On absolutely continuous invariant measures and Krieger-type of Markov subshifts

Nachi Avraham-Re’em1
1The Hebrew university of Jerusalem

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Tài liệu tham khảo

J. Aaronson, An Introduction to Infinite Ergodic Theory, American Mathematical Society, Providence, RI, 1997.

M. Björklund and Z. Kosloff, Bernoulli actions of amenable groups with weakly mixing Maharam extensions, arXiv:1808.05991 [math.DS]

M. Björklund, Z. Kosloff and S. Vaes, Ergodicity and type of nonsingular Bernoulli actions, Invent. Math. 224 (2020), 573–625.

D. Blackwell and D. Freedman, The tail σ-field of a Markov chain and a theorem of Orey, Ann. Math. Stat. 35 (1964), 1291–1295.

F. Blanchard, E. Glasner, S. Kolyada and A. Maass, On Li-Yorke pairs, J. Reine Angew. Math. 547 (2002), 51–68.

L. Bowen, The type and stable type of the boundary of a Gromov hyperbolic group, Geom. Dedicata 172 (2014), 363–386.

J. Choksi, J. Hawkins and V. Prasad, Abelian cocycles for nonsingular ergodic transformations and the genericity of type III1transformations, Monatsh. Math. 103 (1987), 187–205.

N.-P. Chung and H. Li, Homoclinic groups, IE groups, and expansive algebraic actions, Invent. math. 199 (2015), 805–858.

H. Cohn, On a paper by Doeblin on non-homogeneous Markov chains, Adv. in Appl. Probab. 13, 2 (1981), 388–401.

Y. Coudene, The Hopf argument, J. Mod. Dyn. 1 (2007), 147.

A. Danilenko, Weak mixing for nonsingular Bernoulli actions of countable amenable groups, Proc. Amer. Math. Soc. 147 (2019), 4439–4450.

A. I. Danilenko and M. Lemańczyk, K-property for Maharam extensions of non-singular Bernoulli and Markov shifts, Ergodic Theory Dynam. Systems 39 (2019), 3292–3321.

A. I. Danilenko and C. E. Silva, Ergodic theory: Nonsingular transformations, in Mathematics of Complexity and Dynamical Systems, Springer, New York, 2012, pp. 329–356.

R. L. Dobrushin, Central limit theorem for nonstationary Markov chains. I, Theory Probab. Appl. 1 (1956), 65–80.

J. Feldman and C. C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), 289–324.

H.-O. Georgii, Gibbs Measures and Phase Transitions, Walter de Gruyter, Berlin-New York, 2011.

P. Hall and C. C. Heyde, Martingale Limit Theory and its Application, Academic Press, New York, 2014.

P. R. Halmos, Invariant measures, Ann. of Math. (2) 48 (1947), 735–754.

T. Hamachi, Ergodic Groups of Automorphisms and Krieger’s Theorems, Keio University, Yokohama, 1981.

T. Hamachi, On a Bernoulli shift with nonidentical factor measures, Ergodic Theory Dynam. Systems 1 (1981), 273–283 (1982).

T. Hamachi, Y. Oka and M. Osikawa, Flows associated with ergodic non-singular transformation groups, Publ. Res. Inst. Math. Sci. 11 (1975), 31–50.

S. Kakutani, On equivalence of infinite product measures, Ann. of Math. (2) (1948), 214–224.

Y. Katznelson and B. Weiss, The classification of non-singular actions, revisited, Ergodic Theory Dynam. Systems 11 (1991), 333–348.

Z. Kosloff, Conservative Anosov diffeomorphisms of $${\mathbb{T}^2}$$ without an absolutely continuous invariant measure, Ann. Sci. Éc. Norm. Supér. (4) 54 (2021), 69–131.

Z. Kosloff, On the K property for Maharam extensions of Bernoulli shifts and a question of Krengel, Israel J. Math. 199 (2014), 485–506.

Z. Kosloff, On manifolds admitting stable type Anosov diffeomorphisms, J. Mod. Dyn. 13 (2018), 251–270.

Z. Kosloff, Proving ergodicity via divergence of time averages, StudiaMath. 248 (2019), 191–215.

U. Krengel, Transformations without finite invariant measure have finite strong generators, in Contributions to Ergodic Theory and Probability, Springer, Berlin-Heidelberg, 1970, pp. 133–157.

W. Krieger, On the Araki-Woods asymptotic ratio set and non-singular transformations of a measure space, in Contributions to Ergodic Theory and Probability, Springer, Berlin-Heidelberg, 1970, pp. 158–177.

R. LePage and V. Mandrekar, On likelihood ratios of measures given by Markov chains, Proc. Amer. Math. Soc. 52 (1975), 377–380.

A. A. Lodkin, Absolute continuity of measures corresponding to Markov processes with discrete time, Theory Probab. Appl. 16 (1971), 690–694.

D. Maharam, Incompressible transformations, Fund. Math. 56 (1964), 35–50.

O. Sarig, Lecture Notes on Ergodic Theory, available at http://www.weizmann.ac.il/math/sarigo/

K. Schmidt, Cocycles on Ergodic Transformation Groups, Macmillan India, Delhi, 1977.

S. Sethuraman and S. Varadhan, A martingale proof of Dobrushin’s theorem for non-homogeneous Markov chains, Electron. J. Probab. 10 (2005), 1221–1235.

A. N. Shiryaev, Absolute continuity and singularity of probability measures in functional spaces, in Proceedings of the International Congress of Mathematicians, Helsinki, World Scientific, 1978, pp. 209–225.

A. N. Shiryaev, Probability, Springer, New York, 2013.

C. E. Silva and P. Thieullen, A skew product entropy for nonsingular transformations, J. Lond. Math. Soc. (2) 52 (1995), 497–516.

S. Vaes and J. Wahl, Bernoulli actions of type III1 and l 2-cohomology. Geom. Funct. Anal. 28 (2018), 518–562.

L. Wen and Y. Weiguo, An extension of Shannon-Mcmillan theorem and some limit properties for nonhomogeneous Markov chains. Stochastic Process. Appl. 61 (1996), 129–145.

L. Wen and Y. Weiguo, The asymptotic equipartition property for mth-order nonhomogeneous markov information sources, IEEE Trans. Inform. Theory 50 (2004), 3326–3330.