2018, 13: 251-270. doi: 10.3934/jmd.2018020

On manifolds admitting stable type Ⅲ$_{\textbf1}$ Anosov diffeomorphisms

Einstein Institute of Mathematics, Edmond J. Safra Campus (Givat Ram), The Hebrew University, Jerusalem 91904, Israel

Dedicated to the memory of Roy Adler, whose work has been and still is a great inspiration for me

Received  January 22, 2017 Revised  June 28, 2017 Published  December 2018

We prove that for every $d≠3$ there is an Anosov diffeomorphism of $\mathbb{T}^{d}$ which is of stable Krieger type ${\rm III}_1$ (its Maharam extension is weakly mixing). This is done by a construction of stable type ${\rm III}_1$ Markov measures on the golden mean shift which can be smoothly realized as a $C^{1}$ Anosov diffeomorphism of $\mathbb{T}^2$ via the construction in our earlier paper.

Citation: Zemer Kosloff. On manifolds admitting stable type Ⅲ$_{\textbf1}$ Anosov diffeomorphisms. Journal of Modern Dynamics, 2018, 13: 251-270. doi: 10.3934/jmd.2018020
References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory, Amer. Math. Soc., Providence, R.I., 1997. Google Scholar

[2]

J. AaronsonM. Lin and B. Weiss, Mixing properties of Markov operators and ergodic transformations, and ergodicity of Cartesian products, A collection of invited papers on ergodic theory, Israel J. Math., 33 (1979), 198-224 (1980). doi: 10.1007/BF02762161. Google Scholar

[3]

J. AaronsonH. Nakada and O. Sarig, Exchangeable measures for subshifts, Ann. Inst. H. Poincaré Probab. Statist., 42 (2006), 727-751. doi: 10.1016/j.anihpb.2005.10.002. Google Scholar

[4]

R. L. Adler, Symbolic dynamics and Markov partitions, Bull. Amer. Math. Soc. (N.S.), 35 (1998), 1-56. doi: 10.1090/S0273-0979-98-00737-X. Google Scholar

[5]

R. L. Adler and B. Weiss, Similarity of Automorphisms of the Torus, Memoirs of the American Mathematical Society, No. 98, American Mathematical Society, Providence, R.I. 1970. Google Scholar

[6]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, second revised edition, with a preface by David Ruelle, edited by Jean-René Chazottes, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin, 2008. Google Scholar

[7]

J. M. ChoksiJ. H. Hawkins and V. S. Prasad, Abelian cocycles for nonsingular ergodic transformations and the genericity of type Ⅲ1 transformations, Monatsh. Math., 103 (1987), 187-205. doi: 10.1007/BF01364339. Google Scholar

[8]

A. Danilenko and M. Lemanczyk, K-property for Maharam extensions of nonsingular Bernoulli and Markov shifts, arXiv: 1611.05173.Google Scholar

[9]

J. Feldman and C. C. Moore, Ergodic equivalence relations, cohomology and von Neumann algebras, I, Trans. Amer. Math. Soc., 234 (1977), 289-324. doi: 10.1090/S0002-9947-1977-0578730-2. Google Scholar

[10]

H. Furstenberg and B. Weiss, The finite multipliers of infinite ergodic transformations, The structure of attractors in dynamical systems, (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), 127–132, Lecture Notes in Mathematics, 668, Springer-Verlag, Berlin, 1978. doi: 10.1007/BFb0101785. Google Scholar

[11]

J. M. Hawkins, Amenable relations for endomorphisms, Trans. Amer. Math. Soc., 343 (1994), 169-191. doi: 10.1090/S0002-9947-1994-1179396-3. Google Scholar

[12]

A. Gorodnik, Open problems in dynamics and related fields, J. Mod. Dyn., 1 (2007), 1-35. doi: 10.3934/jmd.2007.1.1. Google Scholar

[13]

Z. Kosloff, Conservative Anosov diffeomorphisms of the two torus without an absolutely continuous invariant measure, arXiv: 1410.7707.Google Scholar

[14]

Z. Kosloff, On the K property for Maharam extensions of Bernoulli shifts and a question of Krengel, Israel J. Math., 199 (2014), 485-506. doi: 10.1007/s11856-013-0069-9. Google Scholar

[15]

W. Krieger, On non-singular transformations of a measure space, Ⅰ, Ⅱ, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 11 (1969), 83-97. doi: 10.1007/BF00531812. Google Scholar

[16]

R. LePage and V. Mandrekar, On likelihood ratios of measures given by Markov chains, Proc. Amer. Math. Soc., 52 (1975), 377-380. doi: 10.1090/S0002-9939-1975-0380964-0. Google Scholar

[17]

D. A. Levin, Y. Peres and E. L. Wilmer, Markov Chains and Mixing Times, with a chapter by James G. Propp and David B. Wilson, American Mathematical Society, Providence, RI, 2009. Google Scholar

[18]

W. Parry, Ergodic and spectral analysis of certain infinite measure preserving transformations, Proc. Amer. Math. Soc., 16 (1965), 960-966. doi: 10.1090/S0002-9939-1965-0181737-8. Google Scholar

[19]

K. Schmidt, Cocycles on Ergodic Transformation Groups, Macmillan Lectures in Mathematics, Vol. 1, Macmillan Company of India, Ltd., Delhi, 1977. Google Scholar

[20]

A. N. Shiryayev, Probability, second edition, translated from the Russian by R. P. Boas, Graduate Texts in Mathematics, 95, Springer-Verlag, New York, 1984. Google Scholar

[21]

C. E. Silva and P. Thieullen, A skew product entropy for nonsingular transformations, J. London Math. Soc.(2), 52 (1995), 497-516. doi: 10.1112/jlms/52.3.497. Google Scholar

[22]

J. G. Sinai, Markov partitions and $U$-diffeomorphisms, Funkcional. Anal. i Priložen., 2 (1968), 64-89. Google Scholar

[23]

M. Thaler, Transformations on [0, 1] with infinite invariant measures, Israel J. Math., 46 (1983), 67-96. doi: 10.1007/BF02760623. Google Scholar

show all references

References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory, Amer. Math. Soc., Providence, R.I., 1997. Google Scholar

[2]

J. AaronsonM. Lin and B. Weiss, Mixing properties of Markov operators and ergodic transformations, and ergodicity of Cartesian products, A collection of invited papers on ergodic theory, Israel J. Math., 33 (1979), 198-224 (1980). doi: 10.1007/BF02762161. Google Scholar

[3]

J. AaronsonH. Nakada and O. Sarig, Exchangeable measures for subshifts, Ann. Inst. H. Poincaré Probab. Statist., 42 (2006), 727-751. doi: 10.1016/j.anihpb.2005.10.002. Google Scholar

[4]

R. L. Adler, Symbolic dynamics and Markov partitions, Bull. Amer. Math. Soc. (N.S.), 35 (1998), 1-56. doi: 10.1090/S0273-0979-98-00737-X. Google Scholar

[5]

R. L. Adler and B. Weiss, Similarity of Automorphisms of the Torus, Memoirs of the American Mathematical Society, No. 98, American Mathematical Society, Providence, R.I. 1970. Google Scholar

[6]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, second revised edition, with a preface by David Ruelle, edited by Jean-René Chazottes, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin, 2008. Google Scholar

[7]

J. M. ChoksiJ. H. Hawkins and V. S. Prasad, Abelian cocycles for nonsingular ergodic transformations and the genericity of type Ⅲ1 transformations, Monatsh. Math., 103 (1987), 187-205. doi: 10.1007/BF01364339. Google Scholar

[8]

A. Danilenko and M. Lemanczyk, K-property for Maharam extensions of nonsingular Bernoulli and Markov shifts, arXiv: 1611.05173.Google Scholar

[9]

J. Feldman and C. C. Moore, Ergodic equivalence relations, cohomology and von Neumann algebras, I, Trans. Amer. Math. Soc., 234 (1977), 289-324. doi: 10.1090/S0002-9947-1977-0578730-2. Google Scholar

[10]

H. Furstenberg and B. Weiss, The finite multipliers of infinite ergodic transformations, The structure of attractors in dynamical systems, (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), 127–132, Lecture Notes in Mathematics, 668, Springer-Verlag, Berlin, 1978. doi: 10.1007/BFb0101785. Google Scholar

[11]

J. M. Hawkins, Amenable relations for endomorphisms, Trans. Amer. Math. Soc., 343 (1994), 169-191. doi: 10.1090/S0002-9947-1994-1179396-3. Google Scholar

[12]

A. Gorodnik, Open problems in dynamics and related fields, J. Mod. Dyn., 1 (2007), 1-35. doi: 10.3934/jmd.2007.1.1. Google Scholar

[13]

Z. Kosloff, Conservative Anosov diffeomorphisms of the two torus without an absolutely continuous invariant measure, arXiv: 1410.7707.Google Scholar

[14]

Z. Kosloff, On the K property for Maharam extensions of Bernoulli shifts and a question of Krengel, Israel J. Math., 199 (2014), 485-506. doi: 10.1007/s11856-013-0069-9. Google Scholar

[15]

W. Krieger, On non-singular transformations of a measure space, Ⅰ, Ⅱ, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 11 (1969), 83-97. doi: 10.1007/BF00531812. Google Scholar

[16]

R. LePage and V. Mandrekar, On likelihood ratios of measures given by Markov chains, Proc. Amer. Math. Soc., 52 (1975), 377-380. doi: 10.1090/S0002-9939-1975-0380964-0. Google Scholar

[17]

D. A. Levin, Y. Peres and E. L. Wilmer, Markov Chains and Mixing Times, with a chapter by James G. Propp and David B. Wilson, American Mathematical Society, Providence, RI, 2009. Google Scholar

[18]

W. Parry, Ergodic and spectral analysis of certain infinite measure preserving transformations, Proc. Amer. Math. Soc., 16 (1965), 960-966. doi: 10.1090/S0002-9939-1965-0181737-8. Google Scholar

[19]

K. Schmidt, Cocycles on Ergodic Transformation Groups, Macmillan Lectures in Mathematics, Vol. 1, Macmillan Company of India, Ltd., Delhi, 1977. Google Scholar

[20]

A. N. Shiryayev, Probability, second edition, translated from the Russian by R. P. Boas, Graduate Texts in Mathematics, 95, Springer-Verlag, New York, 1984. Google Scholar

[21]

C. E. Silva and P. Thieullen, A skew product entropy for nonsingular transformations, J. London Math. Soc.(2), 52 (1995), 497-516. doi: 10.1112/jlms/52.3.497. Google Scholar

[22]

J. G. Sinai, Markov partitions and $U$-diffeomorphisms, Funkcional. Anal. i Priložen., 2 (1968), 64-89. Google Scholar

[23]

M. Thaler, Transformations on [0, 1] with infinite invariant measures, Israel J. Math., 46 (1983), 67-96. doi: 10.1007/BF02760623. Google Scholar

Figure 4.1.  The construction of the Markov partition
Table 1.   
$c_{M_{t-1}-3}$ $w$ $d_{M_{t-1}}$
$1$ or $2$ $11$ $1$ or $3$
1 or 2 13 2
3 21 1 or 3
3 23 2
$c_{M_{t-1}-3}$ $w$ $d_{M_{t-1}}$
$1$ or $2$ $11$ $1$ or $3$
1 or 2 13 2
3 21 1 or 3
3 23 2
[1]

João P. Almeida, Albert M. Fisher, Alberto Adrego Pinto, David A. Rand. Anosov diffeomorphisms. Conference Publications, 2013, 2013 (special) : 837-845. doi: 10.3934/proc.2013.2013.837

[2]

Yair Daon. Bernoullicity of equilibrium measures on countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4003-4015. doi: 10.3934/dcds.2013.33.4003

[3]

Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Thermodynamic formalism for random countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 131-164. doi: 10.3934/dcds.2008.22.131

[4]

Kengo Matsumoto. On the Markov-Dyck shifts of vertex type. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 403-422. doi: 10.3934/dcds.2016.36.403

[5]

Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Corrigendum to: Thermodynamic formalism for random countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 593-594. doi: 10.3934/dcds.2015.35.593

[6]

Maria Carvalho. First homoclinic tangencies in the boundary of Anosov diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 765-782. doi: 10.3934/dcds.1998.4.765

[7]

Christian Bonatti, Nancy Guelman. Axiom A diffeomorphisms derived from Anosov flows. Journal of Modern Dynamics, 2010, 4 (1) : 1-63. doi: 10.3934/jmd.2010.4.1

[8]

Matthieu Porte. Linear response for Dirac observables of Anosov diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1799-1819. doi: 10.3934/dcds.2019078

[9]

Jean-Pierre Conze, Y. Guivarc'h. Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4239-4269. doi: 10.3934/dcds.2013.33.4239

[10]

Brian Marcus and Selim Tuncel. Powers of positive polynomials and codings of Markov chains onto Bernoulli shifts. Electronic Research Announcements, 1999, 5: 91-101.

[11]

Wolfgang Krieger, Kengo Matsumoto. Markov-Dyck shifts, neutral periodic points and topological conjugacy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 1-18. doi: 10.3934/dcds.2019001

[12]

Andrey Gogolev. Smooth conjugacy of Anosov diffeomorphisms on higher-dimensional tori. Journal of Modern Dynamics, 2008, 2 (4) : 645-700. doi: 10.3934/jmd.2008.2.645

[13]

Andrey Gogolev, Misha Guysinsky. $C^1$-differentiable conjugacy of Anosov diffeomorphisms on three dimensional torus. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 183-200. doi: 10.3934/dcds.2008.22.183

[14]

Marc Kesseböhmer, Sabrina Kombrink. A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theory. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 335-352. doi: 10.3934/dcdss.2017016

[15]

Kengo Matsumoto. K-groups of the full group actions on one-sided topological Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3753-3765. doi: 10.3934/dcds.2013.33.3753

[16]

Thomas French. Follower, predecessor, and extender set sequences of $ \beta $-shifts. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4331-4344. doi: 10.3934/dcds.2019175

[17]

John Banks, Thi T. D. Nguyen, Piotr Oprocha, Brett Stanley, Belinda Trotta. Dynamics of spacing shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4207-4232. doi: 10.3934/dcds.2013.33.4207

[18]

John Banks, Piotr Oprocha, Brett Stanley. Transitive sofic spacing shifts. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4743-4764. doi: 10.3934/dcds.2015.35.4743

[19]

Meera G. Mainkar, Cynthia E. Will. Examples of Anosov Lie algebras. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 39-52. doi: 10.3934/dcds.2007.18.39

[20]

Jacek Serafin. A faithful symbolic extension. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1051-1062. doi: 10.3934/cpaa.2012.11.1051

2018 Impact Factor: 0.295

Article outline

Figures and Tables

[Back to Top]