Mentor: Prof. Elizabeth Sattler (Lawerence University)

Title: Complexity of 2-Dimensional Symbolic Spaces

Abstract:

In this talk, we will introduce symbolic dynamics and topological entropy of higher-dimensional shift spaces from a combinatorial perspective. We will look to examples in the 1-dimensional case and their natural extensions to 2-dimensions as well as connections to tiling systems to illustrate the general difficulties of computing entropy in higher dimensions. In an attempt to attack this seemingly straight-forward, yet computationally intractable problem, we look at a proposed method of reducing the combinatorial problem in higher dimensions to the 1-dimensional case by using paths, or unary trees, and a complexity measure associated to those paths, which we call path entropy. We will end the talk by comparing the two complexity measures and introduce further questions that arise from this experimental dive into symbolic dynamics.