@ Appl. Gen. Topol. 22, no. 1 (2021), 67-77doi:10.4995/agt.2021.13446 © AGT, UPV, 2021 Equicontinuous local dendrite maps Aymen Haj Salem a, Hawete Hattab b,a, Tarek Rejeiba a,∗ a Laboratory of Dynamical Systems and Combinatorics, Sfax University, Tunisia (hajsalem@hotmail.com, tarekhamzah28@gmail.com) b Umm Alqura University Makkah, Saudi Arabia. (hshattab@uqu.edu.sa) Communicated by F. Balibrea Abstract Let X be a local dendrite, and f : X → X be a map. Denote by E(X) the set of endpoints of X. We show that if E(X) is countable, then the following are equivalent: (1) f is equicontinuous; (2) ∞⋂ n=1 f n (X) = R(f); (3) f| ∞⋂ n=1 f n (X) is equicontinuous; (4) f| ∞⋂ n=1 f n (X) is a pointwise periodic homeomorphism or is topologically conjugate to an irrational rotation of S 1 ; (5) ω(x, f) = Ω(x, f) for all x ∈ X. This result generalizes [17, Theorem 5.2], [24, Theorem 2] and [11, Theorem 2.8]. 2010 MSC: 37B05; 37B20; 37B45. Keywords: dendrite; equicontinuity; local dendrite; recurrent point. ∗This work is supported by the laboratory of research ”Dynamical systems and combina- torics” University of Sfax, Tunisia Received 11 April 2020 – Accepted 26 September 2020 http://dx.doi.org/10.4995/agt.2021.13446 A. H. Salem, H. Hattab and T. Rejeiba 1. Introduction A map is a continuous function between two topological spaces. A topological dynamical system is a pair (X, f), where X is a compact metric space (d is a metric on X) and f is a map from X to itself. Let N be the set of positive integers. Let f0 be the identity map of X. Define, inductively, fn = f ◦ fn−1 for any non-zero positive integer n. For x ∈ X, {fn(x) : n ∈ N} is called the orbit of x and is denoted by O(x, f). For any x ∈ X, write ω(x, f) = {y : ∃ nk ∈ N, nk → ∞, lim k→∞ fnk(x) = y} called the ω-limit set of x under f, and write Ω(x, f) = {y : ∃ xk ∈ X and nk ∈ N, nk → ∞, lim k→∞ xk = x and lim k→∞ fnk(xk) = y}. x is a periodic point if fn(x) = x for some non-zero positive integer n. Note that, if n = 1, then x is a fixed pint. Also, x is called a recurrent point of f if for any neighborhood U of x and any m ∈ N there exists n > m such that fn(x) ∈ U. Note that, x is a recurrent point of f if and only if x ∈ ω(x, f). Let Fix(f), P(f) and R(f) denote the set of fixed points, periodic points and recurrent points, respectively. We say that the map f is pointwise periodic if P(f) = X. Also, f is said to be pointwise recurrent if R(f) = X. A subset A of X is called f-invariant if f(A) is a subset of A. It is called a minimal set of f if it is non-empty, closed, f-invariant and minimal (in the sense of inclusion) for these properties. If X is a minimal set, then f is called a minimal map. f is said to be equicontinuous (with respect to d) if for each ε > 0, there exists 0 < α < ε such that for any non zero-integer n and any x, y ∈ X with d(x, y) < α, one has d(fn(x), fn(y)) < ε. It is interesting to give some characterizations of equicontinuous maps [12, 13, 16, 17, 24]. In [3, Theorem 2.1], by means of the orbit map Of : X → C(N, X) and the metric df , Akin, Auslander and Berg gave some necessary and sufficient conditions for a map f of a compact metric space (X, d) to be equicontinuous. In [6, Proposition 2.2], Blanchard, Host and Maass discussed the topological complexity, and showed that a surjective map f of a compact metric space X is equicontinuous if and only if any finite open cover of X under f has bounded complexity. On one-dimensional spaces, one has some still finer results. In [12], Cano proved that if f is an equicontinuous map from the interval I = [0, 1] to itself, then Fix(f) is connected, and furthermore, if Fix(f) is non-degenerate, then f has no periodic points except fixed points. Bruckner and Hu (only if) and Boyce (if) proved that a map f : I → I is equicontinuous if and only if ∞⋂ n=1 fn(I) = Fix(f2); see [9, 10]. This result was also proved by Blokh in [8]. Valaristos [25] described the characters of equicontinuous circle maps: a map f of the unit circle S1 to itself is equicontinuous if and only if one of the following four statements holds: (1) f is topologically conjugate to a rotation; (2) Fix(f) contains exactly two points and Fix(f2) = S1; © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 68 Equicontinuous local dendrite maps (3) Fix(f) contains exactly one point and Fix(f2) = ∞⋂ n=1 fn(S1); (4) Fix(f) = ∞⋂ n=1 fn(S1). In [21], Sun obtained some necessary and sufficient conditions of equicontinuous σ-maps. When X is a graph, Mai proved in [17, Theorem 5.2] that the following properties are equivalent: (1) f is equicontinuous; (2) R(f) = ∞⋂ n=1 fn(X); (3) the restriction of f on ∞⋂ n=1 fn(X) is equicontinuous; (4) the restriction of f on ∞⋂ n=1 fn(X) is a periodic homeomorphism, or is topologically conjugate to the irrational rotation of the unit circle S1. In [11, 22, 23], the authors studied equicontinuous dendrite maps. For a den- drite X with countable set of endpoints, in [22, Theorem 2.8] it is shown that f is equicontinuous if and only if Ω(x, fn) = ω(x, fn) for any x ∈ X and each n ∈ N. For a dendrite X with finite branch points, in [23, Theorem 28] it is proved that the following statements are equivalent: (1) f is equicontinuous; (2) Ω(x, f) = ω(x, f) for any x ∈ X; (3) P(f) = ∞⋂ n=1 fn(X), and ω(x, f) is a periodic orbit for every x ∈ X and the function ωf : x → ω(x, f) (x ∈ X) is continuous; (4) Ω(x, f) is a periodic orbit for any x ∈ T . Recently, for general dendrites, in [11, Theorem 4.12] it is shown that the following statements are equivalent: (1) f is equicontinuous; (2) Ω(x, f) = ω(x, f) for any x ∈ X; (3) ωf is continuous and P(f) = ∞⋂ n=1 fn(X). In this paper we will give some equivalent conditions of equicontinuity for local dendrite maps, whose dynamical behavior is both important and interesting in the study of Discrete Dynamical Systems and Continuum Theory. Our main results are the following: Theorem 1.1. Let X be a compact metric space and f : X → X be a map. Consider the following statement: (1) f is equicontinuous; (2) Ω(x, fn) = ω(x, fn) for all x ∈ X and n ∈ N. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 69 A. H. Salem, H. Hattab and T. Rejeiba Then (1) implies (2) and, if ∞⋂ n=1 fn(X) = P(f), then (2) implies (1). Theorem 1.1 generalizes [22, Theorem 2.8]. Proposition 1.2. Every pointwise recurrent local dendrite map is equicontin- uous. The following result generalizes [17, Theorem 5.2], [24, Theorem 2] and [11, Theorem 2.8]. Theorem 1.3. Let f : X → X be a local dendrite map. If E(X) is countable, then the following are equivalent: (1) f is equicontinuous; (2) ∞⋂ n=1 fn(X) = R(f); (3) f| ∞⋂ n=1 fn(X) is equicontinuous and surjective; (4) f| ∞⋂ n=1 fn(X) is either a pointwise periodic homeomorphism or is topo- logically conjugate to an irrational rotation of S1; (5) ω(x, f) = Ω(x, f) for all x ∈ X. Corollary 1.4. Under the assumptions of Theorem 1.3, if ω(x, f) = Ω(x, f) for all x ∈ X, then ω(x, fn) = Ω(x, fn) for all x ∈ X and every n ∈ N. Recently, several authors have been interested in studying local dendrite maps (for example one can see [1, 2, 4]). 2. Preliminaries A compact connected metric space is called a continuum. A Peano con- tinuum is a locally connected continuum. An arc is any space homeomorphic to the interval I = [0, 1]. We mean by a simple closed cure every continuum homeomorphic to the circle S1. Recall that a graph is a continuum which can be written as the union of finitely 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). A tree is a graph which contains no simple closed curves. A dendrite is a Peano continuum which contains no simple closed curves. Not that every dendrite is uniquely arcwise connected continuum, that is any two distinct points in a dendrite D can be joined by a unique arc [x, y]. We define by (x, y) = [x, y] \ {x, y}. For more properties of dendrites (see [19, Chapter X]). Let D be a dendrite and x ∈ D, x is called anendpoint of D if D \ {x} is connected. The set of all endpoints of D is denoted by E(D). We say that x is a cut point of D if x ∈ D \ E(D) [19, Theorem 10.7, p. 168]. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 70 Equicontinuous local dendrite maps By a local dendrite, we mean a continuum every point of which has a dendrite neighborhood. Each local dendrite is a Peano continuum which has a finite number of circles [15]. Since every subcontinuum of a dendrite is a dendrite [15, º51, VI, Theorem 4], a subcontinuum of a local dendrite is a local dendrite. The simple examples of local dendrites are graphs and dendrites. Let X be a local dendrite. A point e ∈ X is called an endpoint of X if it admits a neighborhood U in X such that U is an arc and U \ {e} is connected. The set of endpoints of X is denoted by E(X). A point x ∈ X is called a branch point of X if there exists a closed neighborhood D of x which is a dendrite such that x is a branch point of D (i.e. D \{x} has more than two connected components). We denoted by B(X) the set of branch points of X. By [15, Theorem 6,304 and Theorem 7, 302], B(X) is at most countable. A local dendrite map is a map from a local dendrite into itself. Lemma 2.1 ([1]). Let X be a local dendrite and Y be a sub-local dendrite of X distinct of X such that any arc J of X joining two distinct points of Y is included in Y . Then for any connected component C of X \ Y , C ∩ Y is degenerate (i.e. reduced to a point). If S1, S2, . . ., Sr are the circles in a local dendrite X, then Γ(X) is the intersection of all subgraphs in X containing the union of S′is. Therefore, Γ(X) is the smallest graph containing all circles of the local dendrite X. Define X \ Γ(X) = ⋃ i∈A Ci where Ci are the connected components of X \ Γ(X). Since B(X) is at most countable, the set A is at most countable. By Lemma 2.1, for any i ∈ A, Ci ∩ Γ(X) is reduced to a point zi. Let Ak be a subset of A such that, for each i ∈ Ak, Ci ∩ Γ(X) = {zk}. Put C k = ⊔ i∈Ak Ci. Since Γ(X) contains all circles of X, by [1] and [4, Lemma 2.4], we obtain the following lemma. Lemma 2.2. Under the notation above, the (Ck)k are pairwise disjoint sub- dendrites of X. Lemma 2.3. If X is a local dendrite with E(X) is countable then every sub- local dendrite of X has a countable set of endpoints. Proof. Let Y be a sub-local dendrite of X. If E(Y ) is uncountable, then there exists a connected component C of Y \ Γ(Y ) such that C is a dendrite with uncountable set of endpoints. Since Y ∩ Γ(X) = Γ(Y ), Y \ Γ(Y ) = Y \ Γ(X) ⊂ X \ Γ(X). Therefore, there exists a connected component L of X \ Γ(X) such that C ⊂ L. Now we define a one-to-one function f : E(C) → E(L) by: let e ∈ E(C). If e ∈ E(L), then f(e) = e. If e /∈ E(L), let Ce be a connected component of L \ C such that Ce ∩ C = {e}. We take f(e) ∈ E(Ce) \ {e}. It is easy to see that f is a one-to-one function. Thus the cardinality of E(C) is less than or equal to the cardinality of E(L). Consequently, E(L) is uncountable which implies that E(X) is uncountable, a contradiction. � © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 71 A. H. Salem, H. Hattab and T. Rejeiba Lemma 2.4 ([2, Lemma 4.3]). Let (X,d) be a local dendrite. Thus, for every ε > 0, there exists δ = δ(ε) > 0 such that, for any x and y in X with d(x, y) < δ, the diameter diam([x, y]) < ε. 3. Proof of Theorem 1.1 To prove Theorem 1.1 we will use the following two results. The first one is proved in [11, Lemma 3.5]. Lemma 3.1. Let X be a compact metric space and f : X → X be an equicon- tinuous map. Then ω(x, fn) = Ω(x, fn) for all x ∈ X and for all n ∈ N. Part (2) of the following result is shown in [17, Proposition 2.4], so we prove only part (1). Lemma 3.2. Let X be a compact metric space and f : X → X be a map. Then the following assertions hold: (1) If ω(x, f) = Ω(x, f) for all x ∈ X, then ∞⋂ n=1 fn(X) = R(f); (2) If f is equicontinuous, then ∞⋂ n=1 fn(X) = R(f). Proof. Since X is compact and f(R(f)) = R(f), ⋂ n∈N fn(X) ⊇ R(f). Con- versely, given x ∈ ⋂ n∈N fn(X), there is a sequence xk ∈ X and x0 ∈ X with xk → x0 and nk → +∞ such that f nk(xk) = x for all k ∈ N. Hence x ∈ Ω(x0, f) = ω(x0, f). Since x ∈ ω(x0, f), there exists a sequence (pk)k ∈ N, pk → ∞ such that f pk(x0) → x. By choosing a subsequence, we can suppose that pk+1 − pk > k for all k ∈ N. Then f pk+1−pk (fpk (x0)) = f pk+1(x0) → x. Therefore, x ∈ Ω(x, f) = ω(x, f) which implies that x ∈ R(f). Consequently, ∞⋂ n=0 fn(X) = R(f). � Remark that, in the above Lemma, if we suppose that f is an onto map, we infer that f is pointwise recurrent that is, R(f) = X. Proof of Theorem 1.1. By Lemma 3.1 we have (1) ⇒ (2). To show that (2) ⇒ (1), assume that ∞⋂ n=1 fn(X) = P(f) and that (2) holds. Note that, in [22, Lemma 2.4] it is proved that (2) ⇒ (1) if X is a dendrite. We will extend this result for every compact metric space X. Since X is compact then Ω(x, fn) = ω(x, fn) ⊂ ∞⋂ n=1 fn(X) = P(f) for all x ∈ X and every n ∈ N. If f is not equicontinuous, then there exist x ∈ X, a sequence (xn)n in X with xn → x and a sequence (pn)n in N, © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 72 Equicontinuous local dendrite maps pn → ∞ such that f pn(xn) → a ∈ X and f pn(x) → b ∈ X with a 6= b. Then a ∈ Ω(x, f) ⊂ P(f) and b ∈ ω(x, f) ⊂ P(f). Since a and b are in P(f), there exists k ∈ N such that a, b ∈ Fix(fk). We can assume, without loss of generality, that modulus k, pn is congruent with r (pn = kqn + r), for every n ∈ N. Note that fr(xn) → f r(x), kqn → ∞ and f kqn(fr(xn)) → a. Thus a ∈ Ω(fr(x), fk) = ω(fr(x), fk). Consequently, there is a sequence sn → ∞ (by choosing a subsequence, we can suppose sn−qn > n) such that (f k)sn(fr(x)) → a. Since r = pn − kqn, f ksn(fpn−kqn (x)) = fksn−kqn(fpn(x)) → a. Since fpn(x) → b, a ∈ Ω(b, fk) = ω(b, fk) = {b}. Therefore, a = b which leads to a contradiction. Then f is equicontinuous. � The following examples show that the condition ∞⋂ n=1 fn(X) = P(f) of The- orem 1.1 is essential. Example 3.3 is inspired from [14, page 92]. Example 3.3. There exist a compact metric space X and a homeomorphism fX → X such that ∞⋂ n=1 fn(X) 6= P(f), Ω((x, y), fn) = ω(x, fn) for all x ∈ X and n ∈ N, and f is not equicontinuous. Let yn ∈ R \ Q be a sequence which converges to y∞ ∈ R \ Q, S 1 = R/Z be a unit circle, and X = S1 × {yn : n ∈ N ∪ {∞}}. Then X is a compact metric space with the metric d induced from one of the Euclidean space R2. Let f : X → X by f(x, y) = (x + y, y). Note that, f is a homeomorphism. Thus ∞⋂ n=1 fn(X) = X. Since {yn : n ∈ N∪{∞}} ⊂ R\Q, P(f) = ∅. Therefore, ∞⋂ n=1 fn(X) 6= P(f). Let (x, y) ∈ X and n ∈ N. Since fn(x, y) = (x + ny, y), on S1, x 7→ x + ny is an irrational rotation. Then ω((x, y), fn) = S1 × {y} and Ω((x, y), fn) = S1 × {y}. Consequently, Ω((x, y), fn) = ω(x, fn) for all x ∈ X and n ∈ N. f is not equicontinuous; indeed, fix a small ε > 0 (e.g. ε = 1 4 ) and arbitrary δ > 0. Choose a pair p = (x, y); q = (x, y∞) with 0 < |y − y∞| < ε and d(q, p) < δ. Fix an integer N > 1 such that k|y − y∞| < ε < N|y − y∞| < 1 for any integer k < N. Then d(fN(p), fN(q)) > N|y − y∞| > ε. Consequently, f is not equicontinuous. Remark that the action of f is distal; indeed, let (x, y) 6= (x′, y′) be two points of X. If limfki(x, y) = limfki(x′, y′), then lim(x + kiy, y) = lim(x ′ + kiy ′, y′) which implies that y = y′ and x = x′ which is impossible. Example 3.4. There exist a compact metric space X and a non surjective map f : X → X such that ∞⋂ n=1 fn(X) 6= P(f), Ω((x, y), fn) = ω(x, fn) for all x ∈ X and n ∈ N, and f is not equicontinuous. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 73 A. H. Salem, H. Hattab and T. Rejeiba As in the Example 3.3, we put X = S1 × {yn : n ∈ N ∪ {∞}}. Let f : X → X by f(x, y) = (x + y, y∞). Note that, f is a non surjective map and ∞⋂ n=1 fn(X) = S1 × {y∞}. Since {yn : n ∈ N ∪ {∞}} ⊂ R \ Q, P(f) = ∅. Therefore, ∞⋂ n=1 fn(X) 6= P(f). Let (x, y) ∈ X and n ∈ N. Since fn(x, y) = (x + ny, y∞), on S 1, x 7→ x + ny is an irrational rotation. Then ω((x, y), fn) = S1 × {y∞} and Ω((x, y), f n) = S1 × {y∞}. Consequently, Ω((x, y), f n) = ω(x, fn) for all x ∈ X and n ∈ N. As in the Example 3.3, f is not equicontinuous. 4. Proofs of Proposition 1.2 and Theorem 1.3 Lemma 4.1. Let X be a compact metric space and f : X → X be a map. Consider the following statements: (1) R(f) = X; (2) f is an equicontinuous homeomorphism. Then (2) implies (1) and if X is a dendrite, then (1) implies (2). Proof. By [18, Lemma 2.4], a homeomorphism of a compact metric space is equicontinuous if and only if it is distal and locally almost periodic. The last condition implies that R(f) = X. Hence (2) implies (1). (1) ⇒ (2) From [7, Theorem 1.18] we have f is a homeomorphism and X \ E(X) ⊂ P(f) hence, by the equivalence between clauses (1) and (8) of [20, Theorem 3.8], f is equicontinuous. � According to [4, Lemma 2.8], we get the following lemma. Lemma 4.2. Let f : X → X be a local dendrite map. If R(f) = X, then f(Γ(X)) = Γ(X). 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. A space X is said to be almost totally disconnected if the set of its degenerate components, considered as a subset of X, is dense in X. A compact metric space X is called a cantoroid if it is almost totally disconnected and has no isolated point. A generalized brain is a cantoroid whose nondegenerate components form a null family, they are local dendrites and only finitely many of them contain circles. Lemma 4.3. Let f : X → X be a local dendrite map. If X 6= S1 then f is not minimal. Proof. If X 6= S1, by [16, Theorem 3.2] and [19, Theorems 10.31], the result holds whenever X is either a graph or a dendrite. If X is neither a dendrite nor a graph, then X contains at least an attached dendrite. According to [5, Theorem C], a minimal set on local dendrites is either a finite set or a finite © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 74 Equicontinuous local dendrite maps union of disjoint circles or a generalized brain. Since a generalized brain is not connected, a local dendrite can not be a minimal set. This ends the proof of Lemma 4.3. � According to [17, Proposition 2.5], we get the following lemma. Lemma 4.4. Let f : X → X be a local dendrite map and let p ∈ N. Then f is equicontinuous if and only if fp is equicontinuous. According to [4, Theorem 2.1], we have the following result. Theorem 4.5. Let f : X → X be a local dendrite map. Then f is pointwise recurrent if and only if f is a homeomorphism such that one of the following statements holds: (1) If f is not minimal, then every non endpoint has a finite orbit; (2) If f is minimal, then X = S1 and f is topologically conjugate to an irrational rotation. Proof of Proposition 1.2. Assume that R(f) = X. By [17, Corollary 5.1] and Lemma 4.1, the result holds whenever X is either a graph or a dendrite. Assume that X is neither a dendrite nor a graph. Assume that X is neither a dendrite nor a graph. If f is minimal, then, by Theorem 4.5 f is topologically conjugate to an irrational rotation. It is well known that every rotation is an isometric, conse- quently, it is equicontinuous. Thus, by [17, Lemma 3.1] f is equicontinuous. If f is not minimal, then, by Theorem 4.5, f is a homeomorphism and every non endpoint has finite orbit. By Lemma 4.2, Γ(X) is invariant. Then, by [17, Corollary 5.1], f|Γ(X) is equicontinuous. We further have X \ Γ(X) is the union of pairwise disjoint dendrites (Ck) such that Ck ∩ Γ(X) = {zk} and as fn0(zk) = zk hence by [1, Lemma 3.8] one has f n0(Ck) = Ck and since R(f) = R(fn0) = X, fn0 : Ck → Ck is pointwise recurrent. Thus, by Lemma 4.1 and Lemma 4.4, f|Ck is equicontinuous for each k which implies f|X\Γ(X) is equicontinuous ((Ck) are pairwise disjoint dendrites). Consequently, f is equicontinuous on X. � According to [4, Corollary 2.2] we obtain the following result. Proposition 4.6. Let X be a local dendrite and f : X → X be a local dendrite map. Assume that E(X) is countable. Then f is pointwise recurrent if and only if one the following statements holds: (1) f is a pointwise periodic homeomorphism; (2) X = S1 and f is topologically conjugate to an irrational rotation. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 75 A. H. Salem, H. Hattab and T. Rejeiba Proof of Theorem 1.3. Since X is compact and connected, for every n ∈ N, fn(X) is also compact and connected. Let M(f) = ∞⋂ n=1 fn(X). It follows from X ⊃ f(X) ⊃ f2(X) ⊃ · · · that M(f) is a nonempty sub-local dendrite of X. By [11, Lemma 3.1], f(M(f)) = M(f). Since M(f) is an f-invariant sub-local dendrite, by Proposition 4.6, (2) is equivalent to (4). If R(f) = M(f), then f|M(f) : M(f) → M(f) is pointwise recurrent. Con- sequently, by Proposition 1.2, f|M(f) is equicontinuous. Therefore, (2) ⇒ (3). By Lemma 3.2, (1) implies (2). (3) ⇒ (1). If U = M(f) ∩ X \ M(f), then U ⊂ E(M(f)). By Lemma 2.3, E(M(f)) is countable. Since U is compact, it is a finite set. Obviously, for every k ∈ N, there exists pk ∈ N such that f pk(X) ⊂ M(f) ∪ B(U, 2−k). Since f|M(f) is equicontinuous, by [17, Theorem 5.1], f is equicontinuous. By Lemma 3.1, (1) ⇒ (5). By Lemma 3.2, (5) ⇒ (2). � The following example shows that E(X) is countable cannot be removed from the hypothesis of Theorem 1.3. Example 4.7. 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