# Recurrence and Chaos of Local Dendrite Maps

Authors: Hawete Hattab
Journal: Journal of Mathematics
 (2022)

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## Abstract

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


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## Body

## 1. Introduction

A metric space that is compact and connected is calleda 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 pointsx 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).Letx be a point of a dendrite D. We consider the order of x in D as defined in [1], Definition 9.3, p. 141. If the order of x in D is finite, then it is equal to the number of connected components of D\x. If the order of x in D is infinite, 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 ED the subset of all endpoints of D. A point x∈D\ED is called a cut point (see ([1], Theorem 10.7, p. 168)). A branching point of D is a point of order ≥3. The set of all branching points will be denoted by BD. According to [1], Theorem 10.23, BD is at most countable.A local dendrite is a continuum such that every point of it has a dendrite neighborhood. We consider thatX is a local dendrite. The 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. The set of all branching points of X will be denoted by BX. According to Theorem 10.23 of [1] and Theorem 4 of [3], BX 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. EX is the set of endpoints of X.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).Letℕ be the set of positive integers. A map is a continuous function. If X is a topological space, then a self-mapping is a map from X to itself.Letf denote a self-mapping of a continuum X and x∈X. The orbit of the point x is Ox,f=fnx:n∈ℕ. The point x is periodic if fnx=x for some nonzero integer n. The subset of all periodic points of f will be represented by Pf. The point x is recurrent if there exists an increasing sequence mk∈ℕ verifying fmkx converges to x. Note that every iterated of a recurrent point is also recurrent. The subset of all recurrent points of f will be represented by Rf. The 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, fn+ix:i=0,1,…,N∩U≠∅. The subset of all almost periodic points of f will be represented by APf. By definition, we get the following inclusions:(1)Pf⊂APf⊂Rf.Such inclusions are not reversible.We say thatf is as follows:(i)
Pointwise periodic ifPf=X(ii)
Pointwise recurrent ifRf=X(iii)
Relatively recurrent ifRf¯=XA compact metric spaceX has the APR property if, for every continuous self-mapping f of X, we get the equality APf¯=Rf¯ (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 ED is countable ([6], Theorem 1.2).In this study, we show that a local dendriteX has the APR property if and only if EX is countable (see Theorem 4).Many works studied maps such thatPf or Rf verifies additional properties. For example, [7–9] studied homeomorphisms satisfying Pf=X. In [10], it is proved that, for interval map, if Pf is closed then Pf is equal to the set non-wandering points Ωf. For graph maps, Mai proved in [11], Theorem 4.4, the following theorem.Theorem 1.
Rf is the whole graph X if and only if either X=S1, and f is a homeomorphism topologically conjugate to an irrational rotation, or f is a periodic homeomorphism.
Also, for graph maps, Hattab proved in [5], Main Theorem, the following theorem.Theorem 2.
Rf¯ is the whole graph X if and only if exactly one of the following properties holds:(a)
X=S1, and f is a homeomorphism topologically conjugate to an irrational rotation(b)
Pf¯=X
For a dendrite with countable set of endpoints, Naghmouchi ([12], Theorem 1.6) proved that a self-mapping is pointwise recurrent if and only if it is a pointwise periodic homeomorphism.
For monotone local dendrite maps, in [4], Theorems 5.1 and 5.4, it was shown that Rf¯=X if and only if Rf=X. The following result is an immediate of [4], Theorems 5.1 and 5.4.Theorem 3.
Rf¯ is the whole local dendrite X if and only if exactly one of the following properties holds:(a)
X=S1, and f is a homeomorphism topologically conjugate to an irrational rotation(b)
Every cut point is periodic
By item (b) of Theorem3 and the fact that the set of cut points is dense, Pf¯=X.
In this study, we show that for a local dendrite mapf:X⟶X, such that EX is countable, the condition that f is monotone is not essential in Theorem 3 (see Theorem 6).
Letf 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 x0∈X such that Ox0,f¯=X. The 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 dfnx,fny>α. The map f is chaotic in the sense of Devaney if f is transitive, Pf¯=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 Rf¯=X. Thus, on all compact metric space without isolated points satisfying the PR property: Pf¯=Rf¯ (intervals, trees, dendrites, Warsaw circle), and transitivity = chaos ([15, 16]).
For a graphX≠S1, by item (b) of Theorem 2, a self-mapping f of X is chaotic if and only if f is transitive.
For a local dendriteX≠S1, by item (b) of Theorem 3, a monotone self-mapping f of X is chaotic if and only if f is transitive.
In this study, it is shown that on local dendrites≠S1, transitivity = chaos (see Theorem 7).


## 2. Recurrence and Almost Periodicity on Local Dendrites

Throughout this section, the letterX denotes a local dendrite. In [3], Theorem 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 dendrites D1, D2, …, Dk such that X=D1∪D2∪⋯∪Dk, and for every i,j∈1,2,…,k with i≠j, the set Di∩Dj is finite.IfS1, S2, …, Sr are the circles in the local dendrite X, then ΓX is the intersection of all the subgraphs in X containing the union of Si′ s. Therefore, ΓX is the smallest graph containing all circles of the local dendrite X.Lemma 1.
LetX be a local dendrite. Then, for any connected component C of X\ΓX, C¯∩ΓX is reduced to a branching point.Proof.
By [17], Lemma 2.9, any arc in X with endpoints x,y∈ΓX is included in ΓX. Thus, by [17], Lemma 2.11, for any connected component C of X\ΓX, C¯∩ΓX is reduced to a point z. If z∉BX, then there exists a unique arc z,y in ΓX emanating from z and y∈BΓX. Therefore, ΓX\a,b is a graph of X, which contains all circles and smaller than ΓX, a contradiction. This ends the proof.
SetX\ΓX=∪i∈ACi, where Ci is the connected components of X\ΓX. By Lemma 1, for any i∈A, Ci¯∩ΓX is reduced to a branching point zi. Since BX is at most countable and the order in X is at most countable, the set A is at most countable. Let Ak be a subset of A such that, for each i∈Ak, Ci¯∩ΓX=zk. Put Ck=⊔i∈AkCi¯. Since ΓX contains all circles of X, by [17], Lemma 2.9, we obtain that Ckk is pairwise disjoint sub-dendrites of X. Suppose now that X≠ΓX. Let A be an arc contained in X\ΓX. The symbols A¯, A∘, and ∂A denote the closure, the interior, and the endpoints of A as an arc, respectively.
We define the mappingrA:X⟶A as follows.
SinceA is an arc contained in X\ΓX, there exists i∈A such that A⊂Ci¯. Since Ci¯ is a dendrite, by [1], Lemma 10.25, there exists a retraction r′:Ci¯⟶A such that for all p∈Ci¯(2)r′p=the single pointa∈Awitha,p∩A=a.
The mapri:X⟶Ci¯ such that the restriction of ri to Ci¯ is the identity and rX\Ci¯=zi is a retraction. We put rA=r′∘ri.
It is easy to see thatrA is a retraction from X onto A. Suppose that rAzi∈∂A.
Note that any arcA is totally ordered by a standard total order <. Consequently, we get a preorder in X defined by p≺q if and only if rAp<rAq.
The main achievements of this section are to prove the following theorem.Theorem 4.
A local dendriteX has the APR property if and only if EX is countable.
IfX=ΓX, then, by [5], Lemma 3.1, X has the APR property. If X≠ΓX, then we need the following lemmas.
Recall that a subsetS of a topological space is dense in itself if S contains no isolated points.Lemma 2.
LetX be a local dendrite and f denote a continuous self-map of X. If there exists p∈Rf\Pf¯ such that Op,f∩X\ΓX≠∅, then EX is uncountable.
Note that the proof of Lemma2 extends [18], Lemmas 2–4, and the proof of [18], Theorem 2, shown originally for dendrites, to the case of local dendrites.Proof.
LetX be a local dendrite. Assume that A⊂X\ΓX is an arc. Let i∈A, and let Ci be a connected component of X\ΓX such that A⊂Ci and Ci¯∩ΓX=zi. Consider the retraction rA=r′∘ri from X onto A, and recall that Ci¯ is a dendrite.
We start by the following claims.Claim 1.
Letε>0 and pick p∈A. Thus, there is δ>0 satisfying rA−1A∩Bδ,p−p contained in Bε,p, where Bδ,p is the open ball of radius δ centered at p.
LetA=x,y. Let p∈A, and let ε>0. Since Ci¯ is a dendrite, by [18], Lemma 2, there is α>0 with r′−1A∩Bα,p−p contained in Bε,p. For δ=infα,dp,y/3,dp,x/3, we obtain(3)rA−1A∩Bδ,p−p⊂r′−1A∩Bα,p−p⊂Bε,p.
This proves Claim1.Claim 2.
Letp0<q0 denotes two points in A such that p0<rAfp0 and rAfq0<q0. Hence, there is a fixed point z of f satisfying p0<rAz<q0.
Letfi:Ci¯⟶Ci¯ be the restriction of ri∘f to Ci¯. Recall that rA=r′∘ri. We claim that(4)p0<r′fip0,r′fiq0<q0,and indeed,(5)r′fip0=r′ri∘fp0=r′∘rifp0=rAfp0>p0.
Similarly,r′fiq0<q0.
By [18], Lemma 3, there is a fixed point z∈Ci¯ of fi with p0<r′z<q0. Since riz=z, rAz=r′∘riz=r′z. Therefore, p0<rAz<q0.
Iffz=z′∈Ci¯, then(6)z′=rCi¯z′=rifz=fiz=z,which implies that z=z′ and so z is a fixed point of f.
Iffz=z′∉Ci¯, then(7)zi=riz′=rifz=fiz=z,which implies that z=zi, but rAzi is an endpoint of A, which contradicts p0<r′z<q0. This completes the proof of Claim 2.Claim 3.
Suppose that there isS⊂X a nonempty dense in itself subset satisfying the following properties:(1)
S∩X\ΓX≠∅(2)
The subsetB∩S is discrete, for every arc B⊂X\ΓX
Then,EX is uncountable.
By (1), there isi0 such that Si0=S∩Ci0≠∅. Since Ci0 is open, Si0 is a dense in itself subset of the dendrite Ci0¯. Then, for any arc B⊂Ci0¯, the set B∩Si0⊂B∩S, and so, by (2), B∩Si0 is discrete. Consequently, by [18], Lemma 4, ECi0¯ is uncountable. Since ECi0¯\zi0⊂EX, EX is uncountable. This ends the proof of Claim 3.
Now, we use Claims1–3 to prove the lemma.
Letp∈Rf\Pf¯ be such that Op,f∩X\ΓX≠∅. Define S=Op,f. Clearly, S∩X\ΓX≠∅. By [19], Rf is invariant, which implies that S⊂Rf. By the same proof given in [18], (2), p. 233, we have S∩Pf¯=∅. Let n∈ℕ and ε>0. Since S⊂Rf, there exists m∈ℕ such that fmfnp∈Bε,fnp. Since S∩Pf¯=∅, fnp∉Pf. Then, fm+np≠fnp. Therefore, S is dense in itself.
ConsiderB=a,b⊂X\ΓX an arc. Let i∈A, and let Ci be a connected component of X\ΓX such that B⊂Ci and Ci¯∩ΓX=zi. Suppose that B∩S contains a point, which is not isolated. Consequently, B∩S≠∅. Pick x=fnp∈B∩S.Case 1.
We suppose thatrBzi is endpoint of B.
The fact thatS⊂Rf\Pf¯ implies that there is ε>0 with Bx,ε∩Pf=∅. By Claim 1, there is 0<α<infdx,a/3,dx,b/3 satisfying rB−1B∩Bx,α\x⊂Bx,ε.
Sincefnp is a non-isolated point of B∩S, we can pick m>n verifying fmp∈B∩Bx,α\x and the arc x,fmp⊂Bx,α (note that local dendrites are uniformly locally arcwise connected). Assume, for example, that x<fmp. Let g=fm−n. Note that g:X⟶X, gx∈B, and x<gx.
Using an inductive argument, overk, we will show that(8)y≤rBgky,for eachy∈x,gxand everyk∈N∗.
To prove the initialization of∗, we argue by contradiction; indeed, we assume that there is y∈x,gx satisfying rBgy<y. Since x<gx=rBgx and rBgy<y, x<y, and by Claim 2, there is a fixed point z of g satisfying x<rBz<y≤gx. Thus, z∈rB−1B∩Bx,α\x⊂Bx,ε. However, z=fm−nz implies that z∈Pf. This leads to a contradiction with the selection of εBx,ε∩Pf=∅ and ends the initialization step of the proof of ∗.
Let us now assume thaty≤rBgky, for all y∈x,gx, particularly gx≤rBgk+1x. Since x<gx, we have x<rBgk+1x. If there is y∈x,gx such that rBgk+1y<y, then, by the fact that x<rBgk+1x, we have x<y. By applying Claim 2 to gk+1, there is a fixed point z of gk+1 satisfying x<rBz<y<gx. Since gk+1z=z, fk+1m−nz=z. Consequently, z∈Pf, which leads to a contradiction. Thus, y≤rBgk+1y for all y∈x,gx. This ends the proof of ∗.
Consequently,gx≤rBgkx for every k∈ℕ. Therefore, rB−1w∈B:w<gx is an open neighborhood of x disjoint from the subset gkx:k∈ℕ. Thus, x∉Rg=Rfm−n, but according to [20], Theorem I, Rf=Rfm−n. Therefore, x∉Rf. This leads to a contradiction, since x∈S⊂Rf. Therefore, for every arc B in X\ΓX, such that, for all i∈A, rBzi∈∂B, the set B∩S is discrete.Case 2.
We suppose now thatrBzi=c∈B∘. Since B∩S contains a non-isolated point, B∘∩S is infinite. Therefore, we can fix x=fnp in the open arcs a,c or c,b and we proceed as the Case 1 to lead to a contradiction.
Thus, for all arcB⊂X\ΓX, the set B∩S is discrete.
Finally, a direct application of Claim3 implies that EX is uncountable.Lemma 3.
Retractions preserve theAPR 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.
Letg denote a self-mapping of Y, and r:X⟶Y be a retraction. The composition f=g∘r is a self-mapping of X. Thus, by [21], Lemma 3.1, for all n∈ℕ, fn=gn∘r. Therefore, by the proof of [21], Proposition 3.2, Rf=Rg. If x∈X\Y, then fnx=gn∘rx∈Y, and since X\Y is open, x∉APf. Thus, APf⊂Y. If x∈Y, then, for all n∈ℕ, fnx=gnx. Therefore, APf=APg. Consequently, if the continuum X has the APR property, then APf¯=Rf¯, which implies that APg¯=Rg¯. Thus, Y has the APR property.
The following result is due to [22], Lemma 2.4. In this study, we complete its proof.Lemma 4.
LetX be a graph, and C⊂X be a Cantor set. Then, 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.
SinceC is homeomorphic to the Cantor ternary set, C will be totally ordered. For convenience, we may assume that 0=minC and 1=maxC. By [23], Theorem 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 ψ0=0 and ψ1=1. Put BEX=x∈X:xis a branching point or an end point ofX. We define h:C∪BEX⟶X as follows: hx=ψx if x∈C and hx=x if x∈BEX\C. It is easy to see that h is continuous. The set X\C∪BEX is a union of countably many open arcs An=an1,an2 whose endpoints belong to C∪BEX. 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∪BEX, which is disjoint with C∪BEX. Define g:X⟶X as follows. If x∈F∪BEX, put gx=hx. Let x∈An where An is an open arc with endpoints an1,an2∈C∪BEX (note that An is an open set in X). Consider all arcs in X\C∪BEX emanating from ga and disjoint out of ga. There are only finitely many such arcs. Let Bn be one with minimal diameter. Let r:An⟶Bn be a homeomorphism sending an1 to gan1 and an2 to the other endpoint of B. Finally, put gx=rx on A.
We need to prove thatg is continuous. The proof of the continuity of g in F∪BEX is the same as in [22], Lemma 2.4.
Now letx∈X\F∪BEX. Let xn be a sequence of X, which converges to x. Let A be an open arc with endpoints a,b∈F∪BEX containing x. There exists N∈ℕ such that, for all n≥N, xn∈A. If ga=gb, then, for all n≥N, gxn=ga=gx. If ga≠gb, then, for all n≥N, gxn=rxn. Since r is a homeomorphism, rxn converges to rx. In both cases, g is continuous.Proof.
LetX be a local dendrite such that X\ΓX≠∅.
Suppose thatX has an uncountable subset of endpoints EX. Since EX=∪i∈AECi¯\zi and A is countable, there exists i∈A such that ECi¯ is uncountable. By [6], Theorem 1.2, Ci¯ does not have the APR property. Since ri:X⟶Ci¯ is a retraction, by Lemma 3, X does not have the APR property. Therefore, if X has the APR property, then the set EX is countable.
Conversely, we suppose thatEX is countable. Let f be a self-mapping of X. If Rf⊂APf¯, then Rf¯=APf¯ and we are done. Assume then that Rf\APf¯ is nonempty.Case 1.Rf\APf¯∩X\ΓX≠∅SinceRf\APf¯⊂Rf\Pf¯, Rf\Pf¯∩X\ΓX≠∅, and consequently, by Lemma 2, EX is uncountable, which leads to a contradiction.Case 2.Rf\APf¯⊂ΓXLetw∈Rf\APf¯. Since Rf and APf are invariant (see ([24], Propositions 4.1.2 and 4.2.4)), Ow,f⊂Rf\APf¯⊂ΓX. Since Ow,f¯ is not a minimal set (see ([24], Theorem 4.2.2)), there is a nonempty closed f-invariant subset M such that M⊊Ow,f¯. Since ΓX is closed, the set F=Ow,f¯ is a closed f-invariant subset of ΓX. By Lemma 4, there exists a continuous map g:ΓX⟶ΓX such that g is an extension of f:F⟶ΓX. It is easy to see that Rf⊂Rg; consequently, w∈Rg. M is also a nonempty closed g-invariant subset such that M⊊Ow,g¯. Thus, Ow,g¯ is not a minimal set of g; consequently, by [24], Theorem 4.2.2, w∉APg.Ifw∈APg¯\APg, then there exists a minimal set M⊊Ow,g¯. We distinguish two subcases.Case 2.1.M is finiteLeta,b be a connected component of ΓX\M∪BEΓX containing w. Since M∪BEΓX is a finite subset, the restriction of g to M is a homeomorphism.Case 2.1.M is infiniteWe recall the construction ofg given in the proof of [22], Lemma 4. Let h be the function defined by hx=fx if x∈F and hx=x if x∈BEΓX\F, where BEΓX is the set of branching points or endpoints of ΓX. It is easy to see that h is continuous. The set ΓX\F∪BEΓX is a union of countably many open arcs Inn in ΓX with boundary ∂In belonging to F∪BEΓX (note that A is an open set in ΓX). Define g:ΓX⟶ΓX as follows. If x∈F∪BEΓX, put gx=hx. Consequently, F∪BEΓX is g-invariant. If x∈ΓX\F∪BEΓX, there is an open arc A with endpoints a,b∈F∪BEΓX such that x∈A. Consider all arcs in ΓX\F∪BEΓX whose endpoints are ga and gb. There are only finitely many such arcs. Let B be one with minimal diameter. Denote by r a homeomorphism A⟶B sending a to ga and b to gb. Finally, put gx=rx on A. It is easy to see that ΓX\F∪BEΓX is g-invariant.Ifw∈APg¯, then one of the connected components of ΓX\F∪BEΓX, say A, contains a point p of APg and has w as an endpoint, and the other endpoint is w′∈F∪BEΓX. Since A is an open set, there exists n∈ℕ such that gnp∈A. Since ΓX\F∪BEΓX is g-invariant, gnA is a connected subset of ΓX\F∪BEΓX, which intersects A. Consequently, gnA⊂A. Since A∩F∪BEΓX=∅, gnA=A. Since g is continuous, gnA¯⊂gnA¯=A¯. Consequently, gnw,w′⊂A¯∩F∪BEΓX=w,w′ and so gnw,w′⊂w,w′. If gnw,w′=w, then gnw=w, which contradicts the fact that w∉APg. We suppose that gnw,w′=w′. Let vk be a sequence of A, which converges to w. Since gnA=A, there exists a sequence uk of A such that, for all k, gnuk=vk. Since A¯ is compact, uk has at least an accumulation point u∈A¯. Since g is continuous, gnu=w. Since F∪BEΓX and ΓX\F∪BEΓX are g-invariant, u∈F∪BEΓX∩A¯, which implies that u∈w,w′. The fact that gnw,w′=w′ implies that gnu=w′, which contradicts the fact that w≠w′. Consequently, gnw,w′=w,w′. Thus, gnw=w and gnw′=w′ or gnw=w′ and gnw′=w, which implies that g2nw=gngnw=gnw′=w, which contradicts the fact that w∉APg. Therefore, w∉APg¯, which implies that w∈Rg¯\APg¯, which contradicts [5], Lemma 3.1.In both cases, we getRf¯=APf¯.


## 3. Pointwise Recurrent Local Dendrite Map

It was shown, in [4], Theorems 5.1 and 5.4, that for a monotone local dendrite map f:X⟶X we get the following equivalence:(9)Rf¯=X,if and only ifRf=X∗.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 Theorem 5.1 of [26], there exists a transitive nonrecurrent dendrite map.The main results of this section are the following theorems.Theorem 5.
Letf be a self-continuous mapping of a local dendrite X such that EX is countable. Rf=X if and only if exactly one of the following properties holds:(a)
X=S1, and f is a homeomorphism topologically conjugate to an irrational rotation(b)
Pf=XTheorem 6.
Letf be a self-continuous mapping of a local dendrite X such that EX is countable. Rf¯=X if and only if exactly one of the following properties holds:(a)
X=S1, and f is a homeomorphism topologically conjugate to an irrational rotation(b)
Pf¯=X
We need the following results.Proposition 1.
LetX be a local dendrite, which is not a circle. Then, there exists no minimal self-map of X.Proof.
IfX≠S1, then X has a cut point; by [27], Theorem 4.8, X does not have a minimal map.Proposition 2.
Letf:X⟶X be a pointwise recurrent local dendrite map, and let pnn∈ℕ⊂X be a sequence of periodic points of f such that pn+1 is in an arc with endpoints pn and pn+2 for all n∈ℕ and limn⟶∞pn=p∞. Then, p∞ is an almost periodic point.Proof.
LetU be a dendrite neighborhood of p∞. We may assume without loss of generality that U contains all points of pnn∈ℕ. Let pn,pn+2U be the arc in U with endpoints pn and pn+2. Consequently, pn+1∈pn,pn+2U. By [12], Lemma 2.2, the connected component Un of U\pn,p∞ that contains the open arc pn,p∞U satisfies limn⟶∞diamUn=0. Let Vn=Un∪pn,p∞. Let Nn be the period of the point pn. By a similar proof of [12], Lemma 2.7, the orbit of the point p∞ under the map fNn+1 is contained in the set Vn. Since limn⟶∞diamVn=limn⟶∞diamUn=0, p∞ will be regularly recurrent, which implies that p∞ is almost periodic.Proposition 3.
LetX be a local dendrite with a countable set of endpoints. Then, every sub-local dendrite of X has a countable set of endpoints.Proof.
LetY be a sub-local dendrite of X such that EY is uncountable. Set X\ΓX=∪i∈ACi and Y\ΓY=∪i∈A′Ci′, where Ci and Ci′ are the connected components of X\ΓX and Y\ΓY, respectively. It is easy to see that ΓY⊂ΓX and every connected component of Y\ΓY is contained in a connected component of X\ΓX. Since EY=∪i∈A′ECi′¯\zi′, where zi′=Ci′¯∩ΓY, there exists j∈A′ such that ECj′¯ is uncountable. Since there exists k∈A such that Cj′¯⊂Ck¯, by [12], Lemma 4.4, ECk¯ is uncountable. Since EX=∪i∈AECi¯\zi, EX is uncountable.Proposition 4.
LetX be a local dendrite such that EX is countable, f:X⟶X be a pointwise recurrent continuous map, and let W be a minimal set of f. If W≠X, then W is a periodic orbit of f.Proof.
SinceW≠X, X\W is a nonempty open set. Let U be a connected component of X\W. Then, U¯\U=U¯∩W≠∅. It follows from U⊂Rf that there exists n∈ℕ such that fnU∩U≠∅. Consequently, fnU∪U is connected, and fnU∩W=∅ (if a point of U gets mapped into W, then this point is not recurrent). Then, fnU⊂U, and hence, fnU¯⊂U¯. This with fnW⊂W implies fnU¯∩W⊂U¯∩W. We claim that U¯∩W is a finite set. Indeed, it is easy to see that U¯∩W is a minimal set of fn:U¯⟶U¯. Since U¯∩W⊂EU¯ and EU¯ is countable (Proposition 3), U¯∩W is countable and compact. Thus, U¯∩W is finite, because otherwise U¯∩W would be perfect and hence uncountable, which is a contradiction.
We have(10)W∩Pf=W∩Pfn⊃U¯∩W∩Pfn≠∅.
Thus, the minimal setW is a periodic orbit of f.Proposition 5.
LetX≠S1 be a local dendrite, and f:X⟶X be a pointwise recurrent continuous map. If X contains a nonperiodic point, then it contains a sub-dendrite with uncountable set of endpoints.Proof.
SinceRf=X, by Theorem 4 and Proposition 4, Pf¯=X. Then, by Propositions 2–4 and an analogous proof of [12], Theorem 1.6, we get Proposition 5.Proof.
The two statements imply thatRf=X.
Conversely, (a) iff is a minimal mapping, then firstly, by Proposition 1, X is a circle, and secondly, [28], Corollary 1, implies that f is a homeomorphism. Reference [5] (Proposition 2.4) implies that f is a pointwise non-wandering circle map without periodic points. Therefore, f is topologically conjugate to an irrational rotation.
(b) Suppose now thatf is not a minimal map. Since X is a local dendrite with EX countable, by Propositions 3 and 5, Pf=X.Proof.
The two statements imply thatRf¯=X.
Conversely, (a) iff is a minimal mapping, then firstly, by Proposition 1, X is a circle, and secondly, [28], Corollary 1, implies that f is a homeomorphism. Reference [5] (Proposition 2.4) implies that f is a pointwise non-wandering circle map without periodic points. Therefore, f is topologically conjugate to an irrational rotation.
(b) Suppose now thatf is not a minimal map. Since Rf¯=X, by Theorem 4 and Proposition 4, Pf¯=X.
Iff is a transitive local dendrite map, then Rf¯=X. By Theorem 6, we get the following theorem at once.Theorem 7.
LetX≠S1 be a local dendrite, and f:X⟶X be a continuous mapping. If EX is countable, then f is chaotic in the sense of Devaney if and only if f is transitive.



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