Poisson suspensions and entropy for infinite transformations

Abstract

The Poisson entropy of an infinite-measure-preserving transformation is defined in the 2005 thesis of Roy as the Kolmogorov entropy of its Poisson suspension. In this article, we relate Poisson entropy with other definitions of entropy for infinite transformations: For quasi-finite transformations we prove that Poisson entropy coincides with Krengel’s and Parry’s entropy. In particular, this implies that for null-recurrent Markov chains, the usual formula for the entropy, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus sigma-summation q Subscript i Baseline p Subscript i comma j Baseline log p Subscript i comma j"> <mml:semantics> <mml:mrow> <mml:mo> − </mml:mo> <mml:mo> ∑ </mml:mo> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mi>log</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">-\sum q_i p_{i,j}\log p_{i,j}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , holds for any definitions of entropy. Poisson entropy dominates Parry’s entropy in any conservative transformation. We also prove that relative entropy (in the sense of Danilenko and Rudolph) coincides with the relative Poisson entropy. Thus, for any factor of a conservative transformation, difference of the Krengel’s entropies equals difference of the Poisson entropies. In case there already exists a factor with zero Poisson entropy, we prove the existence of a maximum (Pinsker) factor with zero Poisson entropy. Together with the preceding results, this answers affirmatively the question raised by Aaronson and Park about existence of a Pinsker factor in the sense of Krengel for quasi-finite transformations.