# American Institute of Mathematical Sciences

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. 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
The construction of the Markov partition
 $c_{M_{t-1}-3}$ $w$ $d_{M_{t-1}}$ $1$ 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] Thomas Barthelmé, Andrey Gogolev. Centralizers of partially hyperbolic diffeomorphisms in dimension 3. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021044 [2] Azmeer Nordin, Mohd Salmi Md Noorani. Counting finite orbits for the flip systems of shifts of finite type. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021046 [3] Ajay Jasra, Kody J. H. Law, Yaxian Xu. Markov chain simulation for multilevel Monte Carlo. Foundations of Data Science, 2021, 3 (1) : 27-47. doi: 10.3934/fods.2021004 [4] Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934/dcdsb.2020186 [5] Kuei-Hu Chang. A novel risk ranking method based on the single valued neutrosophic set. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021065 [6] Hideaki Takagi. Extension of Littlewood's rule to the multi-period static revenue management model with standby customers. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2181-2202. doi: 10.3934/jimo.2020064 [7] Brandy Rapatski, James Yorke. Modeling HIV outbreaks: The male to female prevalence ratio in the core population. Mathematical Biosciences & Engineering, 2009, 6 (1) : 135-143. doi: 10.3934/mbe.2009.6.135 [8] Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243 [9] Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, 2021, 15 (3) : 387-413. doi: 10.3934/ipi.2020073 [10] Liqiang Jin, Yanqing Liu, Yanyan Yin, Kok Lay Teo, Fei Liu. Design of probabilistic $l_2-l_\infty$ filter for uncertain Markov jump systems with partial information of the transition probabilities. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021070 [11] Wen-Bin Yang, Yan-Ling Li, Jianhua Wu, Hai-Xia Li. Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2269-2290. doi: 10.3934/dcdsb.2015.20.2269 [12] Qian Cao, Yongli Cai, Yong Luo. Nonconstant positive solutions to the ratio-dependent predator-prey system with prey-taxis in one dimension. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021095 [13] Kehan Si, Zhenda Xu, Ka Fai Cedric Yiu, Xun Li. Open-loop solvability for mean-field stochastic linear quadratic optimal control problems of Markov regime-switching system. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021074 [14] Prabir Panja, Soovoojeet Jana, Shyamal kumar Mondal. Dynamics of a stage structure prey-predator model with ratio-dependent functional response and anti-predator behavior of adult prey. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 391-405. doi: 10.3934/naco.2020033

2019 Impact Factor: 0.465