Chapter 13 Pointwise ergodic theorems for actions of groups

Abstract

This chapter discusses pointwise ergodic theorems for general measure-preserving actions of locally compact second countable (lcsc) groups. Ergodic theorems for actions of connected Lie groups and, particularly, equidistribution theorems on homogeneous spaces and moduli spaces, have been developed and used in a rapidly expanding array of applications. Equidistribution for every orbit does not hold in many cases, such as for the case of geodesic flows on compact surfaces of constant negative curvature. Convergence for every orbit fails even if the function is assumed continuous or even smooth. Thus, the restriction to almost every starting point is essential in the pointwise ergodic theorem. Furthermore, Calderon's original formulation of his pointwise ergodic theorem did not prove or assume strict polynomial volume growth, but instead noted that the doubling condition implies the following property for the volume of the balls. Similarly, subexponential growth (which is equivalent to polynomial volume growth in the connected Lie group case but not in general) implies that a subsequence of the sequence of balls is asymptotically invariant. Hence, particularly, the mean ergodic theorem is true for the subsequence.