This research will be conducted in several areas of probability and ergodic theory: 1) Construction and fundamental properties of Gibbs states on sequence spaces with countable alphabets; 2) applications to dynamical systems with countable symbolic dynamics, e.g., geodesic flows on noncompact manifolds and conformal dynamical systems with parabolic points; 3) related problems concerning statistical regularities of orbits in flows; and 4) temporal and spatial growth in branching diffusion processes and branching random walks. This research is a blend of statistical theory, dynamical systems, and ergodic theory. There is a broad class of dynamical systems for which the well-developed machinery of symbolic dynamics is available. This class is characterized by the fact that the associated symbolic dynamics needs only a finite alphabet (so one has to deal with finite state Markov processes). This research extends this class to those systems for which an infinitely countable alphabet becomes necessary and develops the corresponding machinery. This will provide the tools for obtaining new results about the clustering of orbits at limit sets and about the statistics of closed geodesics on Riemannian manifolds. The outcome of this work will be to unite many known results which were won by very different methods, so unifying, thus simplifying, areas with quite separate results.
