Bernoullicity of some skew products with hyperbolic base and Kochergin flow in the fiber

TL;DR

This paper proves that certain skew product systems with hyperbolic bases and Kochergin flows in the fibers are Bernoulli, under specific conditions on the cocycle and singularity growth, indicating strong stochastic properties.

Contribution

It establishes Bernoulli property for skew products with hyperbolic diffeomorphisms and Kochergin flows, extending understanding of their ergodic and statistical behavior.

Findings

Skew products with hyperbolic base and Kochergin flow are Bernoulli under certain conditions.

Bernoulli property holds for almost every rotation when the singularity growth exponent is less than 1/2.

The results connect singularity behavior with stochastic properties of complex dynamical systems.

Abstract

We study the Bernoulli property for skew products with hyperbolic diffeomorphisms equipped with a Gibbs measure in the base and Kochergin flows in the fiber, when the cocycle is aperiodic and of zero mean. The flow in the fiber can be represented as a special flow over an irrational rotation and a roof function with power singularity. We show that if the growth near the singularity is given by an exponent smaller than $\frac{1}{2}$, then for almost every rotation the resulting skew product is Bernoulli.