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

TL;DR

This paper explores polymorphisms related to measure-preserving transformations, introducing new types of operators for finite and infinite measure spaces, and constructs a functor connecting these polymorphisms via convolution measures over Poisson matchings.

Contribution

It introduces R-polymorphisms and v-polymorphisms as analogues of Markov operators for different measure spaces and constructs a functor linking them through convolution measures.

Findings

Defined R-polymorphisms and v-polymorphisms for measure spaces.

Constructed a functor from v-polymorphisms to R-polymorphisms.

Described the functor in terms of summation of convolution products over Poisson matchings.

Abstract

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 with finite measure is a partial case of Markov transition operators. A Markov transition operator also can be considered as a map (polymorphism) A to A, which spreads points of A into measures on A. In this paper, we discuss R-polymorphisms and $\vee$-polymorphisms, who are analogues of the Markov transition operators for the groups of bijections A to A leaving the measure quasiinvariant; two types of the polymorphisms correspond to the cases, when A has finite and infinite measure respectively. We construct a functor from $\vee$-polymorphisms to R-polymorphisms, it is described in terms of summation of convolution products of measures over matchings of Poisson configurations.