Doctor of Philosophy (PhD)



Document Type



Our main focus will be to investigate the various facets of what are commonly called dynamical systems or flows, which are triples $(S,X,\pi)$, where $X$ is a compact Hausdorff space and $\pi:S \times X \longrightarrow X$ is a separately continuous action of a semigroup $S$ on $X$. Historically, as was introduced by R.Ellis 1960, the enveloping semigroup, which is a closure of the set of continuous functions on a compact space $X$, was discovered to be an important tool to study dynamical systems. Soon, a realization of the existence of a universal compactification of a phase semigroup with an extended homomorphism onto the enveloping semigroup lead to an alternate approach to study these systems via this compactification. The importance of this alternative approach in this respect derives from the fact that the dynamical and many topological properties of $S$ can be translated into properties of its compactification. In Chapter one we will present a brief summary of notations and basic results from topological algebra as well as some basic information on the Stone-\v{C}ech Compactification, $\beta S$. In Chapter two we will expand on some of the work in chapter one and recall the necessary background from topological dynamics. We will define the enveloping semigroup and review some of the well known results concerning its structure. Fundamental and well known theorems which lead to the assertion as to the existence of a universal system will be presented. Utilizing this universal property we will further explain how all other dynamical systems arise as quotients of this universal system via suitable closed left congruences. In Chapter three, we concentrate on the special case where the phase semigroup is the set of natural numbers $N$ under addition and treat its compactification as the set of ultrafilters on $N$ and the extended action as that of $\beta N$. In so doing we will present results which relate notions of proximality and almost periodicity in a dynamical system to combinatorially rich central subsets of $N$. In lieu of an appendix we have also included in this chapter a rather deeper exposition of an example in symbolic dynamics arising from an action of $N$ on the product space $X=\prod\limits_{i=1}^{\infty}\{0,1\}$, which is isomorphic to a variety of other dynamic mappings, like the quadratic map on the cantor set which has significant applications in data storage and transmission, linear algebra and many other areas \cite{Dev89}.



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Committee Chair

Lisan, Amha