Global observables and the question of mixing in infinite ergodic theory

Finding a satisfactory definition of mixing for dynamical systems preserving an infinite measure (in short, infinite mixing) is an important open problem. Virtually all the definitions that have been attempted so far use 'local observables', that is, functions that essentially only "see" finite portions of the phase space. We introduce the concept of 'global observable', a function that gauges a certain quantity throughout the phase space. This concept is based on the notion of infinite-volume average, which plays the role of the expected value of a global observable. Endowed with these notions, which are to be specified on a case-by-case basis, we give a number of definitions of infinite mixing. These fall in two categories: global-global mixing, which expresses the "decorrelation" of two global observables, and global-local mixing, where a global and a local observable are considered instead. Time permitting, we will see how these definitions respond on some examples of infinite-measure-preserving dynamical systems.