4.3). Let X be a local dendrite with metric d. Then for any ε > 0 there is 0 < δ < ε such that if d(x, y) < δ then diam([x, y]) < ε.Lemma 2.4.…”