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

Adam Kanigowski; Federico Rodriguez Hertz; Kurt Vinhage

doi:10.3934/jmd.2018019

Article Contents

2018, Volume 13: 221-250. Doi: 10.3934/jmd.2018019

This volume Previous Article Continuity of Lyapunov exponents for cocycles with invariant holonomies Next Article Teichmüller geodesics with $ d$-dimensional limit sets

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

1.
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
2.
Department of Mathematics, The University of Chicago, 5734 S University Ave, Chicago, IL 60637, USA

Dedicated to the memory of Roy Adler

Received: December 30, 2016

Revised: June 08, 2017

Published: December 2018

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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Abstract

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.

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

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

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

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

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Figure 4. Breaking up $[0,N]$

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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	$\not= 0$

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References

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