ABSTRACT
In large parts of the book we have only considered measure preserving transformations on a probability space and in fact several fundamental results we have seen, such as the Pointwise and Mean Ergodic Theorems in particular, do not work in case the underlying measure space is infinite. In this chapter we discuss the dynamics on infinite measure systems in more detail. We first consider some more examples and elaborate on the recurrence behaviour of infinite measure systems, in particular we discuss the conservative and dissipative parts. We then introduce jump and induced transformations. For these transformations, one considers the dynamics only at the time steps in which the system visits a well chosen finite measure subset of the space. Then many of the tools developed for finite measure systems in the previous chapters can be used to derive properties of the infinite system using these transformations. At the end of the chapter we discuss ergodic theorems in relation to infinite measure systems, in particular the Ratio Ergodic Theorem is stated and proved, and as a result it is shown that the ergodic averages of an L1 function defined on an infinite space converge to zero almost everywhere.
