@ Appl. Gen. Topol. 15, no. 2(2014), 167-174doi:10.4995/agt.2014.3050 c© AGT, UPV, 2014 On the topology of the chain recurrent set of a dynamical system Seyyed Alireza Ahmadi Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran. (sa.ahmdi@gmail.com, sa.ahmadi@math.usb.ac.ir) Abstract In this paper we associate a pseudo-metric to a dynamical system on a compact metric space. We show that this pseudo-metric is identically zero if and only if the system is chain transitive. If we associate this pseudo-metric to the identity map, then we can present a characteriza- tion for connected and totally disconnected metric spaces. 2010 MSC: 337B35; 54H20. Keywords: chain recurrent; chain transitive; chain component; inverse limit space. 1. Introduction One of the main problems in dynamical systems is the description of the or- bit structure of a system from a topological point of view [1, 3, 4]. Recurrence behavior is one of the most important concepts in topological dynamics. Vari- ous notions of recurrence have been considered in dynamics such as recurrent points, chain recurrent points and non-wandering points [7]. In this paper (X, d) is a compact metric space and f : X → X is a continuous map. An ǫ-pseudo-orbit (or ǫ-chain) of f from x to y is a sequence {xi} n i=0 with x0 = x, xn = y and d(f(xk), xk+1) < ǫ for k = 0, 1, ..., n − 1. A point x in X is called chain recurrent if there is an ǫ-chain from x to itself. We can define an equivalence relation on the set of chain recurrent points in such Received 3 October 2013 – Accepted 20 May 2014 http://dx.doi.org/10.4995/agt.2014.3050 S. A. Ahmadi a way that two points x and y are said to be equivalent if for every ǫ > 0 there exist an ǫ-chain from x to y and an ǫ-chain from y to x. The equivalence classes of this relation are called chain components. These are compact invariant sets and cannot be decomposed into two disjoint compact invariant sets, hence serve as building blocks of the dynamics. The topology of chain recurrent set and chain components have been always in particular interest [2, 5, 6, 8]. We use the symbol Oδ(f, x, y) for the set of δ-pseudo-orbits {xi} n i=0 of f with x0 = x and xn = y. For given points x, y ∈ X we write x ǫ y if Oǫ(f, x, y) 6= ∅ and we write x y if Oǫ(f, x, y) 6= ∅ for each ǫ > 0. We write x ! y if x y and y x. The set {x ∈ X : x ! x} is called the chain recurrent set of f and is denoted by CR(f). If we define a relation R on X ×X with x R y ⇔ x ! y, then R is an equivalence relation on CR(f). A dynamical system f is called chain recurrent if CR(f) = X. A dynamical system f is called chain transitive if for each x, y ∈ X we deduce x ! y. We say that a dynamical system f has the pseudo-orbit tracing property (POTP) on X if for each ǫ > 0 there is δ > 0 so that for a given sequence ξ = {xk}n∈Z with d(f(xk), xk+1) < δ for k ∈ N there exists a point x ∈ X such that d(fk(x), xk) < ǫ for k ∈ N (in this case we say that p ∈ X ǫ-shadowed ξ). Let X0 be a nonempty set. Then a map d0 : X0 × X0 → R is called pseudo- metric if for all x, y ∈ X0 the following hold (1) d0(x, x) = 0; (2) d0(x, y) = d(y, x) ≥ 0; (3) d0(x, y) ≤ d0(x, z) + d0(z, y). The pair (X0, d0) is called a pseudo-metric space. Let (X0, d0) be a pseudo- metric space. Then, the open balls in X0 together with the empty set form a basis for a topology on X0. This topology is first countable and in it closed balls are closed. Moreover, this topology is a Hausdorff topology if and only if X0 is a metric space. Now we are going to present a pseudo-metric on CR(f). Definition 1.1. Let x, y ∈ CR(f), we define df,ǫ(x, y) = inf{ k∑ i=0 d(pi, qi) : p0 = x, qk = y, k ∈ N} where the infimum is taken over all choices of pi and qi so that qi ǫ ! pi+1 for all i = 0, 1, ..., k − 1. x = p0 p1 p2 p3 ... pk q0 ǫ ;{ ;{ ;{ ;{ ;{ q1 ǫ >~>~ >~ >~ q2 ǫ >~>~ >~ >~ q3 ǫ >~ >~ >~ >~ >~ ... qk = y c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 168 On the topology of the chain recurrent set of a dynamical system We also define df (x, y) = inf{ k∑ i=0 d(pi, qi) : p0 = x, qk = y, k ∈ N} where the infimum is taken over all choices of pi and qi so that qi ! pi+1 for all i = 0, 1, ..., k − 1. x = p0 p1 p2 p3 ... pk q0 ;{ ;{ ;{ ;{ ;{ q1 >~ >~ >~ >~ q2 >~ >~ >~ >~ q3 >~ >~ >~ >~ >~ ... qk = y The straightforward calculations imply that for ǫ1 ≤ ǫ2 we deduce df,ǫ2(x, y) ≤ df,ǫ1(x, y) ≤ df (x, y) ≤ d(x, y) and (CR(f), df ) is a pseudo-metric space. If we define Bfr (x) = {y ∈ X; df(x, y) < r}, then the collection τf = {B f r (x) : x ∈ X, r > 0} ∪ {∅} is a basis of a topology on CR(f) which is finer than τd. So (CR(f), df ) is a compact space. Obviously ! is an equivalence relation on CR(f). Let C̃R(f) = CR(f)/R, and π : CR(f) → C̃R(f) be the quotient map, i.e. π(x) = {y ∈ CR(f) : x ! y}. Then we can define a metric d̃f (π(x), π(y)) = df (x, y) for x, y ∈ CR(f) on C̃R(f). With this metric π is a distance preserving map. The topology induced by d̃f is denoted by τ̃f . The induced map f̃ : C̃R(f) → C̃R(f) with f̃(π(x)) = π(f(x)) is the identity map. In this paper we are going to prove the following theorems. Theorem 1.2. Let f : X → X be a chain recurrent continuous map. Then f is chain transitive if and only if df (x, y) = 0 for all x, y ∈ X. Theorem 1.3. Let f : X → X be a chain recurrent continuous map. Then df (x, y) = d(x, y) for all x, y ∈ X if and only if f is the identity map and X is totally disconnected. Theorem 1.4. Let (X, d) be a compact metric space. Then the following con- ditions are mutually equivalent: (1) X is connected; (2) The identity map ι : X → X is chain transitive; (3) For each x, y ∈ X, dι(x, y) = 0. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 169 S. A. Ahmadi 2. Proof of theorems For the proof of theorem 1.2 we first prove the following lemma. Lemma 2.1. Let x, y in CR(f) and ǫ > 0 be given. Then df,ǫ(x, y) = 0 if and only if x ǫ ! y. Proof. Clearly if x ǫ ! y then df,ǫ(x, y) = 0. Now let df,ǫ(x, y) = 0. We choose 0 < δ < ǫ so that the inequality d(t, s) < δ implies d(f(t), f(s)) < ǫ/2. Thus there exist sequences {pi} k i=0 and {qi} k i=0 with p0 = x and qk = y so that qi ǫ ! pi+1 for i = 0, 1, ..., k − 1 and k∑ i=0 d(pi, qi) < δ. So d(f(pi), f(qi)) < ǫ/2 for i = 0, 1, ..., k. If {qi,n} li n=0 ∈ Oǫ/2(qi, pi+1), then d(qi,1, f(qi)) < ǫ/2. Hence d(qi,1, f(pi)) < ǫ. Let {yi} m i=0 ∈ Oǫ/2(y, y). Then d(y1, f(pk)) ≤ d(y1, f(y)) + d(f(y), f(pk)) < ǫ. Therefore the sequence {p0, q0,1, q0,2, ..., q0,l0, q1,1, q1,2, ... ..., qk−1,1, qk−1,2, ..., qk−1,lk−1 = pk, y1, ..., ym} is belong to Oǫ(f, x, y). So x ǫ y. Since df,ǫ(y, x) = 0, then y ǫ x. � Corollary 2.2. Let x, y ∈ CR(f). Then df (x, y) = 0 if and only if x ! y. Corollary 2.3. Let f : X → X be a chain recurrent continuous map. Then f is chain transitive if and only if df (x, y) = 0 for all x, y ∈ X. Corollary 2.4. If x ! x′ and y ! y′ for x, x′, y, y′ ∈ X, then df (x, y) = df (x ′, y′) Corollary 2.5. If x ∈ CR(f) then df ≡ 0 on O(f, x) × O(f, x), where O(f, x) = {fn(x); n ∈ N} ∪ {x}. Proof. It is enough to show that x ! f(x). Given ǫ > 0 clearly {x, f(x)} ∈ Oǫ(f, x, f(x)), i.e. x ǫ f(x). We can choose 0 < δ < ǫ/2 so that d(x, y) < δ implies to d(f(x), f(y)) < ǫ/2. Now let {x0, ..., xn} ∈ Oδ(f, x, x), then d(x1, f(x)) < δ implies that d(f(x1), f 2(x)) < ǫ/2. So d(x2, f 2(x)) < ǫ. Thus {f(x), x2, ..., xn} ∈ Oǫ(f, f(x), x). Therefore x ǫ ! f(x). � Corollary 2.6. The map f : (CR(f), df ) → (CR(f), df ) is an isometry. Proof of theorem 1.3. First suppose that for each x, y ∈ X, df (x, y) = d(x, y). Hence d(x, f(x)) = df (x, f(x)) = 0 for each x ∈ X. Let α be a connected component of X and α contains x. Given ǫ > 0 we consider the sets πǫ(x) = {y ∈ α : x ǫ ! y} and π(x) = {y ∈ α : x ! y}. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 170 On the topology of the chain recurrent set of a dynamical system If y ∈ πǫ(x) then Bǫ(y) ⊆ πǫ(x). So πǫ(x) is an open set. Now let y ∈ πǫ(x) then there is a sequence {yn} ⊆ πǫ(x) such that yn → y. So y ∈ Bǫ(yn) for some n ∈ N. Thus y ∈ πǫ(x). Hence πǫ(x) is both open and closed. Since πǫ(x) 6= ∅ then πǫ(x) = α. Therefore α = ∩ǫ>0πǫ(x) = π(x) = {x}. Now let X be totally disconnected and let there exist x, y ∈ X so that dι(x, y) 6= d(x, y) where ι : X → X is the identity map. Then there exist points p0, ..., pn, q0, ..., qn ∈ X so that qi ! pi+1 for all i = 0, 1, ..., k − 1, p0 = x, qk = y and∑k i=0 d(pi, qi) < d(x, y). Hence there is at least one index i such that qi 6= pi+1. So if α is a connected component contains qi, as the same as the first part we deduce π(qi) = α. Hence qi, pi+1 ∈ α which contradicts the totally disconnec- tedness of X. ✷ Let X be a compact metric space. Topological dimension of the space X is said to be less than n if for all ǫ > 0 there exists a cover α of X by open sets with diameter less than ǫ such that each point belongs to at most n + 1 sets of α. We know that X is 0-dimensional if and only if it is totally disconnected [1]. Corollary 2.7. Let ι : X → X be the identity map. Then dι(x, y) = d(x, y) for all x, y ∈ X if and only if X has dimension zero. Proof of theorem 1.4. Clearly 2 is equivalent to 3. Suppose that X is connected and x ∈ X. It is enough to show that for each ǫ > 0 the set πǫ = {y ∈ X : ǫ x ! y} is both open and closed. If y ∈ πǫ(x) then Bǫ(y) ⊆ πǫ(x). So πǫ(x) is open. If {yn} is a sequence with yn → y then y ∈ Bǫ(yN) for some N ∈ N. Hence y ∈ πǫ(x). If X is not connected then there is a nonempty proper subset A of X such that it is both open and closed. Therefore A and Ac are disjoint nonempty compact subsets of X. So ǫ = d(A, Ac)/2 > 0. By assumption there is an ǫ-pseudo orbit x0, x1, ..., xn so that x0 = x and xn = y. Thus there is an index 0 ≤ i ≤ n − 1 such that xi ∈ A and xi+1 ∈ A c, which is a contradiction. ✷ Proposition 2.8. If f : (CR(f), df ) → (CR(f), df ) has the POTP with respect to df , then (CR(f), df ) is complete. Proof. Given ǫ > 0, by assumption there exists δ > 0 so that any δ-pseudo- orbit in CR(f) can be ǫ-shadowed with a point in CR(f). Let {xn} be a Cauchy sequence. So there is N ∈ N such that df (f(xn), xn+1) = df (xn, xn+1) < δ for n ≥ N. Thus there exists x ∈ CR(f) so that df (xn, x) = df (xn, f n(x)) < ǫ for n ≥ N. � Let f : X → X be a continuous surjection. Then if Xf = lim ← (X, f) = {(xi) : xi ∈ X and f(xi+1) = xi, i ≥ 0} and d̄((xi), (yi)) = ∞∑ i=0 d(xi, yi) 2i c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 171 S. A. Ahmadi then (Xf , d̄) is a metric space called inverse limit space. The homeomorphism σ : Xf → Xf with σ((xi) ∞ i=0) = (f(xi)) ∞ i=0 is called the shift map. We know that CR(σ) = lim ← (CR(f), f) [1]. Proposition 2.9. Let f be a continuous surjection on a compact metric space X to itself and CR(σ) be the chain recurrent set for the shift map σ : Xf → Xf . Then we deduce 2df(x0, y0) ≤ d̄σ((xi), (yi)) Proof. Suppose that (xi), (yi) ∈ CR(σ) and (p j i ), (q j i ) ∈ CR(σ), j = 0, 1, ..., k so that (p0i ) = (xi), (q k i ) = (yi) and (q j i ) ! (p j+1 i ) for j = 0, 1, ..., k − 1. We show that for each m ≥ 0 and j = 0, 1, ...k − 1, qjm ! p j+1 m . Fixed m, j ≥ 0, for given ǫ > 0 Oǫ/2m(σ, (q j i ), (p j+1 i )) 6= ∅. Let {(r 0 i ), (r 1 i ), ..., (r n i )} ∈ Oǫ/2m(σ, (q j i ), (p j+1 i )). Then d(f(rlm), r l+1 m ) 2m = d(rlm−1, r l+1 m ) 2m ≤ d̄(σ(rli), (r l+1 i )) < ǫ/2 m for l = 0, 1, ..., n − 1. Thus {r0m, r 1 m, ..., r n m} ∈ Oǫ(f, q j m, p j+1 m ) so q j m ǫ ! pj+1m . Since ǫ > 0 is arbitrary then qjm ! p j+1 m . Hence df (xi, yi) ≤ ∑k j=0 d(p j i , q j i ). Therefore ∞∑ i=0 df (xi, yi) 2i ≤ k∑ j=0 ∞∑ i=0 d(p j i , q j i ) 2i = k∑ j=0 d̄((p j i ), (q j i )). So ∞∑ i=0 df (xi, yi) 2i ≤ d̄σ((xi), (yi)). Corollary 2.5 implies 2df (x0, y0) = ∞∑ i=0 df (xi, yi) 2i ≤ d̄σ((xi), (yi)). � Corollary 2.10. Let X be a compact metric space and f be a chain recurrent continuous surjection from X to itself. Then if the shift map σ : Xf → Xf is chain transitive then f : X → X is so. Theorem 2.11. The topology τ̃f coincide with quotient topology on C̃R(f) Proof. Every continuous bijection from a compact topological space to a Haus- dorff space is a homeomorphism. Since X is Hausdorff, any two elements π(x), π(y) ∈ C̃R(f) as compact subsets posses disjoint saturated neighborhood, so C̃R(f) is a Hausdorff space with the quotient topology. Also (C̃R(f), τ̃f ) is compact. Thus the identity map is a homeomorphism. � c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 172 On the topology of the chain recurrent set of a dynamical system Recall that the Hausdorff metric on the compact subsets A, B of X is defined as follows dH(A, B) = max{max a∈A d(a, B), max b∈B d(A, b)}. Let τH be the topology induced by Hausdorff metric dH on C̃R(f). Then we deduce the following proposition. Proposition 2.12. The topology τH is finer than τ̃f . Proof. Suppose that π(x), π(y) ∈ C̃R(f). We can choose y′ ∈ π(y) so that d(x, y′) = d(x, π(y)). Thus d̃f (π(x), π(y)) = df (x, y ′) ≤ d(x, y′) ≤ dH(π(x), π(y)). Therefore τ̃f ⊂ τH. � The next example shows that d̃f and dH are not equal in general. Example 2.13. Let f : [0, 1] → [0, 1] be a strictly increasing continuous map so that • f(x) = x for each x ∈ [2−(2i+1), 2−(2i)], i = 0, 1, ...; • f(x) > x for each x ∈ [2−(2i+2), 2−(2i+1)], i = 0, 1, ...; • f(0) = 0. Then we deduce CR(f) = {0} ∪ ∞⋃ i=0 [2−(2i+1), 2−2i] and C̃R(f) = {[2−(2i+1), 2−2i]; i = 0, 1, ...} ∪ {0}. If x ∈ [2−(2i+1), 2−2i] and y ∈ [2−(2j+1), 2−2j] for somej < i, then df (x, y) = 2 −(2j+1) − i∑ k=j+1 2−(2k+1). Thus we deduce d̃f (π(0), π(1)) = 1/3, but dH(π(0), π(1)) = 1. Proposition 2.14. Let (X, d) and (Y, d′) be two compact metric spaces and f : X → X and g : Y → Y be continuous maps. Then if f and g are topologically conjugate then (C̃R(f), d̃f ) and (C̃R(g), d̃g) are isometric. Proof. Suppose that h : X → Y is a homeomorphism so that h ◦ f = g ◦ h. Given ǫ > 0 there is δ > 0 so that for every x, y ∈ X, the inequality d(x, y) < δ implies to d′(h(x), h(y)) < ǫ and the inequality d′(x, y) < δ implies to d(h−1(x), h−1(y)) < ǫ. If {xi} n i=0 ∈ Oδ(f, p, q) for some p, q ∈ X, then {h(xi)} n i=0 ∈ Oǫ(g, h(p), h(q)). This implies that if p ! q then h(p) ! h(q). Hence for every x, y ∈ CR(f) we deduce df (x, y) ≥ d ′ g(h(x), h(y)). If c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 173 S. A. Ahmadi {xi} n i=0 ∈ Oδ(g, h(p), h(q)), then {h −1(xi)} n i=0 ∈ Oδ(f, p, q). Thus df (x, y) ≤ d′g(h(x), h(y)). So d̃f (π(x), π(y)) = df (x, y) = d ′ g(h(x), h(y)) = d̃′g(π(h(x)), π(h(y))) = d̃ ′ g(h̃(π(x), h̃(π(y)). Therefore h̃ : C̃R(f) → C̃R(g) is an isometry. � 3. Conclusions In this paper we introduce a pseudo-metric df associated to the dynamical system f. We show that the topology induced by df has a significant rela- tion to some dynamical properties of f, such as transitivity and shadowing. 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