# American Institute of Mathematical Sciences

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

## 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 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.

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.
The set $W^f$, with base and roof
Horizontal Separation, $f$ and $\varphi$ have significant differences; the roof is hit a different number of times
Vertical Separation, $f$ and $\varphi$ have moderate differences
Breaking up $[0,N]$
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 $\not= 0$
 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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