On the non-equivalence of the Bernoulli and <inline-formula><tex-math id="M1">\begin{document}$ K$\end{document}</tex-math></inline-formula> properties in dimension four

Adam Kanigowski; Federico Rodriguez Hertz; Kurt Vinhage

doi:10.3934/jmd.2018019

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

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

Adam Kanigowski ^1, , Federico Rodriguez Hertz ^1, and Kurt Vinhage ^2,

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 31, 2016 Revised June 09, 2017 Published December 2018

Fund Project: FRH: Supported by NSF grants DMS 1201326 and DMS 1500947
KV: Supported by the National Science Foundation under Award DMS 1604796.

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.

Keywords: Bernoulli systems, K-property, skew products.

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

Citation: Adam Kanigowski, Federico Rodriguez Hertz, Kurt Vinhage. On the non-equivalence of the Bernoulli and $ K$ properties in dimension four. Journal of Modern Dynamics, 2018, 13: 221-250. doi: 10.3934/jmd.2018019

References:

[1]	L. M. Abramov and V. A. Rohlin, Entropy of a skew product of mappings with invariant measure, Vestnik Leningrad. Univ., 17 (1962), 5-13.
[2]	R. L. Adler and P. C. Shields, Skew products of Bernoulli shifts with rotations, Israel J. Math., 12 (1972), 215-222. doi: 10.1007/BF02790748.
[3]	D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trudy Moskov. Mat. Obšč., 23 (1970), 3-36.
[4]	T. Austin, Scenery entropy as an invariant of RWRS processes, Preprint available at arXiv: 1405.1468.
[5]	A. Avila, M. Viana and A. Wilkinson, Absolute continuity, Lyapunov exponents and rigidity Ⅰ: Geodesic flows, J. Eur. Math. Soc. (JEMS), 17 (2015), 1435-1462. doi: 10.4171/JEMS/534.
[6]	M. Benhenda, An uncountable family of pairwise non-Kakutani equivalent smooth diffeomorphisms, J. Anal. Math., 127 (2015), 129-178. doi: 10.1007/s11854-015-0027-z.
[7]	R. M. Burton and P. C. Shields, A mixing $ T$ for which $ T-T^{-1}$ is Bernoulli, Monatsh. Math., 95 (1983), 89-98. doi: 10.1007/BF01323652.
[8]	R. M. Burton, Jr., A non-Bernoulli skew product which is loosely Bernoulli, Israel J. Math., 35 (1980), 339-348. doi: 10.1007/BF02760659.
[9]	M. Denker and W. Philipp, Approximation by Brownian motion for Gibbs measures and flows under a function, Ergodic Theory Dynam. Systems, 4 (1984), 541-552.
[10]	B. Fayad, G. Forni and A. Kanigowski, Lebesgue spectrum for area preserving flows on the two torus, submitted.
[11]	J. Feldman, New $ K$-automorphisms and a problem of Kakutani, Israel J. Math., 24 (1976), 16-38. doi: 10.1007/BF02761426.
[12]	S. A. Kalikow, $ T,\,T^{-1}$ transformation is not loosely Bernoulli, Ann. of Math. (2), 115 (1982), 393-409. doi: 10.2307/1971397.
[13]	A. Kanigowski, Slow entropy for some smooth flows on surfaces, accepted in Israel J. Math.
[14]	A. B. Katok, Monotone equivalence in ergodic theory, Izv. Akad. Nauk SSSR Ser. Mat., 41 (1977), 104-157. doi: 10.1070/IM1977v011n01ABEH001696.
[15]	A. Katok, Smooth non-Bernoulli $ K$-automorphisms, Invent. Math., 61 (1980), 291-299. doi: 10.1007/BF01390069.
[16]	A. Katok, Combinatorial Constructions in Ergodic Theory and Dynamics, University Lecture Series, 30, American Mathematical Society, Providence, RI, 2003.
[17]	A. B. Katok and E. A. Sataev, Standardness of rearrangement automorphisms of segments and flows on surfaces, Mat. Zametki, 20 (1976), 479-488.
[18]	A. Ya. Khinchin, Continued Fractions, The University of Chicago Press, Chicago, Ill.-London, 1964.
[19]	A. V. Kočergin, Mixing in special flows over a rearrangement of segments and in smooth flows on surfaces, Mat. Sb. (N.S.), 96/138 (1975), 471-502.
[20]	A. Lamotte, Structure de certains produits semi directs, Ergodic Theory Dynam. Systems, 3 (1983), 559-566. doi: 10.1017/S0143385700002145.
[21]	R. Lyons, Strong laws of large numbers for weakly correlated random variables, Michigan Math. J., 35 (1988), 353-359. doi: 10.1307/mmj/1029003816.
[22]	D. Ornstein, Bernoulli shifts with the same entropy are isomorphic, Advances in Math., 4 (1970), 337-352. doi: 10.1016/0001-8708(70)90029-0.
[23]	D. S. Ornstein, An example of a Kolmogorov automorphism that is not a Bernoulli shift, Advances in Math., 10 (1973), 49-62. doi: 10.1016/0001-8708(73)90097-2.
[24]	J. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk, 32 (1977), 55-112,287.
[25]	G. Ponce, A. Tahzibi and R. Varão, On the bernoulli property for certain partially hyperbolic diffeomorphisms, Preprint available at arXiv: 1603.08605.
[26]	M. Ratner, The Cartesian square of the horocycle flow is not loosely Bernoulli, Israel J. Math., 34 (1979), 72-96. doi: 10.1007/BF02761825.
[27]	F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, A survey of partially hyperbolic dynamics, in Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow, Fields Inst. Commun., 51, Amer. Math. Soc., Providence, RI, 2007, 35-87.
[28]	V. A. Rohlin and Ja. G. Sinai, The structure and properties of invariant measurable partitions, Dokl. Akad. Nauk SSSR, 141 (1961), 1038-1041.
[29]	D. J. Rudolph, Classifying the isometric extensions of a Bernoulli shift, J. Analyse Math., 34 (1978), 36-60. doi: 10.1007/BF02790007.
[30]	D. J. Rudolph, Asymptotically Brownian skew products give non-loosely Bernoulli $ K$-automorphisms, Invent. Math., 91 (1988), 105-128. doi: 10.1007/BF01404914.
[31]	P. C. Shields, Weak and very weak Bernoulli partitions, Monatsh. Math., 84 (1977), 133-142. doi: 10.1007/BF01579598.
[32]	P. C. Shields and R. Burton, A skew-product which is Bernoulli, Monatsh. Math., 86 (1978/79), 155-165. doi: 10.1007/BF01320207.
[33]	Ja. G. Sinai, On a weak isomorphism of transformations with invariant measure, Mat. Sb. (N.S.), 63 (1964), 23-42.
[34]	J.-P. Thouvenot, Entropy, isomorphism and equivalence in ergodic theory, in Handbook of dynamical systems, Vol. 1A, 205-238, North-Holland, Amsterdam, 2002.
[35]	B. Weiss, The isomorphism problem in ergodic theory, Bull. Amer. Math. Soc., 78 (1972), 668-684. doi: 10.1090/S0002-9904-1972-12979-3.

show all references

References:

[1]	L. M. Abramov and V. A. Rohlin, Entropy of a skew product of mappings with invariant measure, Vestnik Leningrad. Univ., 17 (1962), 5-13.
[2]	R. L. Adler and P. C. Shields, Skew products of Bernoulli shifts with rotations, Israel J. Math., 12 (1972), 215-222. doi: 10.1007/BF02790748.
[3]	D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trudy Moskov. Mat. Obšč., 23 (1970), 3-36.
[4]	T. Austin, Scenery entropy as an invariant of RWRS processes, Preprint available at arXiv: 1405.1468.
[5]	A. Avila, M. Viana and A. Wilkinson, Absolute continuity, Lyapunov exponents and rigidity Ⅰ: Geodesic flows, J. Eur. Math. Soc. (JEMS), 17 (2015), 1435-1462. doi: 10.4171/JEMS/534.
[6]	M. Benhenda, An uncountable family of pairwise non-Kakutani equivalent smooth diffeomorphisms, J. Anal. Math., 127 (2015), 129-178. doi: 10.1007/s11854-015-0027-z.
[7]	R. M. Burton and P. C. Shields, A mixing $ T$ for which $ T-T^{-1}$ is Bernoulli, Monatsh. Math., 95 (1983), 89-98. doi: 10.1007/BF01323652.
[8]	R. M. Burton, Jr., A non-Bernoulli skew product which is loosely Bernoulli, Israel J. Math., 35 (1980), 339-348. doi: 10.1007/BF02760659.
[9]	M. Denker and W. Philipp, Approximation by Brownian motion for Gibbs measures and flows under a function, Ergodic Theory Dynam. Systems, 4 (1984), 541-552.
[10]	B. Fayad, G. Forni and A. Kanigowski, Lebesgue spectrum for area preserving flows on the two torus, submitted.
[11]	J. Feldman, New $ K$-automorphisms and a problem of Kakutani, Israel J. Math., 24 (1976), 16-38. doi: 10.1007/BF02761426.
[12]	S. A. Kalikow, $ T,\,T^{-1}$ transformation is not loosely Bernoulli, Ann. of Math. (2), 115 (1982), 393-409. doi: 10.2307/1971397.
[13]	A. Kanigowski, Slow entropy for some smooth flows on surfaces, accepted in Israel J. Math.
[14]	A. B. Katok, Monotone equivalence in ergodic theory, Izv. Akad. Nauk SSSR Ser. Mat., 41 (1977), 104-157. doi: 10.1070/IM1977v011n01ABEH001696.
[15]	A. Katok, Smooth non-Bernoulli $ K$-automorphisms, Invent. Math., 61 (1980), 291-299. doi: 10.1007/BF01390069.
[16]	A. Katok, Combinatorial Constructions in Ergodic Theory and Dynamics, University Lecture Series, 30, American Mathematical Society, Providence, RI, 2003.
[17]	A. B. Katok and E. A. Sataev, Standardness of rearrangement automorphisms of segments and flows on surfaces, Mat. Zametki, 20 (1976), 479-488.
[18]	A. Ya. Khinchin, Continued Fractions, The University of Chicago Press, Chicago, Ill.-London, 1964.
[19]	A. V. Kočergin, Mixing in special flows over a rearrangement of segments and in smooth flows on surfaces, Mat. Sb. (N.S.), 96/138 (1975), 471-502.
[20]	A. Lamotte, Structure de certains produits semi directs, Ergodic Theory Dynam. Systems, 3 (1983), 559-566. doi: 10.1017/S0143385700002145.
[21]	R. Lyons, Strong laws of large numbers for weakly correlated random variables, Michigan Math. J., 35 (1988), 353-359. doi: 10.1307/mmj/1029003816.
[22]	D. Ornstein, Bernoulli shifts with the same entropy are isomorphic, Advances in Math., 4 (1970), 337-352. doi: 10.1016/0001-8708(70)90029-0.
[23]	D. S. Ornstein, An example of a Kolmogorov automorphism that is not a Bernoulli shift, Advances in Math., 10 (1973), 49-62. doi: 10.1016/0001-8708(73)90097-2.
[24]	J. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk, 32 (1977), 55-112,287.
[25]	G. Ponce, A. Tahzibi and R. Varão, On the bernoulli property for certain partially hyperbolic diffeomorphisms, Preprint available at arXiv: 1603.08605.
[26]	M. Ratner, The Cartesian square of the horocycle flow is not loosely Bernoulli, Israel J. Math., 34 (1979), 72-96. doi: 10.1007/BF02761825.
[27]	F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, A survey of partially hyperbolic dynamics, in Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow, Fields Inst. Commun., 51, Amer. Math. Soc., Providence, RI, 2007, 35-87.
[28]	V. A. Rohlin and Ja. G. Sinai, The structure and properties of invariant measurable partitions, Dokl. Akad. Nauk SSSR, 141 (1961), 1038-1041.
[29]	D. J. Rudolph, Classifying the isometric extensions of a Bernoulli shift, J. Analyse Math., 34 (1978), 36-60. doi: 10.1007/BF02790007.
[30]	D. J. Rudolph, Asymptotically Brownian skew products give non-loosely Bernoulli $ K$-automorphisms, Invent. Math., 91 (1988), 105-128. doi: 10.1007/BF01404914.
[31]	P. C. Shields, Weak and very weak Bernoulli partitions, Monatsh. Math., 84 (1977), 133-142. doi: 10.1007/BF01579598.
[32]	P. C. Shields and R. Burton, A skew-product which is Bernoulli, Monatsh. Math., 86 (1978/79), 155-165. doi: 10.1007/BF01320207.
[33]	Ja. G. Sinai, On a weak isomorphism of transformations with invariant measure, Mat. Sb. (N.S.), 63 (1964), 23-42.
[34]	J.-P. Thouvenot, Entropy, isomorphism and equivalence in ergodic theory, in Handbook of dynamical systems, Vol. 1A, 205-238, North-Holland, Amsterdam, 2002.
[35]	B. Weiss, The isomorphism problem in ergodic theory, Bull. Amer. Math. Soc., 78 (1972), 668-684. doi: 10.1090/S0002-9904-1972-12979-3.

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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	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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American Institute of Mathematical Sciences

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

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