Induced Dynamics of Non-Autonomous Discrete Dynamical Systems
aa r X i v : . [ m a t h . D S ] M a r INDUCED DYNAMICS OF NON-AUTONOMOUS DISCRETEDYNAMICAL SYSTEMS
PUNEET SHARMAA bstract . In this paper, we investigate the dynamics on the hy-perspace induced by a non-autonomous dynamical system ( X , F ),where the non-autonomous system is generated by a sequence ( f n )of continuous self maps on X . We relate the dynamical behaviorof the induced system on the hyperspace with the dynamical be-havior of the original system ( X , F ). We derive conditions underwhich the dynamical behavior of the non-autonomous system ex-tends to its induced counterpart(and vice-versa). In the process,we discuss properties like transitivity, weak mixing, topologicalmixing, topological entropy and various forms of sensitivities. Wealso discuss properties like equicontinuity, dense periodicity andLi-Yorke chaoticity for the two systems. We also give exampleswhen a dynamical notion of a system cannot be extended to itsinduced counterpart (and vice-versa).
1. INTRODUCTIONFor many years, dynamical systems have been studied to investi-gate many of the physical or natural phenomenon occurring in na-ture. Using dynamical systems, long term behavior of various naturalphenomenon have been predicted to su ffi cient accuracy and many ofthe underlying processes have been investigated using the theory ofdynamical systems[5, 13]. In many cases, the structure of underlyingspace is a pivotal factor in determining the dynamical behavior ofthe system. This has resulted in further investigations in the field oftopological dynamics and a lot of work in this area has already beendone[3, 4, 5]. However, most of the phenomenon occurring in na-ture arise collectively as union of several individual components andhence set valued dynamics plays an important role in understandingany of these phenomenon. Such an approach has found applicationsin various branches of sciences and engineering[6, 9, 14]. Thus therewas a strong need to develop and understand the dynamical behav-ior of the induced set valued systems. As a result, many of the natural Key words and phrases. hyperspaces, non-autonomous dynamical systems, tran-sitivity, weakly mixing, topological mixing, topological entropy, Li-Yorke chaos. questions relating the dynamical behavior of a system and its set val-ued counterpart have been raised and answered[1, 10, 11]. However,most of the investigations have been made when the rule determin-ing the underlying system is time-invariant. Such an approach failsto investigate the dynamics of a general system as the governingrule of a natural phenomenon may change with time. Thus thereis a need to understand the dynamics of the induced system whenthe underlying system is non-autonomous in nature. In this paper,we investigate the dynamical behavior of the induced system, wheninduced by a non-autonomous system. We prove that many of theresults for the non-autonomous case are analogous extensions of theautonomous case. We derive conditions under which the dynamicalbehavior of a system is extended to the hyperspace(and vice-versa).In the process, we discuss properties like dense periodicity, variousforms of mixing and sensitivity, equicontinuity and Li-Yorke chaotic-ity for the two systems. We establish our results when the hyperspaceis endowed with a general admissible hyperspace topology. Beforewe move further, we provide some of the basic concepts required.1.1.
Dynamical Systems.
Let ( X , d ) be a compact metric space and let F = { f n : n ∈ N } be a family of continuous self maps on X . Let ( X , F )denote the non-autonomous system generated by the family F viathe rule x n = f n ( x n − ). For any point x ∈ X , { f n ◦ f n − ◦ . . . ◦ f ( x ) : n ∈ N } defines the orbit of x . The objective of study of any non-autonomousdynamical system is to investigate the orbit of an arbitrary point x in X . For notational convenience, let ω n ( x ) = f n ◦ f n − ◦ . . . ◦ f ( x ) denotethe state of the system after n iterations.A point x is called periodic for F if there exists n ∈ N such that ω nk ( x ) = x for all k ∈ N . The least such n is known as the periodof the point x . The system ( X , F ) is equicontinuous at a point x ∈ X if for each ǫ >
0, there exists δ > d ( x , y ) < δ implies d ( ω n ( x ) , ω n ( y )) < ǫ for all n ∈ N , y ∈ X . The system is called equicon-tinuous if it is equicontinuous at each point of X . The system ( X , F )is uniformly equicontinuous if for each ǫ >
0, there exists δ > d ( x , y ) < δ implies d ( ω n ( x ) , ω n ( y )) < ǫ for all n ∈ N , x , y ∈ X .The system ( X , F ) is transitive (or F is transitive) if for each pair ofopen sets U , V in X , there exists n ∈ N such that ω n ( U ) T V , φ . Let F n = { f kn + ◦ f kn + ◦ . . . ◦ f ( k + n : k ∈ Z + } . The system ( X , F ) is called n -transitive if the system ( X , F n ) is transitive. If ( X , F ) is n -transitivefor each n ∈ N , then the system is called totally transitive. The sys-tem ( X , F ) is said to be weakly mixing if for any two pairs U , U and NDUCED DYNAMICS OF NON-AUTONOMOUS DISCRETE DYNAMICAL SYSTEMS3 V , V of non-empty open subsets of X , there exists a natural number n such that ω n ( U i ) T V i , φ for i = ,
2. Equivalently, we say that thesystem is weakly mixing if F × F is transitive. The system ( X , F ) is saidto be weakly mixing of order k if for any collection U , U , . . . , U k and V , V , . . . , V k of non-empty open subsets of X , there exists a naturalnumber n such that ω n ( U i ) T V i , φ for i = , , . . . , k . The system issaid to be topologically mixing if for every pair of non-empty open sets U , V in X , there exists a natural number K such that ω n ( U ) T V , φ for all n ≥ K . The system is said to be sensitive if there exists a δ > x ∈ X and each neighborhood U of x , there exists n ∈ N such that diam ( ω n ( U )) > δ . If there exists K > diam ( ω n ( U )) > δ , ∀ n ≥ K , then the system is cofinitely sensitive . Asystem ( X , F ) is expansive ( δ -expansive) if for any pair of distinct ele-ments x , y ∈ X , there exists k ∈ Z + such that d ( ω k ( x ) , ω k ( y )) > δ . Aset S is said to be δ - scrambled if for any distict pair of points x , y ∈ S ,lim sup n →∞ d ( ω n ( x ) , ω n ( y )) > δ but lim inf n →∞ d ( ω n ( x ) , ω n ( y )) =
0. A system iscalled Li-Yorke sensitive if there exists δ > x ∈ X and each neighborhood U of x , there exists y ∈ U such that { x , y } is a δ -scrambled set. A system ( X , F ) is said to be Li-Yorke chaotic if it con-tains an uncountable scrambled set. A dynamical system ( X , f ) has chaotic dependence on initial conditions if for any x ∈ X and any neigh-borhood U of x there exists y ∈ U such that lim sup n →∞ d ( f n ( x ) , f n ( y )) > n →∞ d ( f n ( x ) , f n ( y )) =
0. In case the f n ’s coincide, the abovedefinitions coincide with the known notions of an autonomous dy-namical system. See [3, 4, 5] for details.We now define the notion of topological entropy for a non-autonomoussystem ( X , F ).Let X be a compact space and let U be an open cover of X . Then U has a finite subcover. Let L be the collection of all finite subcoversand let U ∗ be the subcover with minimum cardinality, say N U . De-fine H ( U ) = logN U . Then H ( U ) is defined as the entropy associatedwith the open cover U . If U and V are two open covers of X , define, U ∨ V = { U T V : U ∈ U , V ∈ V} . An open cover β is said to be re-finement of open cover α i.e. α ≺ β , if every open set in β is containedin some open set in α . It can be seen that if α ≺ β then H ( α ) ≤ H ( β ).For a self map f on X , f − ( U ) = { f − ( U ) : U ∈ U} is also an opencover of X . Define, PUNEET SHARMA h F , U = lim sup k →∞ H ( U∨ ω − ( U ) ∨ ω − ( U ) ∨ ... ∨ ω − k − ( U )) k Then sup h F , U , where U runs over all possible open covers of X is known as the topological entropy of the system ( X , F ) and is denotedby h ( F ). In case the maps f n coincide, the above definition coincideswith the known notion of topological entropy. See [3, 4] for details.1.2. Hyperspaces.
Let ( X , τ ) be a Haudor ff topological space. A hy-perspace associated with ( X , τ ) is a pair ( Ψ , ∆ ) where Ψ comprises ofa subfamily of all non-empty closed subsets of X and ∆ is a topologyon Ψ generated using topology on X . The set Ψ may comprise ofall compact subsets of X or all compact-connected subsets of X or allclosed subsets of X . A hyperspace topology is called admissible ifthe map x → { x } is continuous. More generally, if Ψ and ∆ are fixed,the induced space ( Ψ , ∆ ) is called the hyperspace generated fromthe space ( X , τ ). Let CL ( X ) and K ( X ) denote set of all non-emptyclosed and non-empty compact subsets of X respectively. We nowgive some of the standard hyperspace topologies.Let I be a finite index set and for all such I , let { U i : i ∈ I } be acollection of open subsets of X . Define for each such collection ofopen sets, < U i > i ∈ I = { E ∈ CL ( X ) : E ⊆ S i ∈ I U i and E T U i , φ ∀ i } The topology generated by such collections is known as the
Vietoristopology .Let ( X , d ) be a metric space. For any two closed subsets A , A of X , define, d H ( A , A ) = inf { ǫ > A ⊆ S ǫ ( B ) and B ⊆ S ǫ ( A ) } It is easily seen that d H defined above is a metric on CL ( X ) and iscalled Hausdor ff metric on CL ( X ). This metric preserves the metric on X , i.e. d H ( { x } , { y } ) = d ( x , y ) for all x , y ∈ X . The topology generated bythis metric is known as the Hausdor ff metric topology on CL ( X ) withrespect to the metric d on X .It is known that the Hausdor ff metric topology equals the Vietoristopology if and only if the space X is compact.Let Φ be a subfamily of the collection of all non-empty closedsubsets of X . The Hit and Miss topology determined by the collection Φ is the topology having subbasic open sets of the form U − where U is open in X and ( E c ) + with E ∈ Φ . As a terminology, U is called thehit set and any member E of Φ is referred as the miss set.For a metric space ( X , d ) and a given collection Φ of closed subsetsof X , the Hit and Far Miss topology or Proximal Hit and Miss Topology
NDUCED DYNAMICS OF NON-AUTONOMOUS DISCRETE DYNAMICAL SYSTEMS5 determined by the collection Φ is the topology having subbasic opensets of the form U − where U is open in X and ( E c ) ++ with E ∈ Φ .Here the collection hits each open set U and far misses the com-plement of each member of Φ and hence forms a hit and far misstopology.A typical member of the base for the Lower Vietoris topology onthe hyperspace CL ( X ) consists of the set, each of whose elementsintersect or hit finitely many open sets U , i.e. a typical basic open setis the intersection of finitely many U − . The Lower Vietoris topologyis the smallest topology on the hyperspace containing all the sets U − where U is open in X .A typical basic open set for the Upper Vietoris topology on the hy-perspace CL ( X ) is of the form U + where U is open in X . Thus, givena closed set C , a typical member of the base in the Upper Vietoristopology is the set whose elements are the elements of the hyper-space disjoint from the closed set C .It is observed that the Vietoris topology equals the join of UpperVietoris and Lower Vietoris topology, and is infact an example of a hitand miss topology. More generally, it is known that any admissiblehyperspace topology is of hit-and-Miss or Hit and Far Miss type.See[2, 7, 8] for details. 2. M ain R esults Let ( X , F ) be a non-autonomous topological dynamical systemand let Ψ ⊂ K ( X ) be a hyperspace admissible with F , i.e. for any A ∈ Ψ , ω k ( A ) ∈ Ψ for all k ∈ N . Then for any initial seed A ∈ Ψ , thenon autonomous system ( X , F ) induces a non-autonomous system( X , F ) on the hyperspace via the relation A n = f n ( A n − ) = ω n ( A ). Let ω k ( A ) denote the state of the point A (in the hyperspace) after k itera-tions. Let the hyperspace be endowed with a topology such that theeach of the induced non-autonomous system ( X , F ) is continuous. Itis intuitive to question the relation between the dynamical behav-ior of the non-autonomous system and its induced counterpart. Forexample, if a given non-autonomous system is transitive / weakly mix-ing / topologically mixing, what can be concluded about its inducedcounterpart (and vice-versa). If a given non-autonomous system ex-hibits sensitivity / strong sensitivity / Li-Yorke sensitivity, what can beconcluded about its induced counterpart (and vice-versa). What dy-namics does the induced system exhibit when the original system isequicontinuous? Such questions have been raised and answered incase of an autonomous system[1, 10, 11]. We prove that answers to
PUNEET SHARMA many of the question remain the same in a non-autonomous settingand hence most of the results obtained for the autonomous case canbe extended analogously to the non-autonomous case. We prove thatthe induced system is topologically mixing if and only if the originalsystem is topologically mixing. We prove that strong sensitivity isalso equivalent for the two systems under consideration. We provethat if F is commutative then a system exhibits weak mixing of allorders if and only if the induced system is weak mixing. We extendour studies to properties like dense periodicity, transitivity, equicon-tinuity, uniform equicontinuity, various notions of sensitivities andLi-Yorke chaos. We now establish the stated results. Proposition 1.
Let F ( X ) ⊆ Ψ and Ψ be endowed with any admissiblehyperspace topology. Then, ( X , F ) has dense set of periodic points ⇒ ( Ψ , F ) has dense set of periodic points.Proof. Let the hyperspace Ψ be endowed with an admissible topology ∆ . As every admissible topology on the hyperspace is of hit andmiss or hit and far miss type, let the topology ∆ be determinedby the collection C . Let U be a non empty basic open set in thehyperspace. Then U hits finitely many open sets, say V , V , . . . , V n and misses(far misses) finitely many elements of C say, C , C , . . . , C m .Let C = S C j . Thus each W i = V i T C c is non-empty, open in X .As periodic points are dense for ( X , F ), there exists x i ∈ W i and aninteger r i ∈ N such that for any i = , , . . . , n , ω r i k ( x i ) = x i ∀ k ∈ N . Let l = lcm { r , r , . . . , r n } . Then the set { x , x , ... x n } is periodic with period l . As { x , x , . . . , x n } ∈ U , the hyperspace has a dense set of periodicpoints. (cid:3) Remark . The above proof establishes that if the hyperspace containsall finite sets then the denseness of periodic points is extended to thehyperspace when the hyperspace is endowed with any admissiblehyperspace topology. The proof is analogous to the autonomous caseand is a natural extension of the result established in [10]. We nowgive an example to show that the converse of the above does not holdgood.
Example 1.
Let Σ be the space of all one-sided sequences of and and let φ : Σ → Σ be defined as f ( x = ( x x . . . )) = x + (100 . . . ) where addition isperformed with carry to the right. It is known that ( Σ , φ ) does not containany periodic point but the cylinder sets [ x x . . . x k ] are periodic and hencethe hyperspace ( K ( Σ ) , φ ) has dense set of periodic points [1] . Let I be theidentity operator on Σ and let the non-autonomous system ( X , F ) definedby the family F = { I , φ, I , φ, . . . , I , φ, . . . } . Then, ( X , F ) fails to contain any NDUCED DYNAMICS OF NON-AUTONOMOUS DISCRETE DYNAMICAL SYSTEMS7 periodic point but ( K ( Σ ) , F ) exhibits dense set of periodic points. Thus theconverse of the above result fails to hold for the non-autonomous system. Proposition 2.
If there exists a base β for the topology on X such that U + isnon empty and U + ∈ ∆ for every U ∈ β , then ( Ψ , F ) is transitive ⇒ ( X , F ) is transitive.Proof. Let U and V be any two non-empty open sets in X . As β forms a base for topology on X , there exists U , V ∈ β such that U ⊆ U and V ⊆ V . As U + and V + are non empty open sets in thehyperspace Ψ and ( Ψ , F ) is transitive, there exists A ∈ U + and k ∈ N such that ω k ( A ) ∈ V + or ω k ( A ) ∈ V + . Consequently for any a ∈ A , a ∈ A ⊂ U ⊂ U and ω k ( a ) ∈ V ⊂ V and hence ( X , F ) is transitive. (cid:3) Remark . The above proof establishes the transitivity of the originalfunction from the transitivity of the induced function. The result is ananalogous extension from the autonomous case and does not comeup as a surprise in the non-autonomous setting. It may be noted thatidentical arguments of the proof establish the n-transitivity of ( X , F )from n-transitivity of the induced system and hence total transitivityon the hyperspace implies total transitivity of ( X , F ). Thus we get thefollowing corollary. Corollary 1.
If there exists a base β for the topology on X such that U + isnon empty and U + ∈ ∆ for every U ∈ β , then ( Ψ , F ) is totally transitive ⇒ ( X , F ) is totally transitive. Proposition 3.
Let F ( X ) ⊆ Ψ . If F is weakly mixing of all orders then F is weak mixing. Further, if F is commutative and there exists a base β fortopology on X such that U + ∈ ∆ for every U ∈ β then F is weak mixing ⇒ F is weakly mixing of all orders.Proof. Let the topology ∆ on the hyperspace be determined by thecollection C and let U , U , V , V be non-empty open sets in thehyperspace. Let W , W , . . . W n ; W , W , . . . W r ; R , R , . . . R n ; R , R , . . . R r define the collection of hit sets and T , T , . . . T m ; T , T , . . . T s ; S , S , . . . S m ; S , S , . . . S s define the collection ofmiss sets for U , U , V , V respectively. Let T i = S j T ji , S i = S j S ji , P ij = W ji T T ci and Q ij = R ji T S ci . As U , U , V , V are non-empty,each of P ij , Q ij are non-empty open sets. Further as F is weakly mixingof all orders, there exists k ∈ N such that ω k ( P ij ) T Q ij , φ , ∀ i , j .Choose x ij ∈ P ij such that ω k ( x ij ) ∈ Q ij . Then A i = { x ij } j ∈ U i such that ω k ( A i ) ∈ V i . Hence F is weakly mixing. PUNEET SHARMA
Conversely, let U , U , . . . , U m and V , V , . . . , V m be non-emptyopen sets in X . As β is the base for the topology on X , ∃ U , U , . . . , U m and V , V , . . . , V mm ∈ β such that U ii ⊆ U i and V ii ⊆ V i for i = , , . . . , m . As f n ’s commute in the original system, f n commutes onthe hyperspace and hence by [12] the induced system is n -transitivefor any n ∈ N . Hence for open sets { U + ii : i = , , . . . , m } and { V + ii : i = , , . . . , m } , there exists k ∈ N such that ω k ( U + ii ) T V + ii , φ for i = , , . . . , m which implies ω k ( U ii ) ∩ V ii , φ and the proof iscomplete. (cid:3) Remark . The above proof establishes that for a commutative family F , weak mixing on the hyperspace is equivalent to weak mixing ofall orders in the original system. It may be noted that the forwardpart of the proof does not require commutativity of the family F andarguments similar to the converse establish weak mixing of order m in the original system from weak mixing of order m of the inducedsystem without using the commutativity of F . Thus the result moregenerally establishes that the original system is weak mixing of allorders if and only if the induced system is weak mixing of all or-ders. Further, as weak mixing of all orders is equivalent to weakmixing of second order for a commutative family F , the result es-tablishes equivalence of weak mixing for the two systems when thenon-autonomous system is induced by a commutative family. Hencewe get the following corollaries. Corollary 2.
Let F ( X ) ⊆ Ψ . If there exists a base β for topology on X suchthat U + ∈ ∆ for every U ∈ β , then ( X , F ) is weakly mixing of all orders ifand only if ( Ψ , F ) is weak mixing of all orders. Corollary 3.
Let F ( X ) ⊆ Ψ . If F is commutative and there exists a base β for topology on X such that U + ∈ ∆ for every U ∈ β , then ( X , F ) is weaklymixing if and only if ( Ψ , F ) is weak mixing.Remark . For the results derived so far, it may be noted that the proofsestablish analogous extensions of results known for the autonomoussystems. Moreover, it is observed that not only the statements butarguments similar to the autonomous case hold good in the non-autonomous case and thus give rise to the analogous extensions. Thishappens due to the fact that many of the proofs for the autonomouscase do not use the fact that the governing rule is constant withrespect to time and hence similar techniques or methodology canbe used to derive the results for the non-autonomous case. Forexample the proof establishing the equivalence of topological mixing
NDUCED DYNAMICS OF NON-AUTONOMOUS DISCRETE DYNAMICAL SYSTEMS9 for the two systems ( X , f ) and ( Ψ , f ) uses the fact that if every pair( U , V ) of non-empty open sets interacts at times n ≥ n U , V , then opensets U , U , . . . , U n and V , V , . . . , V m also interact for n ≥ K where K = max { n U i , V j : 1 ≤ i ≤ n , ≤ j ≤ m } and hence a similar techniqueyields equivalence of topological mixing for the two systems in thenon-autonomous case. Similarly, if for each non-empty open set U there exists n U ∈ N such that diam ω n ( U ) ≥ δ, ∀ n ≥ n U then for anyfinite collection of non-empty open sets U , U , . . . , U m there exists K = max { n U i : i = , , . . . , m } such that diam( ω n ( U i )) ≥ δ, ∀ n ≥ K and i = , , . . . , m and hence a proof similar to the autonomous caseyields equivalence of strong sensitivity for the two systems in thenon-autonomous case(when the hyperspace K ( X ) is equipped withthe Hausdor ff metric). Such arguments does not use the fact that thegoverning rule is constant with time and hence also hold good for thenon-autonomous systems. Similar observations can be made aboutthe proofs involving properties like sensitivity, equicontinuity and Li-Yorke chaoticity. We now state the results for the non-autonomoussystems whose autonomous version do not use constancy of f andhence are trivial extensions of their autonomous versions. Proposition 4.
Let F ( X ) ⊆ Ψ . If ( X , F ) is topologically mixing, then so is ( Ψ , F ) . The converse holds if there exists a base β for topology on X suchthat U + ∈ ∆ for every U ∈ β . Proposition 5. ( K ( X ) , F ) is sensitive ⇒ ( X , F ) is sensitive. Proposition 6. ( K ( X ) , F ) is strongly sensitive ⇔ ( X , F ) is strongly sensi-tive.Remark . The above results analogously relate the dynamical be-havior of a non-autonomous system with its induced counterpartestablishing the equivalence of topological mixing and strong sensi-tivity for the non-autonomous case. The results also establish thatif the induced system is sensitive then the original system is alsosensitive. However, for the converse it is known that there existsensitive autonomous systems ( X , f ) such that induced system isnot sensitive[11]. We now establish existence of a sensitive non-autonomous system ( X , F ) such that the induced system is not sensi-tive. Example 2.
Let ( X , f ) be a sensitive autonomous systems such that inducedsystem is not sensitive and let I be the identity operator on X. Let F = { f , I , f , I , . . . , f , I , . . . } . As ω k − ( A ) = f k ( A ) for any A ⊂ X, sensitivity of ( X , f ) implies sensitivity of ( X , F ) . Further as ( K ( X ) , f ) is not sensitive, ( K ( X ) , F ) is not sensitive and hence there exist sensitive non-autonomousdynamical systems such that their induced counterparts are not sensitive. We now give some more results which appear as trivial analogousextensions of their autonomous counterpart.
Proposition 7. ( K ( X ) , F ) is Li-Yorke sensitive = ⇒ ( X , F ) has chaoticdependence on initial conditions. Further, if ( F ( X ) , F ) is Li-Yorke sensitivethen ( X , F ) is Li-Yorke sensitive. Proposition 8.
Let X be a locally connected. Then, ( X , F ) is sensitive ⇒ ( F ( X ) , F ) is pointwise sensitive. Proposition 9.
Let F ( X ) ⊆ Ψ ⊆ K ( X ) . ( X , F ) is equicontinuous ⇒ ( Ψ , F ) is almost equicontinuous. Proposition 10.
Let F ( X ) ⊆ Ψ . Then, ( Ψ , F ) is equicontinuous ⇒ ( X , F ) is equicontinuous. Proposition 11.
Let F ( X ) ⊆ Ψ ⊆ K ( X ) . ( X , F ) is uniformly equicontin-uous if and only if ( Ψ , F ) is uniformly equicontinuous. Proposition 12. ( X , F ) is uniformly equicontinuous if and only if ( CL ( X ) , F ) is uniformly equicontinuous. Proposition 13.
Let ( X , f ) be a dynamical system and let ( K ( X ) , F ) bethe induced dynamical system on the hyperspace. If F ( X ) ⊆ Ψ , thesystem ( X , F ) has positive topological entropy implies ( Ψ , F ) has a positivetopological entropy. However, the converse is not true. Proposition 14.
Let F ( X ) ⊆ Ψ ⊆ CL ( X ) . Then, ( Ψ , F ) is δ -expansiveimplies ( X , F ) is δ -expansive. Proposition 15.
Let Ψ contain the set of all singletons. If ( X , F ) is Li-Yorke chaotic, so is ( Ψ , F ) . However, the converse is not true.
3. C onclusion
In this work, we have established relation between the dynamicalbehavior of a non-autonomous system and its induced counterpart.We have established that while properties like topologically mixing,strong sensitivity and uniform equicontinuity are equivalent for thetwo systems ( X , F ) and ( Ψ , F ), weak mixing is equivalent for thetwo systems when the family F is commutative. We establish thatif induced system is sensitive or transitive then the original systemexhibits similar dynamical behavior. It is observed that the results NDUCED DYNAMICS OF NON-AUTONOMOUS DISCRETE DYNAMICAL SYSTEMS11 obtained for the non-autonomous case are analogous extensions oftheir autonomous versions. It is further observed that many of theproofs for the autonomous version do not use the autonomous natureof the system and hence many of the results for the autonomous caseextend naturally to the non-autonomous case. Further, as commuta-tivity of the family F establishes equivalence of weak mixing for thetwo systems, it is expected that if the non-autonomous system is ”niceenough”, many of the results from the autonomous case can be ex-tended to the non-autonomous version. As a result one may considerproperties like distality, minimality, syndetic sensitivity and other dy-namical notions for further investigation. It may also be possible toimprove(under additional natural conditions) results for propertieslike Li-Yorke sensitivity, Li-Yorke chaoticity and topological entropy.However we do not consider these questions in this article and leavethem open for further investigation. It is emphasised that if the sys-tem is ”nice enough”, the dynamical behavior between the two sys-tems does not change drastically when compared to the autonomousversion. The result is an indicator of the qualitative stability of thelong term behavior as any autonomous(non-autonomous) systemcan be approximated using non-autonomous(autonomous) system.R eferences [1] Banks John,
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13, no. 10 (2000),40164027.D epartment of M athematics , I.I.T. J odhpur , O ld R esidency R oad , R atanada ,J odhpur -342011, INDIA E-mail address ::