Mathematics Senior Capstone: Conjugacy and S-graph Shifts
For my mathematics senior capstone, I studied conjugacy in symbolic dynamics, focusing on a class of shift spaces called \(\mathcal{S}\)-graph shifts. Conjugacy is a notion of equivalence between shifts that preserves their dynamical behavior, but it is difficult to establish, and a concrete result on conjugacy for \(\mathcal{S}\)-graph shifts remains an open problem.
In the paper, I work through a motivating example of two conjugate \(\mathcal{S}\)-graph shifts, examine a proposed (but unproven) tool called the core graph, and explore new ideas about disjoint cycle sets and Dillon’s entropy formula that suggest directions for future investigation.