  The Poisson entropy of an infinite-measure-preserving transformation is
defined 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 $-\sum q_i
p_{i,j}\log p_{i,j}$ holds in any of the definitions for 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 entropy is equal to the difference
of the Poisson entropies. In case there 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 in arXiv:0705.2148v3 about existence of a Pinsker factor in the
sense of Krengel for quasi-finite transformations.
