42012 mjis lal

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DOI: 10.15415/mjis.2016.42012

Orbit of a point in Dynamical Systems BABU LAL1, ASEEM MIGLANI2 AND VINOD KUMAR3 Department of Applied Sciences, JCDM College of Engineering, Barnala Road, Sirsa (formerly HOD Department of Applied Sciences, JCDM COE, Sirsa) 1

Registrar and Chairperson, Department of Mathematics, CDLU, Sirsa

2

Visiting Professor, centre for advanced study in Mathematics, P.U. Chandigarh (formerly HOD Mathematics, K.U. Kurukshetra) 3

E-mail: jangirbabu29@rediffmail.com Received: October 27, 2014| Revised: September 27, 2015| Accepted: February 11, 2016 Published online: March 30, 2016 The Author(s) 2016. This article is published with open access at www.chitkara.edu.in/publications

Abstract  In this paper, we have proved the necessary and sufficient condition for a weakly mixing and topologically mixing function. Some properties of the monoid, periodic points and eventually periodic points are obtained. Some relations between weakly mixing, transitive and topologically mixing functions are obtained. Some results of considerable importance about the orbit of a point and relation with eventually periodic point are proved. Some results of the set theory that play an important role in our studies are included. Some new terms like singly transitive and lately transitive are introduced.

AMS Subject classification 37A20, 37C25, 54H20 Keywords: Orbit; Eventually periodic point; Monoid; Dynamical system; Transitive; Weakly mixing. 1. INTRODUCTION There are situations or sometimes there arises a situation where the things under consideration are not clear or are vague. Sometimes we come across a ‘setup` where we are unable to say whether we are precise or not; at times we can clearly see that we are not precise. Similarly, sometimes the outcome of our considerations may not be certain. In short we can say that there is an uncertainty or chaos ([1]) in the situation. So there arises a question: Can we handle these terms mathematically? Ideally speaking there should be no place for any one of these terms in the

Mathematical Journal of Interdisciplinary Sciences Vol. 4, No. 2, March 2016 pp. 141–149


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