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. AvilaM. 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. AvilaM. 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
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 $\not= 0$
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 $\not= 0$
[1]

Roy Adler, Bruce Kitchens, Michael Shub. Stably ergodic skew products. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 349-350. doi: 10.3934/dcds.1996.2.349

[2]

Roy Adler, Bruce Kitchens, Michael Shub. Errata to "Stably ergodic skew products". Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 456-456. doi: 10.3934/dcds.1999.5.456

[3]

Imen Bhouri, Houssem Tlili. On the multifractal formalism for Bernoulli products of invertible matrices. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1129-1145. doi: 10.3934/dcds.2009.24.1129

[4]

Àlex Haro. On strange attractors in a class of pinched skew products. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 605-617. doi: 10.3934/dcds.2012.32.605

[5]

Eugen Mihailescu, Mariusz Urbański. Transversal families of hyperbolic skew-products. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 907-928. doi: 10.3934/dcds.2008.21.907

[6]

Jose S. Cánovas, Antonio Falcó. The set of periods for a class of skew-products. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 893-900. doi: 10.3934/dcds.2000.6.893

[7]

Matúš Dirbák. Minimal skew products with hypertransitive or mixing properties. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1657-1674. doi: 10.3934/dcds.2012.32.1657

[8]

Viorel Nitica. Examples of topologically transitive skew-products. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 351-360. doi: 10.3934/dcds.2000.6.351

[9]

Jon Aaronson, Michael Bromberg, Nishant Chandgotia. Rational ergodicity of step function skew products. Journal of Modern Dynamics, 2018, 13: 1-42. doi: 10.3934/jmd.2018012

[10]

Matthew Nicol. Induced maps of hyperbolic Bernoulli systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 147-154. doi: 10.3934/dcds.2001.7.147

[11]

C.P. Walkden. Stable ergodicity of skew products of one-dimensional hyperbolic flows. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 897-904. doi: 10.3934/dcds.1999.5.897

[12]

Kohei Ueno. Weighted Green functions of nondegenerate polynomial skew products on $\mathbb{C}^2$. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 985-996. doi: 10.3934/dcds.2011.31.985

[13]

Kohei Ueno. Weighted Green functions of polynomial skew products on $\mathbb{C}^2$. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2283-2305. doi: 10.3934/dcds.2014.34.2283

[14]

Núria Fagella, Àngel Jorba, Marc Jorba-Cuscó, Joan Carles Tatjer. Classification of linear skew-products of the complex plane and an affine route to fractalization. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3767-3787. doi: 10.3934/dcds.2019153

[15]

Gen Qi Xu, Siu Pang Yung. Stability and Riesz basis property of a star-shaped network of Euler-Bernoulli beams with joint damping. Networks & Heterogeneous Media, 2008, 3 (4) : 723-747. doi: 10.3934/nhm.2008.3.723

[16]

Rui Gao, Weixiao Shen. Analytic skew-products of quadratic polynomials over Misiurewicz-Thurston maps. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2013-2036. doi: 10.3934/dcds.2014.34.2013

[17]

David Färm, Tomas Persson. Dimension and measure of baker-like skew-products of $\boldsymbol{\beta}$-transformations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3525-3537. doi: 10.3934/dcds.2012.32.3525

[18]

Jinjun Li, Min Wu. Generic property of irregular sets in systems satisfying the specification property. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 635-645. doi: 10.3934/dcds.2014.34.635

[19]

Jinjun Li, Min Wu. Divergence points in systems satisfying the specification property. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 905-920. doi: 10.3934/dcds.2013.33.905

[20]

Patrik Nystedt, Johan Öinert. Simple skew category algebras associated with minimal partially defined dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4157-4171. doi: 10.3934/dcds.2013.33.4157

2018 Impact Factor: 0.295

Article outline

Figures and Tables

[Back to Top]