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.

DOI: 10.4236/apm.2012.26067                

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.

Keywords

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

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	I. K. Dontwi, “One Class of Almost-Periodic Distributions and Some Properties,” Proceedings of Special SAMSA International Conference on Computers and Mathematics, University of Zimbabwe, Harrare, 1994, pp. 56-65.
[2]	K. S. Sibiriskii, “Introduction to Topological Dynamics,” Redak-Izdat. Otdel. Akad. Nauk MSSR, Kishinev, 1970.
[3]	B. N. Levitan and V. V. Zhikov, “Almost-Periodic Functions and Differential Equations,” Cambridge University Press, Cambridge, 1982.
[4]	B. A. Shcherbakov, “Topological Dynamics and Poisson Stability of Solutions of Differential Equations,” Shtiintsa. Kishinev, 1972.
[5]	B. A. Shcherbakov, “On Comparison by Character the Recurrence of the Motion of Dynamic Systems,” Journal of Differential Equations, Vol. 2, No. 7, 1975, pp. 1246-1255.
[6]	D. N. Cheban, “On Comparison of Points of Dynamic Systems by the Character of Their Returnability in the Limit,” Journal of Research in Algebra, Mathematical Analysis and Their Applications, Kishinev, Shtiintsa, 1997, pp. 66-71.
[7]	D. N. Cheban, “Non-Autonomous Dissipative Dynamic Systems,” Soviet Mathematics: Doklady, Vol. 33, No. 1, 1986, pp. 207-210.
[8]	I. U. Bronshtein, “Extension of Minimal Transformation Groups,” International Publications, Netherlands, 1975
[9]	V. A. Pliss, “Nonlocal Problems in the Theory of Oscillations,” Academic Press, New York, 1966.
[10]	R. J. Sacker and G. R. Sell, “Dichotomies for Linear Evolutionary Equations in Banach Spaces,” Journal of Differential Equations, Vol. 113, No. 1, 1994, pp. 17-67. doi:10.1006/jdeq.1994.1113
[11]	G. R. Sell, “Topological Dynamics and Differential Equations,” Van Nostrand Reinhold Company, London, 1971.
[12]	J. Hale, “Theory of Functional Differential Equations,” 2nd Edition, Springer-Verlag, New York, 1977. doi:10.1007/978-1-4612-9892-2
[13]	D. N. Cheban, “Relations between the Different Types of Stability of the Linear Almost Periodical Systems in the Banach Space,” Electronic Journal of Differential Equations, Vol. 1999, No. 46, 1999, pp. 1-9. http://ejde.math.swt.edu or http://ejde.math.unt.edu
[14]	D. N. Cheban, “Uniform Exponential Stability of Linear Periodic Systems in a Banach Space,” Electronic Journal of Differential Equations, Vol. 2001, No. 3, 2001 pp. 1-12. http://ejde.math.swt.edu or http://ejde.math.unt.edu
[15]	I. K. Dontwi, “Bounded on R+ Generalized Functions and Some of their Properties,” Journal of Natural Sciences, Vol. 1, No. 2, 2001, pp. 83-92.
[16]	I. Assani, “Spectral Characterisation of Wiener Wintner Dynamical Systems,” Ergordic Theory and Dynamical Systems, Vol. 24, No. 2, 2004, pp. 347-366.
[17]	I. Assani, “Pointwise Convergence of Ergodic Averages along Cubes,” Journal d’Analyse Mathématique, Vol. 110, No. 1, 2010, pp. 241-269. doi:10.1007/s11854-010-0006-3
[18]	I. Assani, Z. Buczolich and D. Mauldin, “An L1 Counting Problem in Ergodic Theory,” Journal d’Analyse Mathematique, Vol. XCV, 2005, pp. 221-241. doi:10.1007/BF02791503
[19]	I. Assani, “Duality and the One-Sided Ergodic Hilbert Transform,” Contemporary Mathematics, 357, Chapel Hill Ergodic Workshops, 2004, pp. 81-90.
[20]	I. Assani, “Pointwise Convergence of Nonconventional Averages,” Colloquium Mathematicum, Vol. 102, No. 2, 2005, pp. 245-262. doi:10.4064/cm102-2-6
[21]	I. Assani, “Averages along Cubes for Not Necessarily Commuting Measure Preserving Transformations,” Contemporary Mathematics, Vol. 430, 2007, pp. 1-19.
[22]	I. Assani and M. Lin, “On the One Sided Ergodic Hilbert Transform,” Contemporary Mathematics, Vol. 430, 2007, pp. 20-39. doi:10.1090/conm/430/08249
[23]	I. Assani and K. Presser, “Pointwise Characteristics Factors for the Multiterm Return Times Theorem,” Ergordic Theory and Dynamical Systems, 2011.
[24]	J. de Vries, “Elements of Topological Dynamics,” Kluwer Academic Publishers, Dordrecht, 1993.
[25]	T. R. Peyam, “Shifts as Dynamical Systems,” 2010. http://math.berkeley.edu./~peyam/shifts.pdf
[26]	W. F. Trench, “Introduction to Real Analysis,” Trinity University, San Antonio, 2003
[27]	K. H. Hofmann, “Introduction to General Topology: An Introductory Course,” 2005.
[28]	W. Obeng-Denteh, “On the Study of Topological Dynamical Systems,” Ph.D. Thesis, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana, 2012.