Dynamic Systems of Shifts in the Space of Piece-Wise Continuous Functions ()
1. Introduction
The interest in the study of Differential Equations with Impulse is increasing. Attempt to extend this study Dontwi [1] to known topological methods of the Theory of Dynamic Systems (DS) (see Sibiriskii [2], Levitan and Zhikov [3], Shcherbakov [4,5], Cheban [6,7]) brings into fore the necessity of studying DS of shifts in the space of piece-wise-continuous functions which are solutions of these equations.
In this paper we extend the study of Dynamical Systems of shifts in the space of piece-wise-continuous functions analogue to Bebutov Systems. We give a formal definition of a topological dynamic system in the space of piece-wise continuous functions and show, by way of an example, stability in the sense of poisson discontinueous function. We prove that a fixed discontinuous function, f, is discontinuous for all its shifts, 
whereas the trajectory of discontinuous function is not a compact set. These should prepare the way for the introduction and application of notions of Recurrence motions of dynamic systems (Bronshtein [8], Pliss [9], Sacker and Sell [10], Sell [11]) to various trajectories of Differential Equations with Impulse (Distributions) (Hale [12], Cheban [13,14], and Dontwi [15]).
Ergodic dynamical system on the finite measure space and its kronecker factor were considered in Assani [16]. Pointwise convergence of ergodic averages along cubes was proved in Assani [17]. In Assani, Buczolich and Mauldin [18], negative solution to counting problem for measure preserving transformation was carried out. Full measures were treated in Assani [19]. A question of H. Furstenburg on the pointwise convergence of the averages was answered in Assani [20]. The pointwise convergence of some weighted averages linked to averages along cubes was studied in Assani [21]. Two questions related to the strongly continuous semigroup were answered in Assani and Lin [22]. Characteristics for certain nonconventional averages were studied in Assani and Presser [23]. Differentiable or smooth instead of topological gives a description of Differentiable Dynamics by Vries in [24].
Concepts such as metric spaces, normed spaces, convergence and homeomorphisms, compactness, and the Heine-Borel Theorem are considered to be known. In terms of discussing shifts this stems from several important applications of symbolic dynamics in the field of dynamical systems. It goes without telling that symbolic dynamics is a strong and formidable tool used in the study of dynamical system in Peyam [25]. The advantages that are gleaned from it are that the technique reduces a complicated system into a set of sequences. Mention should be made in the following passing: invariants, the Zeta function, Markov partition, and Homoclinic orbits.
2. Notions and Preliminaries
Let R and N be the set of real numbers and the set of natural numbers respectively, 
be the left and right sided limits of the function 
at the point 
We consider PC[R]—the space of piece-wise-continuous real-valued functions defined on the number line R with the following properties:
1) The set of points of discontinuity of every function 
PC[R], represented as 
is either empty or has points of discontinuity of the first kind;
2) The point of discontinuity of every function, if it is more than one, is distinct from each other at a distance not less than some fixed positive number for a given function.
The jump or discontinuity of the function 
at the point 
is the number
In PC[R], (or simply PC), we consider countable partitions of family of semi-norms
defined for every function 
PC and induces metrizable topology in this space. Further we shall represent this metrizable space by PC.
Definition 1.
An alphabet A is a set of symbols. A common example is 
and in general
Definition 2.
Given an alphabet A, the full shift space is 
(i.e. the space of sequence is from Z into A)
Definition 3.
A homeomorphism, 
from 
to 
is a continuous function 
such that 
Definition 4.
Two dynamic systems 
and 
are topological conjugates if there exist a homeomorphism 
between them that is also a homeomorphism. This confirms that the subject of dynamical systems studies how a given system behaves throughout time which can be discrete or continuous iterates.
Definition 5.
An infinite subset T of A is compact if and only if every infinite subset of T has a limit point in T [26].
Definition 6.
A function 
between topological spaces is called continuous, if 
is open in X for every open 
In Hoffman [27] the set of all continuous functions 
is often denoted by
Remark 1.
The sequence of functions 
from PC is convergent if in PC there exist a function 
such that 
converges uniformly to 
in every interval 
where 
We write this in the form
The following hold:
Lemma 1.
If the function 
at the point 
is a jump of magnitude 
while the function 
is continuous at this point, then for every 
and 
the following is true:
Lemma 2.
Let 
Then 1) If the function 
is discontinuous at the point 
then all functions 
(except, maybe, for a finite number of points) are also discontinuous at the same point. As a consequence we have the following:
2) If beginning from some number, all functions 
are continuous at the point 
then the function 
is also continuous at this point.
The reverse of the above statements hold.
Example 1.
Let 
and
then 
Remark 2.
The space PC is not complete.
For any 
and 
we represent by the symbol 
the shifts of the function 
by 
that is 
Following Bebutov dynamic systems in the space PC we consider the family of shifts (or translates)
defined by the formula
for all 
Definitions 1 to 5 serve as clues to the concepts discussed.
3. Main Results
Theorem 1.
The mapping 
defined above satisfies the following conditions:
1) 
for any 
2) 
for any 
and 
3) 
is continuous in f for any fixed 
and for a fixed f-continuous function, the mapping 
is continuous in 
however, if for a fixed f it is a discontinuous function then 
is discontinuous at all points of 
Proof.
1) and 2) are obvious.
Continuity of 
in f for a fixed 
by Remark 1 implies uniform convergence of the function 
as 
in every interval
, 
, which in turn implies uniform convergence of the function 
as 
in every interval 
If 
is a continuous function, then 
is continuous in 
by the known property of Bebutov Dynamic System (Sell [11]) and is uniformly continuous in Obeng-Denteh [28].
The motion corresponding to the continuous function f is continuous, and if it is discontinuous it will be discontinuous at every point.
Theorem 2.
For any arbitrary discontinuous function f from PC, its trajectory is not a compact set.
Proof.
Let f be any discontinuous function in PC. Consider
where 
The given sequence converges point-wise to the function f of points of continuity of f. However, no sub-sequence of the given sequence converges to f in PC, and this means, in general it does not converge in PC.
4. Conclusion
The topological dynamical system in the space of piecewise continuous functions has been shown by way of an example, as well as stability in the sense of Poisson discontinuous function. It has also been proved that a fixed discontinuous function, f, is discontinuous for all its shifts, 
, whereas the trajectory of the discontinuous function is not a compact set.