On some kinds of dense orbits in general nonautonomous dynamical systems

We study the theory of universality for the nonautonomous dynamical systems from topological point of view related to hypercyclicity. The conditions are provided in a way that Birkhoff transitivity theorem can be extended. In the context of generalized linear nonautonomous systems, we show that either one of the topological transitivity or hypercyclicity give sensitive dependence on initial conditions. Meanwhile, some examples are presented for topological transitivity, hypercyclicity and topological conjugacy.


Introduction
Nonautonomous dynamical systems provide a successful framework to describe a large variety of phenomena in biology, informatics, economy, medicine, quantum mechanics and etc [2,8,9,17], in the case that they can not be modelled by autonomous systems. A knowledge of the topological and geometrical structure of nonautonomous discrete system is useful both in developing theory and in applications [6,10]. In the last decade chaotic behaviour of nonautonomous discrete systems attracted many researchers [1,3,14,16,18]. Chaos theory deals with complex and unpredictable behaviour of phenomena over time. Even in the context of dynamical systems, di erent de nitions of chaos has been posed which some of them are classi ed by Forti [4]. In most of the de nitions, topological transitivity plays a central role. In this context, Birkho transitivity theorem establishes a straight relationship between topological transitivity and density of orbits on a separable complete metric space without isolated points [5,13].
The present paper focuses on the generalization of nonautonomous dynamical systems and challenges some topological and dynamical characteristics like topological transitivity, sensitivity, hypercyclicity and conjugacy. Under suitable assumptions, some new results have been achieved that Birkho transitivity theorem is one of them. Suppose that X and X are two nonempty topological spaces and fn : X → X, n ∈ N are maps. The orbit of x ∈ X is de ned as orb(x ) := {fn(x )| n ∈ N}. When orb(x ) is a dense subset of X, the point x is called universal [5]. We call the sequence {fn} ∞ n= a generalized nonautonomous discrete system (for short GNDS) on X , that determines an equation in the form The de nition of a GNDS on X is more general than the de nition of a NDS introduced in [7,15] which is as follows: assume that X is a metric space and for each n ∈ N ∪ { }, Xn is its subspace. Also, let gn : Xn → X n+ be an arbitrary map, then they call the family {gn} ∞ n= a NDS. Indeed, such a family generates a NDS via time-dependent di erence equations x n+ = gn(xn). Here, for a given element x ∈ X , a sequence {xn} ∞ n= is called the orbit of x (or an orbit of {gn} ∞ n= in the iterative way) and is de ned as x , x = g (x ), x = g (x ), . . . , x n+ = gn(xn), . . . . Consider fn := g n− • g n− · · · • g , we observe that fn is a map from X to X, and therefore our de nition is a generalization (see Example 1.(h) and the explanation after that). Throughout the paper, we always assume that X is a subspace of X. Hence, for given x ∈ X one may expect two main types of density by restricting to the subspace X . In the other word, for the underlying GNDS we always deal with the subsets Investigating topological structure of these subspaces leads us to achieve some results that are organized as follows: In Section 2, some properties of subspaces O and O are discussed, and then we challenge these subspaces with topological transitivity. In fact we prove that if O = X , the system is topologically transitive. Also, under certain conditions for underlying spaces and maps fn (see Theorem 2), the topological transitivity yields O = X .
In Section 3, the concepts of topological h-conjugacy and topological {hn}-conjugacy are de ned for GNDSs on X . We show that these two de nitions t exactly to De nition 3.6 in [16] and De nition 3.1 in [15], respectively. Then considering the hypothesis hn some classical dynamical properties are preserved under two {hn}-conjugated equations.
In Section 4, assuming underlying space X as a Frechet space we focus on the generalized linear nonautonomous discrete systems (in short GLNDSs). Indeed we prove that either topological transitivity or hypercyclicity always implies sensitive dependence on initial conditions. Also, in this section two examples are posed, the rst system is topologically transitive while O ≠ X , and for the second one (that satis es hypothesis of Theorem 2) O = X holds while it is not topologically transitive.
In the study of dynamical systems, we may deal with results that less hypothesis for their proofs can be written. That is why we choose the underling system in the general form, and then we provide the conditions for the system in a way that some classical results obtained. As we mentioned in above, these results are related to chaos, topological conjugacy and linear systems. To continue this, it seems that we can nd more results around expansiveness, sensitivity and topological entropy. Another extension is to interpret a kind of closeness for two general nonautonomous dynamical systems, and then to achieve the results in some contexts like symbolic dynamic, bifurcation theory and etc.

Hypercyclic Orbits and Topological Transitivity
We begin the section by posing the de nition of a GNDS.
De nition 1. Let X be a topological space and X be a subspace of X. A sequence {fn} ∞ n= of maps fn : X → X is called a GNDS on X and the orbit of x ∈ X is de ned as orb(x ) := {fn(x )| n ∈ N}.
We write f ,∞ instead of {fn} ∞ n= . Obviously, a dynamical system g : X → X can be considered as a GNDS on X by setting fn := g n |X .
Many other examples are presented here that may be useful in the study of the theory of GNDSs.

Example 1. (a)
Let Y be a topological space and F : X × Y → X be an arbitrary map. Also, let {an} ∞ n= be a sequence in Y. Then f ,∞ given by fn(x) = F(x, an) is a GNDS on X .
(b) Let M be a smooth manifold and f ∈Di (M). Also, let p be a hyperbolic point and X = W u loc (p). Then f ,∞ given by fn(x) = f n (x) is a GNDS on W u loc (p). The similar situation can be written for W s loc (p). (c) Let g and g be two discrete dynamical systems on X and A be any subset of natural numbers. Then f ,∞ given by . Also, let D be the di erentiable operator and maps fn de ned by Then f ,∞ is a GLNDS on any given subspace X of H(C). Notice that in the case gn(x) := x n , fn(g) is the map that associates with g its Taylor polynomial of degree n at .
(f) Let {αn} ∞ n= be a sequence of real numbers and X be the space of all continuous functions on R endowed with the sup norm. Also, let fn(g) =g whereg is Birkho 's translationg(x) := g(x + αn). Then f ,∞ is a GLNDS on the subspace X .
(g) Let {αn} ∞ n= be a sequence of real numbers. For each ≤ p < ∞, let X := l p be the space of all psummable sequences. Then f ,∞ given by fn (x , x , · · · , xn , · · · ) = αn(xn , x n+ , · · · ) is a linear GNDS on any subspace X of l p . In general a GNDS can not be stated in the iterative way, but if all maps fn are bijective, by getting g := f and gn := f n+ • f − n as the maps from Xn := fn(X ) to X n+ := f n+ (X ) then this GNDS is in the iterative way. In the case that the maps fn (even one of them) are not injective, maybe GNDS can not be stated in the iterative way. For instance, in Example 1.(h) there is no function gn : Xn → X n+ with Xn = X n+ = [ , ) such that gn • fn = f n+ .
To achieve this, let An := {a| fn(a) = }. Thus, An = k + n | ≤ k < n, k ∈ N ∪ { } . Hence, for every ≤ k < n one follows: Consequently, gn( ) = n + n ( k + ) mod 1, which is in contradiction with the well-de nitive of the function gn. The The smallest such k is called the period of x and the set of weakly periodic points is denoted by WP(fn). Also, weakly k-periodic point x is called k-periodic if f mk+n (x ) = fn(x ), for every m, n ∈ N. The set of periodic points is denoted by P(fn), and so P(fn) ⊂ WP(fn). The point x is called a nite orbit point if the set orb(x ) is nite. The set of nite orbit points is denoted by Fin(fn). Notice that, P(fn) ⊂ Fin(fn) while the following example demonstrates that in general the relation WP(fn) ⊂ Fin(fn) does not hold. Now we are going to generalize some topological concepts and results that is stated in [11], for GNDSs on X and discuss the complicated behaviour that appears in the universality process. There are two kinds of densities of orbits for GNDSs on X as follows:

Example 2. Consider fn(x)
In such a case, x is called a hypercyclic point and the set of hypercyclic points is denoted by • GNDS f ,∞ is called weakly hypercyclic or weakly universal on X , if there exists a point whose orbit's closure contains X . Such a point is called a weakly hypercyclic point and the set of weakly hypercyclic points is denoted by Thus, in dealing with the terminology "density of an orbit" we must specify which one of the above subspaces is intended. When these sets are large subspaces from topological point of view, we will expect a rich dynamic around X . Before going any further, it is important to know the relationship between O and O .
Proposition 1. With notations as above, we have: Proof. The same proof of Theorem 1 in [11].
In the above example, if all the matrixs An are invertible, then by taking gn := A n+ A − n as the functions from An X to A n+ X , this GNDS is stated in the iterative way (see the explanation after Example 1).
Topological transitivity can be extended as below (see De nition 2.3 in [16]).

De nition 2. GNDS f ,∞ is called topologically transitive on X if for any two nonempty open sets U and
Now we are going to state the relation between density and topological transitivity.
This implies the existence of n ∈ N such that fn(x) ∈ V . Hence, fn(U )∩V ≠ ϕ which completes the proof.
The GLNDS appeared in Example 3 is hypercyclic but not topologically transitive. Proof. Since X is a second countable subspace and X is the rst countable at any point of X , it is easy to show the existence of a collection {Um} m∈N of open sets in X such that i) Um ∩ X ≠ ϕ, ii) the family {Um ∩ X } m∈N is a basis for X , iii) for each x ∈ X , the family {Um} m∈N is a local basis for x in X.
We claim that To prove the claim, let  (Um) is a dense G δ -set in X , since X is a Baire space. This also tells us that O = X and the proof of Theorem is complete.
Theorems 1 and 2 enable us to state a generalization of Birkho transitivity theorem.

Theorem 3.
Suppose that X is a second countable and Baire subspace of topological space X. Also, suppose that X is rst countable at any point of X and all the maps fn are continuous. Then, • density of hypercyclic points implies topological transitivity of GNDS f ,∞ on X , • topological transitivity of GNDS f ,∞ on X implies density of a G δ -set of all weakly hypercyclic points.
In particular, when X = X we can drop "weakly" from the statement of theorem.

Topological Conjugacy
The conjugacy is an important feature that worths to mention in the study of GNDSs. We begin by an extension of De nition 3.6 in [16] in the usual way.
De nition 3. Let X and Y be two topological spaces and X ⊂ X, Y ⊂ Y. Let f ,∞ and g ,∞ be two GNDSs on X and Y , respectively. If there exists a homeomorphism h : X → Y such that h • fn = gn • h on X , for each n ∈ N, then f ,∞ is said to be topologically h-conjugate or simply conjugate to g ,∞ .
The conjugacy establishes an equivalence relation on GNDSs. It is easy to show that orbits, invariant sets, weakly k-periodic points, k-periodic points and nite orbit points are preserved by the conjugated GNDSs. Next theorem asserts that hypercyclic points, weakly hypercyclic points and transitivity are preserved under conjugacy too.

Theorem 4.
Let h : X → Y be a homeomorphism and GNDS g ,∞ be topologically h-conjugate to GNDS f ,∞ . Then, i) if x is a hypercyclic point (resp. weakly hypercyclic point) for f ,∞ , then h(x ) is a hypercyclic point (resp. weakly hypercyclic point) for g ,∞ , ii) if f ,∞ is topologically transitive on X , then g ,∞ is topologically transitive on Y .
(ii) Suppose that U and V are two nonempty open subsets of Y . Since, GNDS f ,∞ is topologically transitive on X , there exists a number n ∈ N such that fn h − (U ) ∩ h − (V ) ≠ ϕ. Since h is a homeomorphism and GNDS g ,∞ is h-conjugate to f ,∞ , then Thus, GNDS g ,∞ is topologically transitive on Y that implies the proof for (ii).
In the context of metric spaces, Tian and Chen proved that two uniformly topologically conjugate NDSs share the sensitive dependence on initial conditions (see De nition 5) too. But similar conclusion does not necessarily hold for two conjugate NDSs [16]. As we observed, homeomorphism h plays a key role in transferring the classical dynamical properties of GNDS f ,∞ to GNDS {gn}. Since the maps emerged in GNDSs may be selected randomly, one do not expect such homeomorphism always exists, see Example 4. For the moment, let X * := ∞ n= Xn and Y * := ∞ n= Yn. In the lack of a conjugacy between GNDS {fn} and GNDS {gn}, the existence of a conjugacy between GNDS {Fn} and GNDS {Gn} is possible, where Fn := fn |X * and Gn := gn |Y * . In such cases, most of the dynamical properties are permanent for GNDSs {fn} and {gn}, see Theorem 5. This leads us to another type of conjugacy as below, which completely ts into other De nitions. Two conjugated GNDSs in the sense of De nition 4 may share a given dynamical property under special additional assumptions. For the sake of simplicity, let Xn := fn(X ) and Yn := gn(Y ).  [15]. As we explained in Section 1, by taking fn := g n− • g n− · · · • g andfn :=g n− •g n− · · · •g , f ,∞ andf ,∞ are two GNDSs on X and Y , respectively that are stated in the successive way. Proof. Firstly, assume that h n+ • gn =gn • hn for each n ∈ N ∪ { }. Thus,

De nition 4. Let X and Y be two topological spaces and X
To prove the inverse, let x ∈ Xn. Then there is x ∈ X such that fn(x ) = x. Thus, This completes the proof of the proposition. (iii) Let y := h (x ) and U ∈ τ Y . Then h − (U) ∈ τ X and so This guarantees the existence of xn ∈ orb(x ) such that xn ∈ X ∩ h − (U). Due to the fact xn ∈ X and by the assumption of the theorem, we obtain hn(xn) = h (xn) ∈ Y ∩ U. On the other hand, the sequence {yn} ∞ n= with yn = hn(xn) forms the same orbit of g ,∞ which starts from y = h (x ). Hence, orb(y ) ∩ Y ∩ U ≠ ϕ since yn ∈ Y ∩ U. This implies that y ∈ O in g ,∞ and completes the assertion.
(iv) Suppose that U and V are two nonempty open subsets of Y . Since, f ,∞ is topologically transitive on X , there exists some n ∈ N such that ϕ ≠ fn h − (U ) ∩h − (V ) ∈ Xn ∩X . Since, h is a homeomorphism, by the assumption of the theorem, we have Thus, g ,∞ is topologically transitive on Y that implies the proof for (ii).

By the hypothesis of Theorem 5, if invariant subset
The following example says that periodic points, nite orbit points and O may not remain preserved under a {hn}-conjugacy, even for the linear GNDSs stated in the iterative way with the assumption that hn = h on X ∩ Xn and h − n = h − on Y ∩ Yn.  Strongly {hn}-conjugacy establishes an equivalent relation on GNDSs that shares invariant sets, weakly kperiodic points, periodic points, nite orbit points, transitivity, sensitive dependence on initial conditions in X and the sets O i , i = , .

Generalized linear nonautonomous discrete systems
In this section we discuss the case that underlying space X is a Frechet space, i.e. a vector space endowed with a separating increasing sequence (pn) n∈N of seminorms which is complete with the following metric: x, y ∈ X.
An important feature of this metric is that it is translation-invariant, i.e. d(x, y) = d(x+z, y+z) for all x, y, z ∈ X [5].
De nition 8. Let X be a Frechet space and X be a Frechet subspace of X. A sequence {Tn} ∞ n= of linear maps Tn : X → X is called a generalized linear nonautonomous discrete system (in short GLNDS) on X .
At rst we show that in case of the GLNDSs, sensitive dependence on initial conditions follows from the topological transitivity. Hence, one may omit sensitivity assumption (third condition) from Devaney's and Wiggins's de nitions of chaos in the category of the GLNDSs.

Theorem 7.
Suppose that the GLNDS T ,∞ is topologically transitive on X . Then it has sensitive dependence on initial conditions. Proof. Let δ = and choose x ∈ X , ϵ > . Also, let nonempty open sets U := B d ( , ϵ) and V := {z ∈ X : d(z, ) > }. Since T ,∞ is topologically transitive, there exists some n ∈ N such that Thus, there exists z ∈ U such that Tn(z) ∈ V . Consider y := x+z. By using the translation-invariant property of the metric, we obtain This means that y ∈ B d (x, ϵ). On the other hand, which implies that GLNDS has sensitive dependence on initial conditions with sensitivity constant δ = .
The next theorem states the relation between hypercyclicity and sensitivity.

Theorem 8. Suppose T ,∞ is hypercyclic GLNDS. Then it has sensitive dependence on initial conditions.
Proof. Let δ = , and choose z be a point that orb(z ) ∩ X is dense in X . Let x ∈ X and U be an open set in X containing x . Then there exists a non zero number α small enough such that αz + x belongs to U . Now let y := αz + x . Hence d(xn , yn) = d(αzn , ) for all n ∈ N. We can select the natural number n large enough, such that d(αzn , ) > δ. Thus, d(xn , yn) > δ, that implies it has sensitive dependence on initial conditions.
By a small change in the proof of Theorem 8, it can be proved that the existence of a weakly hypercyclic point always implies the sensitivity. Now we bring two examples of GLNDSs stated in the iterative way to show that the inverse of Theorems 1 and 2 is not always true. Next example is topologically transitive while O ≠ X . Furthermore, O = ∅ that means it is not hypercyclic. where R k : X k+ − → X k+ is the linear transformation that transfers the point (ρ, r k π, ψ)s ∈ X k+ − to (ρ, r k+ π, ψ)s ∈ X k+ . And also, R k : X k+ → X k+ is the linear transformation that rotates the points counterclockwise by the angle α. In other words, R k : X k+ → X k+ is given by (ρ, θ, ψ)s → (ρ, θ, ψ − α)s . Firstly, we show that {Tn} ∞ n= is topologically transitive. It is easy to check that {Tn} ∞ n= satis es the following equalities: (i) T k+ − • T k+ − • · · · • T k = id X k ∀ k ∈ N, (ii) T k +m− • T k +m− • · · · • T k = αmid X k ∀ ≤ m < k − . From relation (i), we obtain: (4.1)

Let U and V be two open sets in X which are bounded regions between two sectors;
(x, y, )| m x < y < m x, r < x + y < r .
Let R ψ be a ψ-valued counter-clockwise rotation on the points X . Thus, there exist p ∈ U , m ∈ N and Ip ⊂ ( , ) such that R πθ (αm .p) ∈ V for all θ ∈ Ip. We identify [ , ] S × { } by θ ; (cos πθ, sin πθ, ). Then there exists k ∈ N such that for each l ∈ N we have S ⊂ l+k i=l R iα (Ip). (4.2)