Dynamics of multidimensional symbolic systems

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project aims to further our understanding of higher-dimensional shifts of finite type (SFTs) and cellular automata (CA). Broadly speaking, the goal is to describe their dynamics and identify types of dynamics that cannot occur in them. In recent work the principal investigator and others have applied methods from recursion theory and topological dynamics to describe the complexity of some classical dynamical invariants associated to SFTs and CA, such as their entropy and directional dynamics (subdynamics). This project represents a continuation of that investigation. The principal investigator hopes to extend the classification of subdynamics to co-rank 1 actions (currently only co-rank 2 and higher are known) and begin a systematic study of the full dynamics, which are related to more quantitative measures of complexity drawn from computer science rather than the qualitative ones of recursion theory. He will also study the relation between SFTs, CA, and other dynamical systems, specifically addressing which actions on the Cantor set can be approximated by certain types of SFTs.The project focuses on the class of dynamical systems arising from infinite collections of discrete elements in which only "nearby" elements interact. Such models are widely used as theoretical models and in simulation in many areas of science and engineering, such as thermodynamics, fluid dynamics, the theory of quasi-crystals, information theory, and computer science, to name just a few. In that context, it is a fundamental problem to determine which dynamics such systems can represent and what their theoretical limitations are and to identify subclasses that are more amenable to study and use. Precise results in this field have come to light recently, and this project aims to continue research in this direction.