Ergodic theorem, reversibility and the filling scheme

Yves Derriennic¹

¹ U.E.B., Mathématiques (U.M.R. CNRS 6205) Université de Brest 6 av. Le Gorgeu 29238 Brest, France

Colloquium Mathematicum, Tome 118 (2010) no. 2, pp. 599-608

Résumé

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.

Détail

DOI : 10.4064/cm118-2-16

Keywords: short note present terse style meaning consequences filling scheme approach probability measure preserving transformation cohomological equation encapsulates argument complete simplify study reversibility ergodic limits integrability assumed short unified proofs known results about behaviour ergodic averages kestens lemma strikingly simple proof ergodic theorem dimension given neveu without maximal inequality nor clever combinatorics followed approach starting point present study