On the non-equivalence of the Bernoulli and K properties in dimension four

2016/12/08 by Adam Kanigowski, Kanigowski, Adam, Federico Rodríguez-Hertz +3
Mathematics · #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Equations Stability Results #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1612.02754

openalex publication_date 2016/12/08 · openalex created_date 2017/01/06 · openalex updated_date 2026/07/28

Abstract

We study skew products where the base is a hyperbolic automorphism of \mathbbT², the fiber is a smooth area preserving flow on \mathbbT² with one fixed point (of high degeneracy) and the skewing function is a smooth non coboundary with non-zero integral. The fiber dynamics can be represented as a special flow over an irrational rotation and a roof function with one power singularity. We show that for a full measure set of rotations the corresponding skew product is K and not Bernoulli. As a consequence we get a natural class of volume-preserving diffeomorphisms of \mathbbT⁴ which are K and not Bernoulli.

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