Recurrence and Chaos of Local Dendrite Maps

LetX be a local dendrite, and letf be a continuous self-mapping ofX. Let E(X) represent the subset of endpoints ofX. LetAP(f) denote the subset of almost periodic points of f, R(f) be the subset of recurrent points of f, and P(f) be the subset of periodic points of f. In this work, it is shown that R(f) AP(f) if and only if E(X) is countable. Also, we show that if E(X) is countable, then R(f) X (respectively, R(f) X) if and only if either X S1, and f is a homeomorphism topologically conjugate to an irrational rotation, or P(f) X (respectively, P(f) X). In this setting, we derive that if E(X) is countable, then, on local dendrites≠S1, transitivity chaos.


Introduction
A metric space that is compact and connected is called a continuum. An arc is any space homeomorphic to the compact interval [0, 1]. A topological space is arcwise connected if any two of its points can be joined by an arc. A circle is any space homeomorphic to a simple closed curve.
A continuum that is locally connected and uniquely arcwise connected is called a dendrite. In this setting, every two distinct points x and y can be joined by a single arc denoted by [x, y]. Note that, in a dendrite, any subcontinuum is also a dendrite ( [1], Corollary 10.6).
Let x be a point of a dendrite D. We consider the order of x in D as de ned in [1], De nition 9.3, p. 141. If the order of x in D is nite, then it is equal to the number of connected components of D\ x { }. If the order of x in D is in nite, then it is countable and the diameters of connected components of D\ x { } tend to zero ( [2], (2.6), p. 92). By an endpoint of D, we mean a point of order 1. We denote E(D) the subset of all endpoints of D. A point x ∈ D\E(D) is called a cut point (see ([1], eorem 10.7, p. 168)). A branching point of D is a point of order ≥3. e set of all branching points will be denoted by B(D). According to [1], eorem 10.23, B(D) is at most countable.
A local dendrite is a continuum such that every point of it has a dendrite neighborhood. We consider that X is a local dendrite. e point x ∈ X is a branching point if it has a closed neighborhood U, which is a dendrite, and x is a branching point of U. e set of all branching points of X will be denoted by B(X). According to eorem 10.23 of [1] and eorem 4 of [3], B(X) is at most countable. A point a ∈ X is an endpoint of X if it admits an arc neighborhood U in X and U\ a { } is connected. E(X) is the set of endpoints of X.
Recall that a graph is a continuum, which can be written as the union of nitely many arcs any two of which are either disjoint or intersect only in one or both of their endpoints (i.e., it is a one-dimensional compact connected polyhedron).
Let N be the set of positive integers. A map is a continuous function. If X is a topological space, then a selfmapping is a map from X to itself.
Let f denote a self-mapping of a continuum X and x ∈ X. e orbit of the point x is O(x, f) f n (x): n ∈ N . e point x is periodic if f n (x) x for some nonzero integer n. e subset of all periodic points of f will be represented by P(f). e point x is recurrent if there exists an increasing sequence (m k ) ∈ N verifying f m k (x) converges to x. Note that every iterated of a recurrent point is also recurrent. e subset of all recurrent points of f will be represented by R(f). e point x is almost periodic provided that for every open set U containing x there is an integer N such that, for all n ∈ N, f n+i (x): i � 0, 1, . . . , N ∩ U ≠ ∅.
e subset of all almost periodic points of f will be represented by AP(f). By definition, we get the following inclusions: Such inclusions are not reversible.
We say that f is as follows: A compact metric space X has the APR property if, for every continuous self-mapping f of X, we get the equality AP(f) � R(f) (see ([4], Definition 1.1)).
In [5], Lemma 3.1, it was shown that the graph has the APR property. A dendrite D has the APR property if and only if E(D) is countable ( [6], eorem 1.2).
In this study, we show that a local dendrite X has the APR property if and only if E(X) is countable (see eorem 4).
Many works studied maps such that P(f) or R(f) verifies additional properties. For example, [7][8][9] studied homeomorphisms satisfying P(f) � X. In [10], it is proved that, for interval map, if P(f) is closed then P(f) is equal to the set non-wandering points Ω(f). For graph maps, Mai proved in [11], eorem 4.4, the following theorem. Also, for graph maps, Hattab proved in [5], Main eorem, the following theorem.  In this study, we show that for a local dendrite map f: X ⟶ X, such that E(X) is countable, the condition that f is monotone is not essential in eorem 3 (see eorem 6).
Let f be a self-mapping of a compact metric space without isolated points (X, d). We say that f is a transitive map if there is x 0 ∈ X such that O(x 0 , f) � X. e map f is sensitive if there is a positive number α with ∀x ∈ X, and for every neighborhood U of x, there exist y ∈ U and an integer n satisfying d(f n (x), f n (y)) > α. e map f is chaotic in the sense of Devaney if f is transitive, P(f) � X, and f is sensitive [13]. To simplify notation, in this study chaotic means chaotic in the sense of Devaney. By [14], the transitivity and the density of periodic points imply the sensitivity condition. Also, it is easy to see that a transitive map satisfies R(f) � X. us, on all compact metric space without isolated points satisfying the PR property: P(f) � R(f) (intervals, trees, dendrites, Warsaw circle), and transitivity � chaos ( [15,16]). For In this study, it is shown that on local dendrites ≠S 1 , transitivity � chaos (see eorem 7).

Recurrence and Almost Periodicity on
Local Dendrites roughout this section, the letter X denotes a local dendrite. In [3], eorem 4, p. 303, it is proved that the following conditions are equivalent: (a) X is a local dendrite. (b) X is a locally connected continuum, which contains at most a finite number of circles. (c) X is a continuum and there exist a finite number of If S 1 , S 2 , . . ., S r are the circles in the local dendrite X, then Γ(X) is the intersection of all the subgraphs in X containing the union of S i ′ s. erefore, Γ(X) is the smallest graph containing all circles of the local dendrite X. Lemma 1. Let X be a local dendrite. en, for any connected component C of X\Γ(X), C ∩ Γ(X) is reduced to a branching point.
Proof. By [17], Lemma 2.9, any arc in X with endpoints x, y ∈ Γ(X) is included in Γ(X). us, by [17], Lemma 2.11, for any connected component C of X\Γ(X), C ∩ Γ(X) is reduced to a point z. If z ∉ B(X), then there exists a unique arc [z, y] in Γ(X) emanating from z and y ∈ B(Γ(X)). erefore, Γ(X)\(a, b] is a graph of X, which contains all circles and smaller than Γ(X), a contradiction. is ends the proof.
Set X\Γ(X) � ∪ i∈A C i , where C i is the connected components of X\Γ(X). By Lemma 1, for any i ∈ A, C i ∩ Γ(X) is reduced to a branching point z i . Since B(X) is at most countable and the order in X is at most countable, the set A is at most countable. Let A k be a subset of A such that, contains all circles of X, by [17], Lemma 2.9, we obtain that (C k ) k is pairwise disjoint sub-dendrites of X. Suppose now that X ≠ Γ(X). Let A be an arc contained in X\Γ(X).
e symbols A, � A, and zA denote the closure, the interior, and the endpoints of A as an arc, respectively.
We define the mapping r A : X ⟶ A as follows.
Since A is an arc contained in X\Γ(X), there exists i ∈ A such that A ⊂ C i . Since C i is a dendrite, by [1], Lemma 10.25, there exists a retraction r ′ : e map r i : X ⟶ C i such that the restriction of r i to C i is the identity and r(X\C i ) � z i is a retraction. We put It is easy to see that r A is a retraction from X onto A.
Note that any arc A is totally ordered by a standard total order < . Consequently, we get a preorder in X defined by p≺q if and only if r A (p) < r A (q).
e main achievements of this section are to prove the following theorem.

Theorem 4. A local dendrite X has the APR property if and only if E(X) is countable.
If X � Γ(X), then, by [5], Lemma 3.1, X has the APR property. If X ≠ Γ(X), then we need the following lemmas.
Recall that a subset S of a topological space is dense in itself if S contains no isolated points. Lemma 2. Let X be a local dendrite and f denote a con- Note that the proof of Lemma 2 extends [18], Lemmas 2-4, and the proof of [18], eorem 2, shown originally for dendrites, to the case of local dendrites.
Proof. Let X be a local dendrite. Assume that A ⊂ X\Γ(X) is an arc. Let i ∈ A, and let C i be a connected component of X\Γ(X) such that A ⊂ C i and C i ∩ Γ(X) � z i . Consider the retraction r A � r ′ ∘ r i from X onto A, and recall that C i is a dendrite.
We start by the following claims. Let A � [x, y]. Let p ∈ A, and let ε > 0. Since C i is a dendrite, by [18], Lemma 2, there is α > 0 with is proves Claim 1.
Claim 2. Let p 0 < q 0 denotes two points in A such that p 0 < r A (f(p 0 )) and r A (f(q 0 )) < q 0 . Hence, there is a fixed point z of f satisfying p 0 < r A (z) < q 0 .
Let f i : C i ⟶ C i be the restriction of r i ∘ f to C i . Recall that r A � r ′ ∘ r i . We claim that and indeed, Similarly, r ′ (f i (q 0 )) < q 0 . By [18], Lemma 3, there is a fixed point z ∈ which implies that z � z ′ and so z is a fixed point of f.
which implies that z � z i , but r A (z i ) is an endpoint of A, which contradicts p 0 < r ′ (z) < q 0 . is completes the proof of Claim 2.
Claim 3. Suppose that there is S ⊂ X a nonempty dense in itself subset satisfying the following properties: is a dense in itself subset of the dendrite C i 0 . en, for any arc B ⊂ C i 0 , the set B ∩ S i 0 ⊂ B ∩ S, and so, by (2), B ∩ S i 0 is discrete. Consequently, by [18], Lemma 4 is ends the proof of Claim 3. Now, we use Claims 1-3 to prove the lemma.
en, f (m+n) p ≠ f n p. erefore, S is dense in itself.

Journal of Mathematics 3
Consider B � [a, b] ⊂ X\Γ(X) an arc. Let i ∈ A, and let C i be a connected component of X\Γ(X) such that B ⊂ C i and C i ∩ Γ(X) � z i . Suppose that B ∩ S contains a point, which is not isolated. Consequently, B ∩ S ≠ ∅. Pick x � f n (p) ∈ B ∩ S.
{ }) and the arc [x, f m (p)] ⊂ B(x, α) (note that local dendrites are uniformly locally arcwise connected). Assume, for example, that x < f m (p). Let g � f m− n . Note that g: X ⟶ X, g(x) ∈ B, and x < g(x).
Using an inductive argument, over k, we will show that y ≤ r B g k (y) , for each y ∈ [x, g(x)] and every k ∈ N( * ).

(8)
To prove the initialization of ( * ), we argue by contradiction; indeed, we assume that there is y ∈ [x, g(x)] satisfying r B (g(y)) < y. Since x < g(x) � r B (g(x)) and r B (g(y)) < y, x < y, and by Claim 2, there is a fixed point z is leads to a contradiction with the selection of ε (B(x, ε) ∩ P(f) � ∅) and ends the initialization step of the proof of ( * ).
) is an open neighborhood of x disjoint from the subset g k (x): k ∈ N .
us, x ∉ R(g) � R(f m− n ), but according to [20], eorem I, is leads to a contradiction, since x ∈ S ⊂ R(f). erefore, for every arc B in X\Γ(X), such that, for all i ∈ A, r B (z i ) ∈ zB, the set B ∩ S is discrete. the open arcs (a, c) or (c, b) and we proceed as the Case 1 to lead to a contradiction.
us, for all arc B ⊂ X\Γ(X), the set B ∩ S is discrete. Finally, a direct application of Claim 3 implies that E(X) is uncountable.

Lemma 3.
Retractions preserve the APR property; i.e., let Y denote a subcontinuum of a continuum X, which is a retract of X. If X has the APR property, then Y has also the APR property.
Proof. Let g denote a self-mapping of Y, and r: X ⟶ Y be a retraction. e composition f � g ∘ r is a self-mapping of X.
us, by [21], Lemma 3.1, for all n ∈ N, f n � g n ∘ r. erefore, by the proof of [21], Proposition 3.2, erefore, AP(f) � AP(g). Consequently, if the continuum X has the APR property, then AP(f) � R(f), which implies that AP(g) � R(g). us, Y has the APR property. e following result is due to [22], Lemma 2.4. In this study, we complete its proof. □ Lemma 4. Let X be a graph, and C ⊂ X be a Cantor set. en, there is a self-mapping g of X such that C is the closure of an orbit of g and X\C is g-invariant.
Proof. Since C is homeomorphic to the Cantor ternary set, C will be totally ordered. For convenience, we may assume that 0 � min C and 1 � max C. By [23], eorem 2.1, there exists a minimal homeomorphism φ: C\ 0, 1 { } ⟶ C\ 0, 1 { } defined on the non-compact locally compact and totally disconnected set C\ 0, 1 { }. It is easy to see that there exists a unique homeomorphism ψ of C whose restriction to C\ 0, 1 { } is φ and ψ(0) � 0 and ψ(1) � 1. Put BE(X) � x ∈ X: x is a branching point or an end point of X . We define h: C ∪ BE(X) ⟶ X as follows: It is easy to see that h is continuous. e set X\(C ∪ BE(X)) is a union of countably many open arcs (A n � (a n1 , a n2 )) whose endpoints belong to C ∪ BE(X). By an open arc, we mean here and below always an arc without its endpoints, which is simultaneously an open set in X, and its endpoints belong to C ∪ BE(X), which is disjoint with C ∪ BE(X). Define g: X ⟶ X as follows. If is an open arc with endpoints a n1 , a n2 ∈ C ∪ BE(X) (note that A n is an open set in X). Consider all arcs in X\(C ∪ BE(X)) emanating from g(a) and disjoint out of g(a). ere are only finitely many such arcs. Let B n be one with minimal diameter. Let r: A n ⟶ B n be a homeomorphism sending a n1 to g(a n1 ) and a n2 to the other endpoint of B. Finally, put g(x) � r(x) on A.
We need to prove that g is continuous. e proof of the continuity of g in F ∪ BE(X) is the same as in [22], Lemma 2.4.
Now let x ∈ X\(F ∪ BE(X)). Let (x n ) be a sequence of X, which converges to x. Let A be an open arc with endpoints a, b ∈ F ∪ BE(X) containing x. ere exists N ∈ N such that, for all n ≥ N, x n ∈ A. If g(a) � g(b), then, for all n ≥ N, g(x n ) � g(a) � g(x). If g(a) ≠ g(b), then, for all n ≥ N, g(x n ) � r(x n ). Since r is a homeomorphism, (r(x n )) converges to r(x). In both cases, g is continuous.
□ Proof. Let X be a local dendrite such that X\Γ(X) ≠ ∅.
Suppose that X has an uncountable subset of endpoints E(X). Since E(X) � ∪ i∈A (E(C i )\ z i ) and A is countable, there exists i ∈ A such that E(C i ) is uncountable. By [6], eorem 1.2, C i does not have the APR property. Since r i : X ⟶ C i is a retraction, by Lemma 3, X does not have the APR property. erefore, if X has the APR property, then the set E(X) is countable.
Conversely, we suppose that E(X) is countable. Let f be a self-mapping of X. If R(f) ⊂ AP(f), then R(f) � AP(f) and we are done. Assume then that R(f)\AP(f) is nonempty.
Since R(f) and AP(f) are invariant (see ([24], Propositions 4.1.2 and 4.2.4)), is a closed f-invariant subset of Γ(X). By Lemma 4, there exists a continuous map g: Γ(X) ⟶ Γ(X) such that g is an extension of f: F ⟶ Γ(X). It is easy to see that R(f) ⊂ R(g); consequently, w ∈ R(g). M is also a nonempty closed g-invariant subset such that M⊊O(w, g). us, O(w, g) is not a minimal set of g; consequently, by [24], eorem 4.2.2, w ∉ AP(g). If w ∈ AP(g)\AP(g), then there exists a minimal set M⊊O(w, g). We distinguish two subcases. Case 2.1. M is finite Let (a, b) be a connected component of Γ(X)\M ∪ BE(Γ(X)) containing w. Since M ∪ BE(Γ(X)) is a finite subset, the restriction of g to M is a homeomorphism.

Case 2.1. M is infinite
We recall the construction of g given in the proof of [22], Lemma 4. Let h be the function defined by h( is the set of branching points or endpoints of Γ(X). It is easy to see that h is continuous. e set Γ(X)\F ∪ BE(Γ(X)) is a union of countably many open arcs (I n ) n in Γ(X) with boundary zI n belonging to F ∪ BE(Γ(X)) (note that A is an open set in Γ(X)). Define g: Γ(X) ⟶ Γ(X) as follows.
there is an open arc A with endpoints a, b ∈ F ∪ BE(Γ(X)) such that x ∈ A. Consider all arcs in Γ(X)\F ∪ BE(Γ(X)) whose endpoints are g(a) and g (b). ere are only finitely many such arcs. Let B be one with minimal diameter. Denote by r a homeo- morphism A ⟶ B sending a to g(a) and b to g(b).
Finally, put g(x) � r(x) on A. It is easy to see that Γ(X)\F ∪ BE(Γ(X)) is g-invariant.
If w ∈ AP(g), then one of the connected components of Γ(X)\F ∪ BE(Γ(X)), say A, contains a point p of AP(g) and has w as an endpoint, and the other endpoint is w ′ ∈ F ∪ BE(Γ(X)). Since A is an open set, there exists n ∈ N such that g n (p) ∈ A. Since Γ(X)\F ∪ BE(Γ(X)) is g-invariant, g n (A) is a connected subset of Γ(X)\F ∪ BE(Γ(X)), which intersects A.
Consequently, BE(Γ(X))) � w, w ′ and so g n ( w, w ′ ) ⊂ w, w ′ . If g n ( w, w ′ ) � w { }, then g n (w) � w, which contradicts the fact that w ∉ AP(g). We suppose that g n ( w, w ′ ) � w ′ . Let (v k ) be a sequence of A, which converges to w. Since g n (A) � A, there exists a sequence (u k ) of A such that, for all k, g n (u k ) � v k . Since A is compact, (u k ) has at least an accumulation point u ∈ A. Since g is continuous, g n (u) � w. Since F ∪ BE(Γ(X)) and Γ(X)\F ∪ BE(Γ(X)) are g-invariant, u ∈ (F ∪ BE(Γ(X))) ∩ A, which implies that u ∈ w, w ′ . e fact that g n ( w, w ′ ) � w ′ implies that g n (u) � w ′ , which contradicts the fact that w ≠ w ′ . Consequently, g n ( w, w ′ ) � w, w ′ . us, g n (w) � w and g n (w ′ ) � w ′ or g n (w) � w ′ and g n (w ′ ) � w, which implies that g 2n (w) � g n (g n (w)) � g n (w ′ ) � w, which contradicts the fact that w ∉ AP(g). erefore, w ∉ AP(g), which implies that w ∈ R(g)\AP(g), which contradicts [5], Lemma 3.1. In both cases, we get R(f) � AP(f).

Pointwise Recurrent Local Dendrite Map
It was shown, in [4], eorems 5.1 and 5.4, that for a monotone local dendrite map f: X ⟶ X we get the following equivalence: Note that the equivalence ( * ) is false for an arbitrary local dendrite mapping. Indeed, on the interval, the tent map is a transitive nonrecurrent map [25]. Also, by eorem 5.1 of [26], there exists a transitive nonrecurrent dendrite map.
e main results of this section are the following theorems.

Conflicts of Interest
e authors declare that they have no conflicts of interest.