Spreading maps (polymorphisms), symmetries of Poisson processes, and matching summation

Yu. A. Neretin

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part VII, Tome 292 (2002), pp. 62-91

Résumé

The matrix of a permutation is a partial case of Markov transition matrices. In the same way, a measure preserving bijection of a space $(A,\alpha)$ with finite measure is a partial case of Markov transition operators. A Markov transition operator also can be considered as a map (polymorphism) $(A,\alpha)\to (A,\alpha)$, which spreads points of $(A,\alpha)$ into measures on $(A,\alpha)$. Denote by $\mathbb R^*$ the multiplicative group of positive real numbers and by $\mathscr M$ the semigroup of measures on $\mathbb R^*$. In this paper, we discuss $\mathbb R^*$-polymorphisms and $\curlyvee$-polymorphisms, who are analogues of the Markov transition operators (or polymorphisms) for the groups of bijections $(A,\alpha)\to (A,\alpha)$ leaving the measure $\alpha$ quasiinvariant; two types of the polymorphisms correspond to the cases, when $A$ has finite and infinite measure respectively. For the case, when the space $A$ itself is finite, the $\mathbb R^*$-polymorphisms are some $\mathscr M$-valued matrices. We construct a functor from $\curlyvee$-polymorphisms to $\mathbb R^*$-polymorphisms, it is described in terms of summations of $\mathscr M$-convolution products over matchings of Poisson configurations.