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Title:	Orbits, entropy and periodic points of shift dynamical systems
Authors:	Alsharari Fahad Mohammed (P60773)
Supervisor:	Mohd Salmi Md Noorani, Prof. Dr.
Keywords:	Universiti Kebangsaan Malaysia -- Dissertations Orbit theorem Topological entropy Zeta function Dissertations, Academic -- Malaysia
Issue Date:	19-Jan-2016
Description:	In the present work in this thesis we count the closed orbits for shift dynamical systems of infinite type called the Dyck and Motzkin shifts. We also define fundamental and essential subshifts of the Dyck and Motzkin shifts and deduce significant results. Additionally we compute the topological entropy and investigate the number of doubly periodic points of their two dimensional versions. In the setting of counting orbits, the Prime Orbit Theorem, Mertens' Theorem, Meissel's theorem for orbits and the dynamical Dirichlet series are proved for the Dyck and Motzkin shifts using some direct methods instead of using the zeta function approach. The results obtained for these shifts are given in such notations derived from asymptotic analysis. By comparing some orbit counting problems solved for the Dyck and Motzkin shifts together with other previous dynamical maps that have used the same method, it could be seen that the orbit counting findings for the Dyck and Motzkin shifts are mainly controlled by lemmas concerning the number of periodic points of the Dyck shift and Motzkin shift, respectively. In the forth chapter, crucial subshifts of the Dyck and Motzkin shift were defined and their entropies computed. The entropy of these subshifts is computed using the two-dimensional patterns. In the setting of higher dimensional shifts, the two dimensional Dyck shift, the vertically independent Dyck shift and the horizontally independent Dyck shift are defined using the forbidden sets approach. A defined mapping that sends every one-row and one-column of the occurring two-dimensional patterns into a one-dimensional monoid is used to give such formulas of these definitions. The two dimensional Motzkin shift, the vertically independent Motzkin shift and the horizontally independent Motzkin shift are then also defined using another approach of defining notions in higher dimensional shifts by the mean of the languages of the shifts. The relationship between these subshifts are given with illustrative examples. The main difficulties raise up in higher dimensional shifts (even for shifts of finite type) do not only lie in generalizing notions from one dimensional setting to higher dimensions, but also to obtain the entropy or periodic points for these generalized shifts. Therefore significant methods for computing the exact entropy for these two-dimensional shifts are obtained. Moreover, non trivial bounds for the number of doubly periodic points for the two dimensional versions of the Dyck and Motzkin shifts are precisely given. In addition, an alternative proof of the topological entropy for the one dimensional Motzkin shift is given by using a new method by counting possible paths without any measure theoretical discussion.,Ph.D.
Pages:	133
Publisher:	UKM, Bangi
Appears in Collections:	Faculty of Science and Technology / Fakulti Sains dan Teknologi

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