On Nonautonomous Discrete Dynamical Systems

We define and study expansiveness, shadowing, and topological stability for a nonautonomous discrete dynamical system induced by a sequence of homeomorphisms on a metric space.

In [16], Liu and Chen studied -limit sets and attraction of nonautonomous discrete dynamical systems.In [8,17] authors studied weak mixing and chaos in nonautonomous discrete systems.In [18] Yokoi studied recurrence properties of a class of nonautonomous discrete systems.Recently in [19] we defined and studied expansiveness, shadowing, and topological stability in nonautonomous discrete dynamical systems given by a sequence of continuous maps on a metric space.
In this paper we define and study expansiveness, shadowing, and topological stability in nonautonomous discrete dynamical systems given by a sequence of homeomorphisms on a metric space.In the next section, we define and study expansiveness of a time varying homeomorphism on a metric space.In section following to the next section, we define and study shadowing or P.O.T.P. for a time varying homeomorphism on a metric space.In the final section, we study topological stability of a time varying homeomorphism on a compact metric space.
Remark 4. Note that expansiveness of a time varying homeomorphism  is independent of the choice of metric for  if  is compact.Definition 5. Let (,  1 ) and (,  2 ) be two metric spaces.Let  = {  } ∞ =0 and  = {  } ∞ =0 be time varying homeomorphisms on  and , respectively.If there exists a homeomorphism ℎ :  →  such that ℎ ∘   =   ∘ ℎ for all  = 0, 1, 2, . . .then  and  are said to be conjugate with respect to the map ℎ or ℎ-conjugate.In particular, if ℎ :  →  is a uniform homeomorphism, then  and  are said to be uniformly conjugate or uniformly ℎ-conjugate.(Recall that homeomorphism ℎ :  → , such that ℎ and ℎ −1 are uniformly continuous, is called a uniform homeomorphism.) Theorem 6.Let (,  1 ) and (,  2 ) be metric spaces.Let  = {  } ∞ =0 and  = {  } ∞ =0 be time varying homeomorphisms on  and , respectively, such that  is uniformly conjugate to .Then  is expansive on  if and only if  is expansive on .
Proof.Since  is uniformly conjugate to , therefore there exists a uniform homeomorphism ℎ :  →  sach that ℎ ∘   =   ∘ ℎ, for all  ≥ 0, which implies that   ∘ ℎ −1 = ℎ −1 ∘   , for all  ≥ 0, and and similarly, for all  ≤ 0, we also have So we get Suppose  is expansive on  with expansive constant  > 0. Since ℎ −1 is uniformly continuous, therefore there exists a  > 0 such that, for any and since  is expansive on ; there exists  ∈ Z such that which implies that  2 (  ( 1 ),   ( 2 )) ≥ .Hence  is expansive on .
Using the above Theorem, analogous to Theorem 2.2 in [19], we have the following result.Theorem 9. Let (, ) be a compact metric space, {  } ∞ =0 an equicontinuous family of self-maps on , and  an integer.Then time varying homeomorphism  = {  } ∞ =0 is expansive if and only if   is expansive for any  ∈ Z − {0}.
We have the following result from the definition of invariance.
Theorem 11.Let (, ) be a metric space,  = {  } ∞ =0 a time varying homeomorphism which is expansive on , and  an invariant subset of ; then restriction of  to , defined by  |  = {  | }, is expansive.Similar to Theorem 2.4 in [19], we have the following result.
Theorem 12. Let (,  1 ) and (,  2 ) be metric spaces and  = {  } ∞ =0 ,  = {  } ∞ =0 time varying homeomorphisms on  and , respectively.Consider the metric  on  ×  defined by Then the time varying homeomorphism is expansive on  ×  if and only if  and  are expansive on  and , respectively.Hence every finite direct product of expansive time varying homeomorphisms is expansive.
We have the following result for time varying homeomorphism similar to that for expansive homeomorphism on compact metric space [21].
Theorem 13.Let (, ) be a compact metric space and  = {  } ∞ =0 a time varying homeomorphism which is expansive on .If  is the least upper bound of the expansive constants for , then  is not an expansive constant for .
The topological analogue of generator was defined and studied by Keynes and Robertson [22].We define and study this notion for invertible nonautonomous discrete dynamical system.(1)  is expansive.
Proof.(2) ⇒ (3) follows by definitions of generator and weak generator.We prove that (3) ⇒ (2).Let  = { 1 ,  2 , . . .,   } be a weak generator for  and  > 0 a Lebesgue number for .Let  be a finite open cover by sets   with diam(  ) ≤ .If {   } is a bisequence of members of , then for every  there is   such that    ⊂    , and so ) also contains atmost one point and hence  is a generator.
Now note that Thus  cannot be a weak generator for .Therefore  has no weak generator and hence by the above result  is a nonexpansive time varying homeomorphism.
The time varying homeomorphism  is said to have shadowing property or pseudo orbit tracing property (P.O.T.P.) if, for every  > 0, there exists a  > 0 such that every pseudo orbit is -traced by some point of .Proof.Given any  > 0, applying the uniform continuity of ℎ implies that there exists 0 <  1 <  such that, for any  1 ,  2 ∈  with  1 ( 1 ,  2 ) <  1 ,  2 (ℎ( 1 ), ℎ( 2 )) < .As  has P.O.T.P., there exists 0 <  1 <  1 such that every  1 -pseudo orbit of  is  1 -traced by some point of .Noting the fact that ℎ −1 is uniformly continuous, there exists 0 <  <  1 such that, for any Now, we assert that every -pseudo orbit of  is -traced by some point of .

Definition 14 .
Let (, ) be a compact metric space and  = {  } ∞ =0 a time varying homeomorphism on .A finite open cover  of  is said to be a generator for  if, for every bisequence {  } of members of , ⋂ ∞ =−∞ (  ) −1 (  ) is at most one point, where   denotes the closure of   .Definition 15.Let (, ) be a compact metric space and  = {  } ∞ =0 a time varying homeomorphism on .A finite open cover  of  is said to be a weak generator for  if for every bisequence {  } of members of , ⋂ ∞ =−∞ (  ) −1 (  ) is at most one point.Theorem 16.Let (, ) be a compact metric space and  = {  } ∞ =0 a time varying homeomorphism on .Then the following are equivalent.

Theorem 23 .
Let  = {  } ∞ =0 ,  = {  } ∞ =0 betime varying homeomorphisms.Then  and  have shadowing property if and only if the time varying homeomorphism  ×  = {  ×   } ∞ =0 has shadowing property on  × .Hence every finite direct product of time varying homeomorphisms having shadowing property, has shadowing property.Theorem 24.Let  = {  } ∞ =0 be the time varying homeomorphism on a metric space (, ).Then  has P.O.T.P. if and only if  −1 has P.O.T.P. Proof.The proof follows observing that {  } is a -pseudo orbit of  if and only if {  =  − } is a -pseudo orbit of  −1 .

Theorem 8 .
Let (, ) be a compact metric space and {  } ∞ =0 a family of self-homeomorphisms on .Then time varying map  = {  } ∞ =0 is expansive if and only if  −1 is expansive.