The past years have seen the rise of various fields within mathematics. These are considered 'sexy' and have aesthetic components. These include the use of of Fractals, Chaos Theory, Complexity Theory, Formal Theories of Linguistics, and Symbolic Dynamical Systems. All of these have been birthed and raised within the Theory of Dynamical Systems.
This talk is expository, elementary, and designed to show how one generates dynamical systems from 'scratch.' A Dynamical System is a set X together with some mathematical structure such as a topology or a probability measure or some algebraic structure. Then we have a transformation T:X → X that preserves the mathematical structure on X. Here T represents the flow of time so that T('today') = 'tomorrow.' The question that we want to address is how do we take finite descriptions and turn them into infinite orbits within Dynamical Systems in an understandable way?