Date of Award


Document Type

Honors Thesis



First Advisor

David S. Richeson




When studying a dynamical system, it is common to partition the space (or a subset of the space) into a finite number of disjoint regions. Associated to each orbit is its itinerary, the sequence of regions it passes through. If the regions in the space overlap, a single orbit can have multiple itineraries. Hence, the itineraries are ambiguous. In order to study such systems, we need a bank of examples. We can represent the example via a directed graph (the transition graph from the dynamical system) and an undirected graph (the intersection graph from the intervals). We will discuss which pairs of transition and intersection graphs can be realized by continuous one-dimensional dynamical systems (on R and on S1). We also count the numbers of possible transition graphs in the case of disjoint intervals. Moreover, we can generate a realization in the form of a piecewise linear function for every such pair of intersection graph and transition graph. We will use techniques from graph coloring, combinatorics, algorithms, and dynamical systems theory.