Burton has contributions to ergodic theory and dynamical systems and has worked in various parts of pure and applied mathematics. These include probability theory, percolation and interacting partical systems, neural networks and data mining, number theory and its interactions with dynamics such continued fractions, point processes and random measures, algorithms and coding theory, substitution process, symbolic systems in one and higher dimensions. His current projects include work with the Center for Gene Research and Biotechnology, symbolic and physical models, and coding and search algorithms.