On Some of the Differences Between Z and Z^2 in Dynamics

Around 1970 it was discovered that measure preserving actions of Z^2 on probability spaces can have remarkably different properties from Z-actions, i.e. from actions of single transformations. Many of these differences can be described as 'rigidity phenomena' and manifest themselves in the scarcity of certain objects for Z^2-actions which are abundant for Z-actions: they may have unexpectedly few invariant probability measures, invariant sets, isomorphisms or automorphisms.

This lecture will aim to explain--in some examples--both the reasons and the ramifications of these rigidity phenomena.