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

Affiliation(s)

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

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

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.

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,

KEYWORDS

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

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

Cite this paper

I. Dontwi and W. Obeng-Denteh, "Dynamic Systems of Shifts in the Space of Piece-Wise Continuous Functions,"*Advances in Pure Mathematics*, Vol. 2 No. 6, 2012, pp. 446-449. doi: 10.4236/apm.2012.26067.

I. Dontwi and W. Obeng-Denteh, "Dynamic Systems of Shifts in the Space of Piece-Wise Continuous Functions,"

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.

[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.