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

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

Zemer Kosloff

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.

Keywords: Ratio set, Markov shifts, Anosov diffeomorphisms, Maharam extension.
Mathematics Subject Classification: Primary: 37A40, 37C40; Secondary: 37A20.
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. Aaronson, M. 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. Aaronson, H. 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. Choksi, J. 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. Aaronson, M. 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. Aaronson, H. 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. Choksi, J. 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
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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
Table Options

Download as excel

$c_{M_{t-1}-3}$ $w$ $d_{M_{t-1}}$
$1$ or $2$ $11$ $1$ or $3$
1 or 2 13 2
3 21 1 or 3
3 23 2
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