POISSON STABILITY IN PRODUCT OF DYNAMICAL SYSTEMS

This paper deals with Poisson stable and distal dynamical system. It is shown that the product of Poisson stable motions is a Poisson stable motion in dynamical systems.

systems [I] is a semidynamical system.In [I] Bajaj has shown that if fIR s e I be the product of positive limit sets of the factor systems and R be the positive x limit set of the product system then R IIR In general, cannot be replaced X X by equality.Indeed may be empty even if each of E I is nonempty.He x x 0 has found the conditions under which I Xl x2 2 x and x could be nonempty and compact for a finite product.
We studied Poisson stable and distal dynamical systems in [2].In this paper we shall mention the conditions under which IIR e I and the product of x x Poisson stable motions is a Poisson stable motion in dynamical systems.

DEFINITIONS AND NOTATIONS.
We shall use the definition and notations of [2] and [3].PROOF.Let the motion (x,t) be Poisson stable and distal then its trajectory y(x) is closed hence y(x) cl y(x) x As the motion is compact each of the above sets is compact and minimal, and thus by Birkhoff recurrence theorem w(x,t) is compact recurrent.
THEOREM 3.3.Let (X,) be a semldynamlcal system, let be Lagrange stable, then is distal iff for every net t i in R +, the phase space X {zeX x3 z for some xeX and some subnet tj of tl}. [4,theorem 2.6].
A. KUMAR AND R.P. BHAGAT One can show that positive distal dynamical system are distal whenever all positive trajectory closures are compact.Thus the above result for semidynamical systems is applicable in present dynamical systems as well.
THEOREM 3.4.A compact motion (x,t) is Poisson stable and distal iff y(x) cly(x) is compact and minimal.x PROOF.If a compact motion (x,t) is Poisson stable and distal then y(x) cly(x) is compact and minimal.x Conversely, let y(x) cly(x) be compact and minimal then it is compact x recurrent and therefore compact Poisson stable.
To show the dynamical system (X,) is distal we shall use theorem 3.3.Here y(x) is compact and minimal, for every x in X the net xt i in (x) has a subset xt.z in y(x).3 Thus X {z e X xt.z for some net t i in T and t. is some subnet of t i in T}.
Which completes the proof.
THEOREM 3.5.The trajectory y(x) of a compact Poisson stable distal motion is complete and totally bounded.
PROOF.Since y(x) is compact in a Hausdorff uniform space X.Now we shall consider product of such dynamical systems.Let (Xa,), ae I be a family of dynamical systems.Let X IIX a be the product space.Let xX, x {x}.Define a map from X x T into X by (x,t) {xat}, ae I then (X,) is a dynamical system.
The dynamical system (X,) obtained above is called direct product or product of the family (Xa,a), a I. PROPOSITION 3.6.Let (Xa,a) a e I be a family of dynamical systems and (X,) the product dynamical system.Let xeX, x {xa}.Then II x x THEOREM 3.7.Let (Xa,), I be a family of {Lagrange stable} {distal} dynamical systems and (X,) is the product dynamical system.Let x E X and x {x then (X,) is {Lagrange stable} {distal}.THEOREM 3.8.Let (Xa,), a E I be Lagrange stable distal dynamical systems.
Let x X, x {x and (X,) their product dynamical system, then the product (x,t) is Poisson stable motion in the product space (X,) iff each (xa,t), a I is Poisson stable motion.
If a compact motion is Poisson stable and distal then it is a compact recurrent motion.