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

TL;DR

This work proves that a broad class of skew products with a hyperbolic base endowed with a Gibbs measure and a Kochergin flow in the fiber are Bernoulli, provided the cocycle is aperiodic with zero mean and the fiber exponent satisfies $\gamma<\tfrac{1}{2}$. The authors reduce the problem to establishing a generating VWB partition and implement a meticulous blockwise matching argument that leverages limit theorems for conditional measures, Denjoy–Koksma bounds, and exponential equidistribution to control both base and fiber dynamics. The key contribution is a robust construction that combines blockwise matching, good/very good control sets, and a final global matching to yield VWB, thereby implying Bernoullicity for almost every rotation parameter $\alpha$. This advances the understanding of Bernoulli behavior in partially hyperbolic skew products with zero-entropy fibers and demonstrates Bernoulli stability under slow fiber dynamics in the Kochergin setting.

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.

Theorems & Definitions (57)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Definition 2.4
• Theorem 2.5
• Lemma 2.6
• Lemma 2.7
• Remark 2.8