# Product of two Kochergin flows with different exponents is not standard

## TL;DR

This paper demonstrates that the product of two Kochergin flows with different singularity exponents on the torus generally does not possess the standard (zero entropy loosely Bernoulli) property, highlighting complex behavior in such dynamical systems.

## Contribution

It proves that for most pairs of rotations and different singularity exponents, the product of two Kochergin flows is not standard, extending understanding of their ergodic properties.

## Key findings

- Product of two Kochergin flows with different exponents is not standard for a full measure set of rotations.
- The result applies to flows represented as special flows over irrational rotations with singular roof functions.
- Shows the non-standard behavior is typical in this class of flows.

## Abstract

We study the standard(zero entropy loosely Bernoulli or loosely Kronecker) property for products of Kochergin smooth flows on $\mathbb{T}^2$ with one singularity. These flows can be represented as special flows over irrational rotations of the circle and under roof functions which are smooth on $\mathbb{T}^2\setminus \{0\}$ with a singularity at $0$. We show that there exists a full measure set $\mathscr{D}\subset\mathbb{T}$ such that the product system of two Kochergin flows with different power of singularities and rotations from $\mathscr{D}$ is not standard.