  In the scope of the statistical description of dynamical systems, one of the
defining features of chaos is the tendency of a system to lose memory of its
initial conditions (more precisely, of the distribution of its initial
conditions). For a dynamical system preserving a probability measure, this
property is named `mixing' and is equivalent to the decay of correlations for
observables in phase space. For the class of dynamical systems preserving
infinite measures, this probabilistic connection is lost and no completely
satisfactory definition has yet been found which expresses the idea of losing
track of the initial state of a system due to its chaotic dynamics. This is
actually on open problem in the field of infinite ergodic theory. 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.
In this note 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, whose rigorous
definition is 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. These definitions are tested on two types of
infinite-measure-preserving dynamical systems, the random walks and the Farey
map.
