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

Dedicated to the memory of Roy Adler

FRH: Supported by NSF grants DMS 1201326 and DMS 1500947
KV: Supported by the National Science Foundation under Award DMS 1604796.
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  • We study skew products where the base is a hyperbolic automorphism of $\mathbb{T}^2$, the fiber is a smooth area preserving flow on $\mathbb{T}^2$ 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 $\mathbb{T}^4$ which are $K$ and not Bernoulli.

    Mathematics Subject Classification: Primary: 37A05, 37C05; Secondary: 37A35, 37C50.


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  • Figure 1.  The set $W^f$, with base and roof

    Figure 3.  Horizontal Separation, $f$ and $\varphi$ have significant differences; the roof is hit a different number of times

    Figure 2.  Vertical Separation, $f$ and $\varphi$ have moderate differences

    Figure 4.  Breaking up $[0,N]$

    Table 1.  Summary of development

    LB Fiber LB Fiber Entropy Smooth $\int \varphi $
    Ornstein [23] N/A Yes N/A No N/A
    Feldman [11] No No 0 No $\not= 0$
    Katok [15] No No 0 Yes $\not= 0$
    Burton [8] Yes Yes Any No $\not= 0$
    Kalikow [12] Yes No $> 0$ No $0$
    Rudolph [30] Yes No $> 0$ Yes $0$
    Theorem 1 Yes Yes 0 Yes $\not= 0$
     | Show Table
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