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Ergodic theorem, reversibility and the filling scheme

100%

Derriennic Y.

Colloquium Mathematicum

2010

tom 118

nr 2

599-608

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.

On the mean ergodic theorem for Cesàro bounded operators

100%

Derriennic Y.

Colloquium Mathematicum

2000

tom 84/85

nr 2

443-455

For a Cesàro bounded operator in a Hilbert space or a reflexive Banach space the mean ergodic theorem does not hold in general. We give an additional geometrical assumption which is sufficient to imply the validity of that theorem. Our result yields the mean ergodic theorem for positive Cesàro bounded operators in $L^{p}$ (1 < p < ∞). We do not use the tauberian theorem of Hardy and Littlewood, which was the main tool of previous authors. Some new examples, interesting for summability theory, are described: we build an example of a mean ergodic operator T in a Hilbert space such that $∥T^{n}∥/n$ does not converge to 0, and whose adjoint operator is not mean ergodic (its Cesàro averages converge only weakly).

Ograniczanie wyników

2 Colloquium Mathematicum

2 Derriennic Y.

1 2010

1 2000

Wyniki wyszukiwania

Ergodic theorem, reversibility and the filling scheme

On the mean ergodic theorem for Cesàro bounded operators