Consequences of Shadowing Property of G-spaces

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1 Int. Journal of Math. Analysis, Vol. 7, 013, no. 1, Consequences of Shadowing Property of G-spaces Ekta Shah and Tarun Das Department of Mathematics, Faculty of Science The Maharaja Sayajirao University of Baroda, Vadodara-39000, India Abstract. We define and study the notion of minimality and specification for a self homeomorphism of a metric G space X. Using G minimality, we obtain a class of maps which do not have the G shadowing property. Further, we obtain a sufficient condition for G expansive homeomorphisms with G shadowing property to have G specification property. Mathematics Subject Classification: 37D05, 37B0, 37C85, 54H0 Keywords: Shadowing Property, Expansive Homeomorphism, Minimal Homeomorphism, Specification Property, Metric G space 1. Introduction A discrete dynamical system (X, f) consists of a topological space X and a continuous self map f. The basic aim of the theory of discrete dynamical system is to recognize the actions of an orbit, O f (x), of a point x X given by {x, f(x),f,.., f n (x),...}. In various circumstances, it is not possible to compute the exact initial value of x. This therefore do not give the exact value of f(x), which further gives us an approximate value of f (x) and so on. So on computation we actually obtain nearby values of the orbit. If these nearby values are respectively given by {x 0,x 1,x,..., x n,...}, then the natural question is to know whether these nearby values are actually close to the actual values in the orbit. This leads to the study of shadowing property of a map. Shadowing is one of the important property in the theory of discrete dynamical systems as it is close to the stability of the system and also to the chaotic behavior of the system. One of the basic problem is to know maps possessing / not possessing shadowing property. For instance, in [1], a class of maps possessing shadowing property on the unit interval is obtained. Similarly, in [15], a family of maps on the unit circle possessing shadowing property is obtained. In [1], Aoki have shown that a minimal homeomorphism on a non degenerate

2 580 E. Shah and T. Das continuum never possesses shadowing property. Recently, in [13], certain implications of shadowing property are discussed. In fact, it is shown that shadowing property serves as an equalizer of various stronger forms of transitivity. Also, in [], Aoki have shown that the restriction of a continuous map f to the set, Ω(f), of the non wandering points possesses shadowing property if so does f. This result turns out to be one of the key result for the proof of Smale Decomposition Theorem. Further, it is known that shadowing property together with expansivity and mixing implies specification [3, Theorem ]. Another important dynamical property is expansivity of a system. For instance, shadowing together with expansivity gives the stability of the system [3]. Both of these properties are studied in various settings. One such setting is the study of these properties on G spaces. A good amount of literature now exists on G expansivity and G shadowing property. For instance refer [8, 9, 10, 11, 18, 19]. In this paper we study several consequences of these two properties on G spaces. The paper is organized in the following manner. In next Section we give necessary terminologies. In Section 3, we show that if f has the G shadowing property, then so does f ΩG (f), where Ω G (f) is the set of G nonwandering points. This is similar to the result of Aoki proved in []. Further, in Section 4, we define the notion of minimality on a metric G space and use it to obtain a class of maps not possessing G shadowing property. This result is similar to that of Aoki proved in [1]. The notion of specification on G spaces is defined and studied in the Section 5. We obtain a sufficient condition for G expansive homeomorphism with G shadowing property to have G specification property.. Preliminaries By a dynamical system we mean a pair (X, f), where X is a metric space and f is a self homeomorphism on X. Throughout the paper maps are self maps. A homeomorphism f is said to be expansive if there exists c>0 such that for distinct x, y X, there is an integer n Z satisfying d(f n (x),f n (y)) >c; c is called an expansive constant for f. If there exists an expansive homeomorphism f on X, then we say (X, f) is an expansive dynamical system. Given a positive real number δ, a sequence of points θ = {x n : n 0} is said to be a δ pseudo orbit for a continuous map f if for each n, d(f(x n ),x n+1 ) <δ. Map f is said to have the shadowing property if for each ɛ>0, there is a δ>0 such that for every δ pseudo orbit θ, there is an x X satisfying d(f n (x),x n ) <ɛ, for all n. For x X, O f (x) ={f n (x) :n Z} denotes an orbit of x. A homeomorphism f on a metric space X is said to be minimal if the orbit of every point of X is dense in X.

3 Consequences of shadowing property of G-spaces 581 By a metric G space X we mean a metric space X on which a topological group G acts continuously by an action ϑ. For g G and x X we denote ϑ(g, x) by gx. The G orbit of a point x, denoted by G(x), is the set {gx : g G}. The set X/G of all G orbits in X with the quotient topology induced by the quotient map π : X X/G defined by π(x) =G(x), is called the orbit space of X and the map π is called the orbit map. Map f is said to be pseudoequivariant if f(g(x)) = G(f(x)), for all x X. For details on G space one can refer to [4, 14, 0]. It is known that if f is pseudoequivariant continuous map, then it induces a continuous map ˆf : X/G X/G given by ˆf(G(x)) = G(f(x)) [7]. We recall the definition of G expansive homeomorphism. Definition.1. [7] A homeomorphism f defined on a metric G space X is said to be a G expansive homeomorphism if there exists a positive real number c such that for each x, y X with G(x) G(y), there is an integer n Z satisfying d(f n (u),f n (v)) >c, for all u G(x) and v G(y); c is called a G expansive constant for f. In [7] it is shown that G expansivity depends on the action of G and that it neither implies expansivity nor is implied by expansivity. Authors in [9] obtains certain interesting applications of results regarding projecting and lifting of G expansive homeomorphisms. Recently in [11], same authors have shown that the problem of studying G expansive homeomorphisms on a bounded subset of a normed linear G space is equivalent to the problem of studying linear G expansive homeomorphisms on a bounded subset of another normed linear G space. Definition.. [18] Let f be a continuous map defined on a metric G space X. A sequence of points θ = {x n : n 0} in X is said to be a (δ, G) pseudo orbit for f if for each n, d(u, x n+1 ) <δ, for some u G(f(x n )). Map f is said to have the G shadowing property if for each ɛ>0 there is a δ>0 such that for every (δ, G) pseudo orbit θ of f, there is a point x in X satisfying for each n 0, d(u, f n (x)) <ɛ, for some u G(x n ). In [18] it is shown that this property is independent of the metric used if the space is compact. Also, it is observed there through examples that G shadowing depends on the action of G and that it neither implies shadowing nor is implied by shadowing. Recently, in [10], the notion of topological transitivity on G space was defined and studied. We recall the definition. Definition.3. Let (X, d) be a metric G space and f : X X be a continuous map then f is called G transitive if for every pair of non-empty open subsets U and V of X, there exists n N and g G such that g.f n (U) V.

4 58 E. Shah and T. Das It was observed that transitivity of a map implies G transitivity of map. But converse in general is not true under the non-trivial action of G. 3. Properties of G nonwandering set Definition 3.1. Let (X, d) be a metric G space and f : X X be a continuous map. A point x in X is said to be a G nonwandering point of f if for every neighbourhood U of x there is an integer n>0 and a g G such that gf n (U) U φ. The set of all G nonwandering points is denoted by Ω G (f). Obviously Ω(f) Ω G (f). That converse need not be true follows from Examples given below. Note that in Example 3., group is a non compact group, whereas in Example 3.3, group is compact. Further, if X is a compact metric space then Ω G (f) is non empty as Ω(f) is non-empty. Also, under the trivial action of G on X, Ω(f) =Ω G (f). If G acts transitively on X (i.e. G(x) =X), then for any f on X, Ω G (f) =X. Example 3.. Consider the subspace X={± 1, ±(1 1 ):n N} of R with n n the usual metric of R and the homeomorphism h : X X defined by x, if x { 1, 0, 1} h(x) = x +, if 0 <x<1, where x + is the element of X right to x x, if 1 <x<0, where x is the element of X left to x Suppose the group G = {h n : n Z} act on X by the usual action. If f is the left shift fixing 1, 0 and 1, then Ω G (f) =X, whereas Ω(f) ={ 1, 0, 1}. Example 3.3. Let X= 1, where i = {0, 1}Z, i =1,, be two disjoint copies having one point common namely 0=(..., 0, 0, 0,...). We denote a point of i by xi, i =1,. Suppose G=Z act on X by 1x = x, 1x 1 = x and 1x = x 1. Define a map f : X X by f(x) ={ σ1 (x), if x 1 σ (x), if x where σ i is the usual shift on i, i =1,. Then Ω G(f) =X, whereas Ω(f) = Per(f). Remark 3.4. Let X be a compact metric G space, with G compact. Then the following holds for a continuous pseudoequivariant onto map f defined on X: 1. Ω G (f) is non empty closed G invariant subset of X.. f(ω G (f)) Ω G (f). 3. If f is a homeomorphism, then f(ω G (f)) = Ω G (f), Ω G (f) =Ω G (f 1 ).

5 Consequences of shadowing property of G-spaces 583 That pseudoequivariancy is a necessary condition in Remark 3.4() is exhibited by the following example: Example 3.5. Let X = S 1, the unit circle of the plane and suppose G = U 4, the group of 4 th roots of unity, acts on X by the usual action of complex multiplication. Define map f : X X by f(e iθ )=e i θ. Then Ω G (f)={e i0,e i π,e iπ,e i 3π } and f(ω G (f))={e i0,e i π 4,e i π, e i 3π 4 }. In order to prove the converse of Remark 3.4(), we recall the following result proved in [17] [Lemma 3.1]. Lemma 3.6. Let X be a compact metric G space, where G is compact. Then for ɛ>0 there are η > 0 and δ > 0 such that for all g in G and x in X U η (gx) gu ɛ (x) and gu δ (x) U ɛ (gx). The following results gives conditions under which Ω G (f) isanf invariant set. Proposition 3.7. Let X be a compact metric G space, with G compact. Suppose a pseudoequivariant onto map f defined on X has the G shadowing property. Then f(ω G (f)) = Ω G (f). Proof. In view of Theorem 3.4() it is sufficient to show that Ω G (f) f(ω G (f)). Assume on the contrary. Then for an x in Ω G (f) f(ω G (f)) there is an ɛ>0 such that U ɛ (x) X f(ω G (f)). By Lemma 3.6, choose an η, 0< η < ɛ, such that for all y X and g G, U η (gy) gu ɛ (y). By the G shadowing property of f there is a δ > 0 such that every (δ, G) pseudo orbit for f is η shadowed by a point of X. Also f is uniformly continuous, therefore there is a ν > 0 such that d(y, z) < ν d(f(y),f(z)) < δ. Let U be the ν- neighbourhood of x. Then there is an integer k>0, a g G and an element z in gf k (U) U such that z = f k (gt), where t U. Construct a (δ, G) pseudo orbit θ = {x n : n 0} = {x, f(t),..., f k 1 (t), x,...}. Since f has G shadowing θ is η shadowed by a point of X, sayy. Therefore, for each j 0, there is g kj G, satisfying d(f kj (y),g kj x) <η. This implies {f kj (y) :j 0} G(U ɛ (x)), where G(U ɛ (x)) = g G gu ɛ(x). If a subsequence of {f kj (y) :j 0} converge to the point ỳ of X then it is now easy to verify that f kj (mỳ) U ɛ (x) as well as f kj (mỳ) f(ω G (f)), which is not possible. One of the important result in the topological theory of dynamical systems is Spectral decomposition theorem due to Smale [3, Theorem ]. A similar type of Theorem for G spaces was first proved in [16] and subsequently also in [6]. One of the main ingredients in both the proof is the following Theorem, which is similar to the one proved by Aoki in [].

6 584 E. Shah and T. Das Theorem 3.8. Let f : X X be a pseudoequivariant onto map defined on a compact metric G space X, where G is compact. If f has G shadowing property then so does f ΩG (f). Proof. Let ɛ>0 be given. By the G shadowing property of f, there is a δ>0 such that every (δ, G) pseudo orbit for f is ɛ shadowed by a point of X. Let {B 1,B,..., B n } be a finite cover of Ω G (f) and set η = min{d(b i,b j ):i j}. We complete the proof by showing that every τ Gpseudo orbit for f ΩG (f) is ɛ shadowed by a point of Ω G (f), where 0 < τ < min{δ, η}. Let {x i : i 0} be a τ Gpseudo orbit for f ΩG (f) in Ω G (f). Then for each i, x i B k, for some k. Take x p,x q {x i : i 0} such that p>qand let {x p = z 0,z 1,..., z n = x q } and {x q = y 1,y,..., y m = x p } be finite (δ, G) pseudo orbits from x p to x q and x q to x p respectively. Put k = n + m and construct a (δ, G) pseudo orbit {t i : i 0} = {x p = z 0,z 1,..., z n = x q,y,..., y m = x p,...}. Since f has G shadowing {t i : i 0} is ɛ shadowed by a point of X, sayx p,q. Therefore, d(f n (x p,q ), g n t j ) < ɛ, where n = ki + j, i>0, 0 <j<k. Let T = cl{f ki (x p,q ):i 0}. If T is discrete then there is an r>0such that f r (x p,q ) = x p,q. Therefore x p,q Ω G (f). Suppose T is not discrete, then there is a subsequence in T which is convergent, say, it converges `x p,q. Then `x p,q Ω G (f). Consider the sequence {`x p,q }. Then {x i : i 0} is ɛ shadowed by a point y of Ω G (f), where y is a limit point of the subsequence of {`x p,q }. 4. G Minimality Definition 4.1. A self homeomorphism f defined on a compact metric G space X( is said to be a G minimal homeomorphism if for each x X, ) cl g G O f(gx) = X. Minimality of f implies G minimality of f but that the converse in general need not be true is justified by Example 4.. If the action of G on X is trivial then the notion of G minimality of f coincides with that of minimality of f. Also, a G minimal map is always topologically G transitive. Real numbers {β 1,β,...β n } are said to be rationally independent if {β 1,β,..., β n, 1} are linearly independent over Q. Recall that a rotation f on n dimensional Torus T n defined by f ((θ 1,θ,..., θ n )) = (θ 1 + β 1,θ + β,..., θ n + β n ) is a minimal map if and only if {β 1,β,...β n } is rationally independent [0] Example 4.. Consider n dimensional Torus T n = S 1 S 1... S 1. Suppose G = T k, k < n, acts on T n by the action (g 1,g,..., g k ) (θ 1,θ,..., θ k,θ k+1,..., θ n )= (θ 1 + g 1,θ + g,..., θ k + g k,θ k+1,..., θ n ), where (g 1,g,..., g k ) G are represented by arguments. Define a map h : T n T n

7 Consequences of shadowing property of G-spaces 585 by (θ 1,θ,..., θ k,θ k+1,..., θ n )=(θ 1,θ,..., θ k,θ k+1 + β k+1,..., θ n + β n ), where {β k+1,β k+,..., β n } are rationally independent. Then h is a T k minimal map but is not minimal, as O h (0, 0,..., 0) is not dense in T n. Lemma 4.3. Let X be a compact metric G space, with G compact. A pseudoequivariant homeomorphism f is G minimal if and only the only f invariant G invariant closed subset of X is either X or empty set. Proof. Suppose f is G minimal and E is a non empty f invariant G invariant closed subset of X. Then for x E, g G O f(gx) E. Therefore E = X. Conversely suppose the only f invariant G invariant ( closed non empty subset ) of X is X. But this implies f is G minimal as cl g G O f(gx) is a non empty closed f invariant G invaraint subset of X. We use the following lemma to prove the main theorem of the section, which is similar to that of Aoki proved in [1]. Lemma 4.4. Suppose f in H(X) is pseudoequivariant and G is compact. If f has the G shadowing property( then for given ) ɛ>0 and x Ω G (f) there exists y X and k>0 such that cl O ˆf k(g(y)) U ɛ (G(x)), where U ɛ (G(x)) is the ɛ neighbourhood of G(x) with respect to metric d 1 on X/G induced by d. Proof. Let ɛ>0 be given. By uniform continuity of the orbit map π there is a β>0such that d(x, y) <β d 1 (π(x),π(y)) <ɛ. Choose an η, 0<η< β, such that for all g G and x X, gu η (y) U β (gy). G shadowing of f implies there is a δ, 0<δ< β, such that every (δ, G) pseudo orbit for f is η traced 3 by a point of X. Forx Ω G (f), let U denote the δ neighbourhood of x. Then there is k>0and ǵ G such that ǵf k (U) U φ. Ifz ǵf k (U) U, then for t U there is g G with z = f k (gt). Construct a k periodic (δ, G) pseudo orbit θ = {y i : i Z} = {..., f k 1 (gy), y,f(gy),..., f k 1 (gy), y...}. Since f has the G shadowing property, θ is η traced by a point of X, say, z. Therefore, for all n Z there is g nk G such that d(f nk (z), g nk y) <η. This( further implies ) d(g 1 nk f nk (z), x) <β. By uniform continuity of π we have cl O ˆf k(g(z)) U ɛ (G(x)). Theorem 4.5. Let X be a non degenerate compact connected metric G space, where G is compact. Then a pseudoequivariant G minimal homeomorphism does not have the G shadowing property.

8 586 E. Shah and T. Das Proof. Let l = diamx/g and ɛ = l. Suppose f has the G shadowing property. 3 ( ) Then for each x X by Lemma 4.4 there is y X such that cl O ˆf k(g(y)) U ɛ (G(x)). G minimalty of f implies X = ( ) k cl (f i (gy)). i=0 g G ( ) Therefore by connectedness and G minimality we have cl O ˆf k(g(y)) = X. But this further implies X/G = O ˆf k(g(y)). Therefore l ɛ, which is not possible. Therefore f does not have the G shadowing property. 5. G specification One of the important property in the theory of dynamical system is the specification property. For instance, existence of mixing properties and large number of periodic orbits is possible in the presence of specification property. If one desires to approximate two finite pieces of two distinct finite/ infinite orbits by one periodic orbit, then one can do it using specification property. The notion was first defined by Bowen in [5]. In this Section we define the notion of specification on G spaces. Definition 5.1. Let f : X X be a homeomorphism of a compact metric G space X. Then f is said to have G specification if for any ɛ>0 there exists M = M(ɛ) > 0, such that for any finite sequence of points g 1 x 1,g x,...g k x k X, for some g 1,g,...g k G and for j k choosing any sequence of integers a 1 b 1 < a b <... < a k b k such that a j b j 1 M ( j k) and an integer p with p M(b k a 1 ) there exists a point x X with f p (x) =gx, for some g G and satisfying d(f i (x),p i f i (x j )) <ɛ, for some p i G and a j i b j, 1 j k. Following is an example of a map which has G specification but not specification. Example 5.. Recall space and map of Example 3.3. The map f being shift map on each of i, i =1,, f is Z expansive homeomorphism and also f has the Z shadowing property. Further, observe that the set of G periodic point is dense in X and also there is a point x X such that g Z (O f (gx)) is dense in X. Therefore, for any open sets U and V in X there is an integer N such that for all n N there exists g n G satisfying U g n f n (V ) φ. Therefore by Theorem 5.3 f has the Z specification.

9 Consequences of shadowing property of G-spaces 587 In the following theorem we obtain a sufficient condition for a G expansive homeomorphism possessing the G shadowing property to have G specification. This result is similar to the result proved by Aoki and Hiraide in [3, Theorem ]. Theorem 5.3. Let (X, d) be a compact metric G space with G compact and d be an invariant metric. Suppose f : X X is a G expansive pseudoequivariant homeomorphism having the G shadowing property. If for non empty open sets U, V in X there is an N>0such that for all n N there exists g n G satisfying U g n f n (V ) φ then f has the G specification. Proof. Let e>0beag expansive constant for f and let ɛ be such that 0 <ɛ< e. Since f has the G shadowing property, there exists δ>0such that every (δ, G) pseudo orbit for f is ɛ traced by a point of X. If I = {U 1,U,...U m } is a finite subcover of X with diam U i < δ, for each i {1,,..., m}, then by hypothesis for each U i,u j I, there is M i,j > 0 such that for all n M i,j there is ǵ n G satisfying U j ǵ n f n (U i ) φ (*) Let M=max{M i,j : 1 i, j m} and g 1 x 1,g x,..., g k x k X, for some g 1,g,..., g k G and for j k, choose any sequence of integers a 1 b 1 < a b <... < a k b k such that a j b j 1 M and an integer p with p M + (b k a 1 ). Define a k+1 = b k+1 = p+a 1, x k+1 = f a 1 a k+1(g1 x 1 ). By U(z) we mean an open ball U in I containing z. Since a j+1 b j M, by (*) there is ǵ aj+1 b j G such that U(f a j+1 (g j+1 x j+1 )) ǵ aj+1 b j f a j+1 b j (U(f b j (g j x j ))) φ. This implies there is y j f a j+1 b j (U(f b j (g j x j ))) such that f a j+1 b j (y j )=ḱa j+1 b j ý j, where y j U(f a j+1 (g j+1 x j+1 )) ǵ aj+1 b j f a j+1 b j (U(f b j (g j x j ))), ḱa j+1 b j G. Construct a (δ, G) pseudo orbit {z i : i Z} for f in X as follows: z i = f i (g j x j ), if a j i b j z i = f i b j (y j ), if b j i a j+1 z i+p = z i, for all i Z Then {z i : i Z} is ɛ traced by a point of X, sayx. Therefore for each i Z, there is l i, l i+p G such that d(f i (x), l i z i ) <ɛand d(f i+p (x), l i+p z i+p ) <ɛ. This implies for each i Z, there exists l i, l i+p G satisfying d(l 1 i+p f i+p (x), l 1 i f i (x)) < ɛ <e. G expansivity of f further implies G(f p (x)) = G(x). Therefore there is g G such that f p (x) =gx. Also, for a j i b j, z i = f i (g j x j ) implies d(f i (x),l i z i )=d(f i (x),l i f i (g j x j )) <ɛand f p (x) =gx. Hence by definition f has G specification.

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