The notion of a normal number and the Normal Number Theorem date back over 100 years. Émile Borel first stated his Normal Number Theorem in 1909. Despite their seemingly basic nature, normal numbers are still engaging many mathematicians to this day. In this paper, we provide a reinterpretation of the concept of a normal number. This leads to a new proof of Borel's classic Normal Number Theorem, and also a construction of a set that contains all absolutely normal numbers. We are also able to use the reinterpretation to apply the same definition for a normal number to any point in a symbolic dynamical system. We then provide a proof that the Fibonacci system has all of its points being normal, with respect to our new definition. / Graduation date: 2012

In recent years, combinatorial properties of finite and infinite words have become increasingly important in fields of physics, biology, mathematics, and computer science. In particular, the fascinating family of Sturmian words has become an extremely active subject of research. These infinite binary sequences have numerous applications in various fields of mathematics, such as symbolic dynamics, the study of continued fraction expansion, and also in some domains of physics ( quasicrystal modelling ) and computer science ( pattern recognition, digital straightness ). There has also been a recent surge of interest in a natural generalization of Sturmian words to more than two letters - the so - called episturmian words, which include the well - known Arnoux - Rauzy sequences. This thesis represents a significant contribution to the study of Sturmian and episturmian words, and related objects such as generalized Thue - Morse words and substitutions on a finite alphabet. Specifically, we prove some new properties of certain palindromic factors of the infinite Fibonacci word; establish generalized ' singular ' decompositions of suffixes of certain morphic Sturmian words; completely describe where palindromes occur in characteristic Sturmian words; explicitly determine all integer powers occurring in a certain class of k-strict episturmian words ( including the k-bonacci word ) ; and prove that certain episturmian and generalized Thue - Morse continued fractions are transcendental. Lastly, we begin working towards a proof of a characterization of invertible substitutions on a finite alphabet, which generalizes the fact that invertible substitutions on two letters are exactly the Sturmian morphisms. / Thesis (Ph.D.)--School of Mathematical Sciences, 2006.