Topics in Symbolic Dynamics

University Of Denver, Denver CO

Investigators

Abstract

In the current project, the PI will explore several questions and open problems in the area of symbolic dynamics. Dynamics is the study of closed systems whose elements evolve over time according to deterministic rules; some examples include planetary motion, or a particle's movement inside a box. Symbolic dynamics is the study of dynamics where the elements in question are infinite sequences of symbols, and the evolution over time comes from horizontally shifting the sequence. Symbolic dynamics was introduced as a tool for studying more complicated dynamical systems, but has become a fundamental area in its own right, with applications and connections ranging from computer science (e.g. data storage in binary) to statistical physics (e.g. the Ising model for magnetism). A current area of high activity is multidimensional symbolic dynamics, in which one considers not sequences, but multidimensional arrays of symbols, which can be shifted in any direction. Though the core definitions and ideas remain the same, there are remarkable changes that occur for multiple dimensions; many questions that were easy to answer in one dimension become quite difficult, and classes of systems whose behavior was relatively simple in one dimension can exhibit strange and complex behaviors. The research supported by this grant will treat a variety of problems in symbolic dynamics, ranging from the more classical one-dimensional case to the less well-understood multidimensional case. A common thread throughout is the use of ideas, techniques, and viewpoints from other areas, including probability theory, statistical physics, and percolation theory, to attack fundamental problems. The two projects on one-dimensional symbolic dynamics propose new directions of research about the classical notions of follower sets and the specification property respectively. The first multidimensional project proposes a new method for the efficient approximation of entropies of multidimensional shifts of finite type (SFT) which, unlike existing techniques, can be used on systems with multiple measures of maximal entropy. The final project will extend recent work by the PI and Kevin McGoff on random multidimensional SFTs, in which a probabilistic framework was defined where a "typically" chosen multidimensional SFT does not exhibit the worst pathological behaviors of the class; we now plan to use this framework to prove multidimensional probabilistic versions of some classical one-dimensional results. All work will be disseminated through publications in peer-reviewed journals, conference presentations, and the author's continued collaboration with young researchers, including his two Ph.D. students.