On pointwise periodicity and expansiveness

Following Kaul, a discrete (topological) group G of transformations of set X is pointwise periodic if the stabilizer of every point is of finite index (co-compact) in G. Equivalently, all G-orbits are finite (compact). Generalizing a result of Montgomery, Kaul showed in the early 70’s that a pointwise periodic transformation group is always compact when the group acts (faithfully) on a connected manifold without boundary. I will discuss implications of expansiveness and pointwise periodicity of certain groups and semigroups of transformations. In particular I’ll state implications for cellular automata and for planner tilings. Based on joint work with Ville Salo.