﻿Dynamic Systems of Shifts in the Space of Piece-Wise Continuous Functions

Vol.2 No.6(2012), Article ID:24984,4 pages DOI:10.4236/apm.2012.26067

Dynamic Systems of Shifts in the Space of Piece-Wise Continuous Functions

Isaac Kwame Dontwi, William Obeng-Denteh

Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana

Email: ikedontwi@hotmail.com, obengdentehw@yahoo.com

Received July 24, 2012; revised September 17, 2012; accepted September 24, 2012

Keywords: Dynamic System; Shifts; Piecewise Continuous Functions; Compact Sets; Poisson Discontinuous Function

ABSTRACT

In this paper we embark on the study of Dynamic Systems of Shifts in the space of piece-wise continuous functions analogue to the known Bebutov system. 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 discontinuous 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.

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.

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