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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 |
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.
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. |



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