Lind/Tuncel Lind will investigate various aspects of several dimensional actions, including zeta functions and directional entropies. Emphasis will be placed on the determination of the expansive sets, how the dynamics of one component of the expansive set may influence the dynamics in another, and how finite type assumptions may change the answers. He will further probe the coincidence entropies of certain actions with the growth rate for spanning trees of related infinite graphs. He will also pursue a link with electrical networks suggested by his discovery of a new invariant for Markov shifts arising from consideration of spanning trees. Tuncel will work on two-dimensional shifts of finite type. He has observed that certain matrix equations determine the entropy whenever they have (positive) solutions, and that similar inequalities give bounds on entropy. He will pursue the characterization of conjugacy classes for which the equations have solutions, the possibility of improving the bounds by state-splitting, and the use of the inequalities in coding algorithms related to holographic data storage. He will also continue his analysis of Markov chains and their classifications, with emphasis on the development of dynamical invariants and coding methods for handling near-boundary constraints. Traditional problems of encoding data have focused on one-dimensional sequences, such as those found on computer disk drives or (in spiral form) on compact audio discs. In such situations physical constraints mean that, for efficient encoding, arbitrary data must be encoded into data subject to constraints. For instance data on a compact audio disk must have the property that between consecutive 1's there are at least two, but no more that seven, 0's. Symbolic dynamics provides that mathematical framework for discovering and investigating methods of encoding that have practical implications. More recently both industrial and research mathematicians have been studying higher-dimen sional versions of these problems. Industry is interested, for example, in holographic data storage where information is spread over a three-dimensional cube. Research mathematicians have been discovering fascinating new phenomena in this area, in particular some rich connections with the seemingly unrelated fields of commutative algebra and algebraic geometry. The investigators will continue their work with the higher-dimensional theory, with emphasis on ways of measuring information and a study of periodic structures.
