Professor Park will continue her ergodic-theoretical study of actions by Z^2 and other groups, including such nonabelian groups as GL(2,Z). Properties of non-cocompact subgroup actions in Z^2-actions are to be examined, in particular when the action has a rigid property like minimal self-joinings or simplicity. She will also study further the centralizers of rank 1 actions of Z^2. She will continue her work on cellular automata, in particular on the continuity of directional entropy. She will seek representations of certain kinds of geodesic flows, from geometry, as flows built under ceiling functions. Classically, ergodic theory is concerned with what happens in the long run when a transformation of a space is iterated many times. (The space in question need not be anything particularly elaborate; something like a line segment will do.) Via the transformation, one can think of the group Z of integers acting on the space, or of time passing in discrete steps. Continuous time, similarly, is modeled by an action of the group R of real numbers, called a flow. This situation is also of interest in ergodic theory. A good example is a geodesic flow on a surface, where points on the surface are imagined as moving at constant speed along paths of shortest distance. A principal goal of research in ergodic theory is to classify the various sorts of long-run behavior that are possible in a given situation. Typically, there will be not too many, and all of them will be realizable within an appropriate standard model, or representation. Professor Park's research is concerned with what happens for other groups beside Z and R. The general framework of ergodic theory still makes sense, but new phenomena appear, even in the case of the group Z^2 of pairs of integers.
