Disjointness of interval exchange transformations from systems of probabilistic origin

TL;DR

This paper proves that most interval exchange transformations with a specific symmetric permutation are disjoint from ELF systems, which are models of probabilistic origin, and explores related disjointness properties of certain special flows.

Contribution

It establishes disjointness results between specific interval exchange transformations and ELF systems, advancing understanding of their probabilistic independence.

Findings

Almost all such transformations are disjoint from ELF systems.

ELF systems include mixing, ergodic Gaussian, Poisson suspensions, and stationary infinitely divisible processes.

Disjointness properties of special flows over interval exchange transformations are analyzed.

Abstract

It is proved that almost every interval exchange transformation given by the symmetric permutation 1->m, 2->m-1,..., m-1->2, m->1, where m>1 is an odd number, is disjoint from ELF systems. The notion of ELF systems was introduced to express the fact that a given system is of probabilistic origin; the following standard classes of systems of probabilistic origin enjoy the ELF property: mixing systems, ergodic Gaussian systems, Poisson suspensions, dynamical systems coming from stationary infinitely divisible processes. Some disjointness properties of special flows built over interval exchange transformations and under piecewise constant roof function are investigated as well.