Dissipative sequences in infinite ergodic theory

AbstractLet T be an invertible, ergodic, measure-preserving transformation on a nonatomic, infinite, σ-finite measure space (Ω,A,μ). A set A∈A is weakly wandering under the sequence of integers W={ni}i=1∞ if the sets TniA are disjoint. If, for a given sequence W, there exists a weakly wandering set of positive measure, then we call W a weakly wandering sequence. A sequence {ki}i=1∞ is dissipative if, for every function f∈L1(Ω,A,μ), we have ∑i=1∞|f(Tkix)|<∞, almost everywhere. Every weakly wandering sequence is dissipative. In 1978, Hajian and Ito constructed a dissipative sequence which is not weakly wandering. An example of a dissipative sequence which is not a finite union of weakly wandering sequences is presented. Also, some further properties of dissipative sequences are studied