Zero Krengel entropy does not kill Poisson entropy.

*(English. French summary)*Zbl 1269.37003For a transformation \(T\) with an infinite invariant measure, there are different notions of entropy which extend the Kolmogorov entropy of a probability measure preserving transformation.

The definition by U. Krengel uses the supremum of the normalized entropy of the transformations induced by \(T\) on sets of finite measure. There are two other definitions, one by W. Parry based on the conditional entropy and a third one, “the Poisson entropy” introduced by E. Roy, using the entropy of a probability measure preserving a transformation \(T_*\) canonically associated to \(T\), its “Poisson suspension”. It is known that the Parry entropy is less than or equal to the Krengel entropy and to the Poisson entropy. For large classes of transformations, the three entropies coincide.

A natural question is the existence of a conservative infinite-measure-preserving transformation \(T\) with zero Krengel entropy, but whose associated Poisson suspension has positive entropy.

The authors give a positive answer by constructing a transformation \(T\), a tower over the Von Neumann-Kakutani odometer, such that the associated \(T_*\) has positive Kolmogorov entropy.

An important tool is the \(\overline d\)-distance between stationary processes introduced by Ornstein. It is shown that, for a suitable choice of the parameters in the construction of \(T\), the stationary process “living” in the Poisson suspension of \(T\) is close for the \(\overline d\)-distance to an i.i.d. sequence of random Poisson variables. This implies the positivity of the entropy of \(T_*\).

Comments and open questions conclude the paper.

The definition by U. Krengel uses the supremum of the normalized entropy of the transformations induced by \(T\) on sets of finite measure. There are two other definitions, one by W. Parry based on the conditional entropy and a third one, “the Poisson entropy” introduced by E. Roy, using the entropy of a probability measure preserving a transformation \(T_*\) canonically associated to \(T\), its “Poisson suspension”. It is known that the Parry entropy is less than or equal to the Krengel entropy and to the Poisson entropy. For large classes of transformations, the three entropies coincide.

A natural question is the existence of a conservative infinite-measure-preserving transformation \(T\) with zero Krengel entropy, but whose associated Poisson suspension has positive entropy.

The authors give a positive answer by constructing a transformation \(T\), a tower over the Von Neumann-Kakutani odometer, such that the associated \(T_*\) has positive Kolmogorov entropy.

An important tool is the \(\overline d\)-distance between stationary processes introduced by Ornstein. It is shown that, for a suitable choice of the parameters in the construction of \(T\), the stationary process “living” in the Poisson suspension of \(T\) is close for the \(\overline d\)-distance to an i.i.d. sequence of random Poisson variables. This implies the positivity of the entropy of \(T_*\).

Comments and open questions conclude the paper.

Reviewer: Jean-Pierre Conze (Rennes)

##### MSC:

37A05 | Dynamical aspects of measure-preserving transformations |

37A35 | Entropy and other invariants, isomorphism, classification in ergodic theory |

37A40 | Nonsingular (and infinite-measure preserving) transformations |

28D20 | Entropy and other invariants |

##### Keywords:

Krengel entropy; Poisson suspension; infinite measure-preserving transformation; \(\overline d\)-distance##### References:

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