| Summary: | In Chapter 1, we consider many topics that are both combinatorial and dynamical in nature. In particular, we study substitution maps, subword complexity, symbolic dynamics and interval-exchange maps. After describing the basic concepts and notation, we study the subword complexity functions that arise from substitutions connected with /3-transformations. We then make some general observations regarding subword complexity functions associated with substitutions, before going on to study some specific examples with quadratic growth in section 1.4. In section 1.5, we study the symbolic dynamics associated with these types of substitutions, generalising the notions of recurrence, minimality etc. In section 1.6, we briefly describe and compute an invariant measure for the substitutions considered in section 1.4. We then prove a result that describes a connection between the symbolic dynamics and interval-exchange maps, and apply it to these substitution maps. In Chapter 2, we study a dynamical skew-product and some of the combinatorial questions that it raises. In sections 2.1 and 2.2 we describe the skew-product, and explain the connection between it and some one-player games. We then describe and analyse a code-word problem, and explain how we can generalise our results. In sections 2.6 and 2.7, we study a continuous version of the problem and prove a result that might shed some light on the original skew-product. At the end of both chapters, we present some problems which we believe to be still open, and suggest ideas for further research into the topics presented in this thesis.
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