## Ergodic theorem, reversibility and the filling scheme

### Volume 118 / 2010

Colloquium Mathematicum 118 (2010), 599-608
MSC: 37A05, 28D.
DOI: 10.4064/cm118-2-16

#### Abstract

The aim of this short note is to present in terse style the meaning and consequences of the “filling scheme” approach for a probability measure preserving transformation. A cohomological equation encapsulates the argument. We complete and simplify Woś' study (1986) of the reversibility of the ergodic limits when integrability is not assumed. We give short and unified proofs of well known results about the behaviour of ergodic averages, like Kesten's lemma (1975). The strikingly simple proof of the ergodic theorem in one dimension given by Neveu (1979), without any maximal inequality nor clever combinatorics, followed this approach and was the starting point of the present study.