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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[11]

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

[12]

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

[13]

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

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[22]

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

[23]

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

show all references

References:
[1]

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

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[11]

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

[12]

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

[13]

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

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[22]

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

[23]

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

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
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