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  2021, 17: 145-182. doi: 10.3934/jmd.2021005

The orbital equivalence of Bernoulli actions and their Sinai factors

Zemer Kosloff 1, and  Terry Soo 2,

1. 

Einstein Institute of Mathematics, Hebrew University of Jerusalem, Edmund J. Safra Campus, Givat Ram. Jerusalem, 9190401, Israel
2. 

Department of Statistical Science, University College London, Gower Street, London WC1E 6BT, United Kingdom

ZK: Funded in part by ISF grant No. 1570/17.

Received  May 07, 2020 Revised  December 28, 2020 Published  March 2021

Given a countable amenable group $ G $ and $ \lambda \in (0,1) $, we give an elementary construction of a type-Ⅲ$ _{\lambda} $ Bernoulli group action. In the case where $ G $ is the integers, we show that our nonsingular Bernoulli shifts have independent and identically distributed factors.

Keywords: Krieger types, Bernoulli shifts, factors, amenable group actions.
Mathematics Subject Classification: Primary: 37A40, 37A20; Secondary: 37A35, 60G09.
Citation: Zemer Kosloff, Terry Soo. The orbital equivalence of Bernoulli actions and their Sinai factors. Journal of Modern Dynamics, 2021, 17: 145-182. doi: 10.3934/jmd.2021005
References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, 50, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050.  Google Scholar

[2]

D. Aldous and J. Pitman, On the zero-one law for exchangeable events, Ann. Probab., 7 (1979), 704-723.  doi: 10.1214/aop/1176994992.  Google Scholar

[3]

H. Araki and E. J. Woods, A classification of factors, Publ. Res. Inst. Math. Sci. Ser. A, 4 (1968), 51-130.  doi: 10.2977/prims/1195195263.  Google Scholar

[4]

N. Avraham-Re'em, On absolutely continuous invariant measures and Krieger-type of Markov subshifts, J. Analyse Math., to appear Google Scholar

[5]

T. Berendschot and S. Vaes, Nonsingular Bernoulli actions of arbitrary Krieger type, Anal. PDE, to appear Google Scholar

[6]

M. Björklund, Z. Kosloff and S. Vaes, Ergodicity and type of nonsingular Bernoulli actions, Invent. Math., (2020). doi: 10.1007/s00222-020-01014-0.  Google Scholar

[7]

J. R. Choksi, J. M. 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]

K. L. Chung and W. H. J. Fuchs, On the distribution of values of sums of random variables, Mem. Amer. Math. Soc., 6 (1951), 12 pp.  Google Scholar

[9]

K. L. Chung and D. Ornstein, On the recurrence of sums of random variables, Bull. Amer. Math. Soc., 68 (1962), 30-32.  doi: 10.1090/S0002-9904-1962-10688-0.  Google Scholar

[10]

A. Connes, J. Feldman and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynam. Systems, 1 (1981), 431-450.  doi: 10.1017/S014338570000136X.  Google Scholar

[11]

A. Connes and E. J. Woods, Approximately transitive flows and ITPFI factors, Ergodic Theory Dynam. Systems, 5 (1985), 203-236.  doi: 10.1017/S0143385700002868.  Google Scholar

[12]

A. I. Danilenko, Z. Kosloff and E. Roy, Generic nonsingular Poisson suspension is of type $III_1$., Ergodic Theory Dynam. Systems, to appear, arXiv: 2002.05094. Google Scholar

[13]

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.  doi: 10.1017/etds.2018.14.  Google Scholar

[14]

A. I. Danilenko and C. E. Silva, Ergodic theory: Non-singular transformations, in Mathematics of Complexity and Dynamical Systems, 1–3, Springer, New York, 2012,329–356. doi: 10.1007/978-1-4614-1806-1_22.  Google Scholar

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A. I. Danilenko and C. E. Silva, Ergodic theory: Non-singular transformations, preprint, arXiv: 0803.2424. Google Scholar

[16]

M. Deijfen and R. Meester, Generating stationary random graphs on $\mathbb Z$ with prescribed independent, identically distributed degrees, Adv. in Appl. Probab., 38 (2006), 287-298.  doi: 10.1239/aap/1151337072.  Google Scholar

[17]

A. H. Dooley, I. Klemeš and A. N. Quas, Product and Markov measures of type Ⅲ, J. Austral. Math. Soc. Ser. A, 65 (1998), 84-110.  doi: 10.1017/S1446788700039410.  Google Scholar

[18]

H. A. Dye, On groups of measure preserving transformations. Ⅰ, Amer. J. Math., 81 (1959), 119-159.  doi: 10.2307/2372852.  Google Scholar

[19]

H. A. Dye, On groups of measure preserving transformations. Ⅱ, Amer. J. Math., 85: 551–576, 1963. doi: 10.2307/2373108.  Google Scholar

[20]

E. Fø lner, On groups with full Banach mean value, Math. Scand., 3 (1955), 243-254.  doi: 10.7146/math.scand.a-10442.  Google Scholar

[21]

P. R. Halmos, Invariant measures, Ann. of Math., 48 (1947), 735-754.  doi: 10.2307/1969138.  Google Scholar

[22]

T. Hamachi, On a Bernoulli shift with nonidentical factor measures, Ergodic Theory Dynam. Systems, 1 (1981), 273-283.  doi: 10.1017/S0143385700001255.  Google Scholar

[23]

E. Hewitt and L. J. Savage, Symmetric measures on Cartesian products, Trans. Amer. Math. Soc., 80 (1955), 470-501.  doi: 10.1090/S0002-9947-1955-0076206-8.  Google Scholar

[24]

A. E. Holroyd, R. Pemantle, Y. Peres and O. Schramm, Poisson matching, Ann. Inst. Henri Poincaré Probab. Stat., 45 (2009), 266-287.  doi: 10.1214/08-AIHP170.  Google Scholar

[25]

A. E. Holroyd and Y. Peres, Extra heads and invariant allocations, Ann. Probab., 33 (2005), 31-52.  doi: 10.1214/009117904000000603.  Google Scholar

[26]

A. del Junco, Finitary codes between one-sided Bernoulli shifts, Ergodic Theory Dynam. Systems, 1 (1981), 285-301.  doi: 10.1017/S0143385700001267.  Google Scholar

[27]

A. del Junco, Bernoulli shifts of the same entropy are finitarily and unilaterally isomorphic, Ergodic Theory Dynam. Systems, 10 (1990), 687-715.  doi: 10.1017/S014338570000585X.  Google Scholar

[28]

S. Kakutani, On equivalence of infinite product measures, Ann. of Math., 49 (1948), 214-224.  doi: 10.2307/1969123.  Google Scholar

[29]

S. Kalikow and B. Weiss, Explicit codes for some infinite entropy Bernoulli shifts, Ann. Probab., 20 (1992), 397-402.  doi: 10.1214/aop/1176989933.  Google Scholar

[30]

O. Kallenberg, Foundations of Modern Probability, 2$^{nd}$ edition, Probability and its Applications (New York), Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-4015-8.  Google Scholar

[31]

A. Katok, Fifty years of entropy in dynamics: 1958–2007, J. Mod. Dyn., 1 (2007), 545-596.  doi: 10.3934/jmd.2007.1.545.  Google Scholar

[32]

M. Keane and M. Smorodinsky, A class of finitary codes, Israel J. Math., 26 (1977), 352-371.  doi: 10.1007/BF03007652.  Google Scholar

[33]

M. Keane and M. Smorodinsky, Bernoulli schemes of the same entropy are finitarily isomorphic, Ann. of Math., 109 (1979), 397-406.  doi: 10.2307/1971117.  Google Scholar

[34]

A. N. Kolmogorov, Foundations of the Theory of Probability, Chelsea Publishing Co., New York, 1956.  Google Scholar

[35]

A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, Dokl. Akad. Nauk SSSR (N.S.), 119 (1958), 861-864.   Google Scholar

[36]

Z. Kosloff, On a type $\rm III_1$ Bernoulli shift, Ergodic Theory Dynam. Systems, 31 (2011), 1727-1743.  doi: 10.1017/S0143385710000647.  Google Scholar

[37]

Z. Kosloff, The zero-type property and mixing of Bernoulli shifts, Ergodic Theory Dynam. Systems, 33 (2013), 549-559.  doi: 10.1017/S0143385711001052.  Google Scholar

[38]

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

[39]

Z. Kosloff, On manifolds admitting stable type $\rm III_1$ Anosov diffeomorphisms, J. Mod. Dyn., 13 (2018), 251-270.  doi: 10.3934/jmd.2018020.  Google Scholar

[40]

Z. Kosloff, Conservative Anosov diffeomorphisms of the two torus without an absolutely continuous invariant measure, Ann. Sci. Éc. Norm. Supér. (4), to appear, arXiv: 1410.7707. Google Scholar

[41]

Z. Kosloff and T. Soo, Some factors of nonsingular Bernoulli shifts, arXiv: 2010.04636. Google Scholar

[42]

U. Krengel, Transformations without finite invariant measure have finite strong generators, in Contributions to Ergodic Theory and Probability (Proc. Conf., Ohio State Univ., Columbus, Ohio, 1970), Springer, Berlin, 1970,133–157.  Google Scholar

[43]

W. Krieger, On entropy and generators of measure-preserving transformations, Trans. Amer. Math. Soc., 149 (1970), 453-464.  doi: 10.1090/S0002-9947-1970-0259068-3.  Google Scholar

[44]

W. Krieger, On the Araki-Woods asymptotic ratio set and non-singular transformations of a measure space, in Contributions to Ergodic Theory and Probability (Proc. Conf., Ohio State Univ., Columbus, Ohio, 1970), Lecture Notes in Math., 160, 1970,158–177.  Google Scholar

[45]

W. Krieger, On ergodic flows and the isomorphism of factors, Math. Ann., 223 (1976), 19-70.  doi: 10.1007/BF01360278.  Google Scholar

[46]

D. Maharam, Incompressible transformations, Fund. Math., 56 (1964), 35-50.  doi: 10.4064/fm-56-1-35-50.  Google Scholar

[47]

L. D. Mešalkin, A case of isomorphism of Bernoulli schemes, Dokl. Akad. Nauk SSSR, 128 (1959), 41-44.   Google Scholar

[48]

D. Ornstein, Newton's laws and coin tossing, Notices Amer. Math. Soc., 60 (2013), 450-459.  doi: 10.1090/noti974.  Google Scholar

[49]

D. S. Ornstein, On invariant measures, Bull. Amer. Math. Soc., 66 (1960), 297-300.  doi: 10.1090/S0002-9904-1960-10478-8.  Google Scholar

[50]

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

[51]

D. J. Rudolph and C. E. Silva, Minimal self-joinings for nonsingular transformations, Ergodic Theory Dynam. Systems, 9 (1989), 759-800.  doi: 10.1017/S0143385700005320.  Google Scholar

[52]

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

[53]

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

[54]

J. Sinaĭ, On the concept of entropy for a dynamic system, Dokl. Akad. Nauk SSSR, 124 (1959), 768-771.   Google Scholar

[55]

J. G. Sinaĭ, On a weak isomorphism of transformations with invariant measure, Mat. Sb. (N.S.), 63 (1964), 23-42.   Google Scholar

[56]

T. Soo, Translation-equivariant matchings of coin flips on $\mathbb{Z}^d$, Adv. in Appl. Probab., 42 (2010), 69-82.  doi: 10.1239/aap/1269611144.  Google Scholar

[57]

S. Vaes and J. Wahl, Bernoulli actions of type Ⅲ1 and $L^2$-cohomology, Geom. Funct. Anal., 28 (2018), 518-562.  doi: 10.1007/s00039-018-0438-y.  Google Scholar

show all references

References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, 50, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050.  Google Scholar

[2]

D. Aldous and J. Pitman, On the zero-one law for exchangeable events, Ann. Probab., 7 (1979), 704-723.  doi: 10.1214/aop/1176994992.  Google Scholar

[3]

H. Araki and E. J. Woods, A classification of factors, Publ. Res. Inst. Math. Sci. Ser. A, 4 (1968), 51-130.  doi: 10.2977/prims/1195195263.  Google Scholar

[4]

N. Avraham-Re'em, On absolutely continuous invariant measures and Krieger-type of Markov subshifts, J. Analyse Math., to appear Google Scholar

[5]

T. Berendschot and S. Vaes, Nonsingular Bernoulli actions of arbitrary Krieger type, Anal. PDE, to appear Google Scholar

[6]

M. Björklund, Z. Kosloff and S. Vaes, Ergodicity and type of nonsingular Bernoulli actions, Invent. Math., (2020). doi: 10.1007/s00222-020-01014-0.  Google Scholar

[7]

J. R. Choksi, J. M. 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]

K. L. Chung and W. H. J. Fuchs, On the distribution of values of sums of random variables, Mem. Amer. Math. Soc., 6 (1951), 12 pp.  Google Scholar

[9]

K. L. Chung and D. Ornstein, On the recurrence of sums of random variables, Bull. Amer. Math. Soc., 68 (1962), 30-32.  doi: 10.1090/S0002-9904-1962-10688-0.  Google Scholar

[10]

A. Connes, J. Feldman and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynam. Systems, 1 (1981), 431-450.  doi: 10.1017/S014338570000136X.  Google Scholar

[11]

A. Connes and E. J. Woods, Approximately transitive flows and ITPFI factors, Ergodic Theory Dynam. Systems, 5 (1985), 203-236.  doi: 10.1017/S0143385700002868.  Google Scholar

[12]

A. I. Danilenko, Z. Kosloff and E. Roy, Generic nonsingular Poisson suspension is of type $III_1$., Ergodic Theory Dynam. Systems, to appear, arXiv: 2002.05094. Google Scholar

[13]

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.  doi: 10.1017/etds.2018.14.  Google Scholar

[14]

A. I. Danilenko and C. E. Silva, Ergodic theory: Non-singular transformations, in Mathematics of Complexity and Dynamical Systems, 1–3, Springer, New York, 2012,329–356. doi: 10.1007/978-1-4614-1806-1_22.  Google Scholar

[15]

A. I. Danilenko and C. E. Silva, Ergodic theory: Non-singular transformations, preprint, arXiv: 0803.2424. Google Scholar

[16]

M. Deijfen and R. Meester, Generating stationary random graphs on $\mathbb Z$ with prescribed independent, identically distributed degrees, Adv. in Appl. Probab., 38 (2006), 287-298.  doi: 10.1239/aap/1151337072.  Google Scholar

[17]

A. H. Dooley, I. Klemeš and A. N. Quas, Product and Markov measures of type Ⅲ, J. Austral. Math. Soc. Ser. A, 65 (1998), 84-110.  doi: 10.1017/S1446788700039410.  Google Scholar

[18]

H. A. Dye, On groups of measure preserving transformations. Ⅰ, Amer. J. Math., 81 (1959), 119-159.  doi: 10.2307/2372852.  Google Scholar

[19]

H. A. Dye, On groups of measure preserving transformations. Ⅱ, Amer. J. Math., 85: 551–576, 1963. doi: 10.2307/2373108.  Google Scholar

[20]

E. Fø lner, On groups with full Banach mean value, Math. Scand., 3 (1955), 243-254.  doi: 10.7146/math.scand.a-10442.  Google Scholar

[21]

P. R. Halmos, Invariant measures, Ann. of Math., 48 (1947), 735-754.  doi: 10.2307/1969138.  Google Scholar

[22]

T. Hamachi, On a Bernoulli shift with nonidentical factor measures, Ergodic Theory Dynam. Systems, 1 (1981), 273-283.  doi: 10.1017/S0143385700001255.  Google Scholar

[23]

E. Hewitt and L. J. Savage, Symmetric measures on Cartesian products, Trans. Amer. Math. Soc., 80 (1955), 470-501.  doi: 10.1090/S0002-9947-1955-0076206-8.  Google Scholar

[24]

A. E. Holroyd, R. Pemantle, Y. Peres and O. Schramm, Poisson matching, Ann. Inst. Henri Poincaré Probab. Stat., 45 (2009), 266-287.  doi: 10.1214/08-AIHP170.  Google Scholar

[25]

A. E. Holroyd and Y. Peres, Extra heads and invariant allocations, Ann. Probab., 33 (2005), 31-52.  doi: 10.1214/009117904000000603.  Google Scholar

[26]

A. del Junco, Finitary codes between one-sided Bernoulli shifts, Ergodic Theory Dynam. Systems, 1 (1981), 285-301.  doi: 10.1017/S0143385700001267.  Google Scholar

[27]

A. del Junco, Bernoulli shifts of the same entropy are finitarily and unilaterally isomorphic, Ergodic Theory Dynam. Systems, 10 (1990), 687-715.  doi: 10.1017/S014338570000585X.  Google Scholar

[28]

S. Kakutani, On equivalence of infinite product measures, Ann. of Math., 49 (1948), 214-224.  doi: 10.2307/1969123.  Google Scholar

[29]

S. Kalikow and B. Weiss, Explicit codes for some infinite entropy Bernoulli shifts, Ann. Probab., 20 (1992), 397-402.  doi: 10.1214/aop/1176989933.  Google Scholar

[30]

O. Kallenberg, Foundations of Modern Probability, 2$^{nd}$ edition, Probability and its Applications (New York), Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-4015-8.  Google Scholar

[31]

A. Katok, Fifty years of entropy in dynamics: 1958–2007, J. Mod. Dyn., 1 (2007), 545-596.  doi: 10.3934/jmd.2007.1.545.  Google Scholar

[32]

M. Keane and M. Smorodinsky, A class of finitary codes, Israel J. Math., 26 (1977), 352-371.  doi: 10.1007/BF03007652.  Google Scholar

[33]

M. Keane and M. Smorodinsky, Bernoulli schemes of the same entropy are finitarily isomorphic, Ann. of Math., 109 (1979), 397-406.  doi: 10.2307/1971117.  Google Scholar

[34]

A. N. Kolmogorov, Foundations of the Theory of Probability, Chelsea Publishing Co., New York, 1956.  Google Scholar

[35]

A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, Dokl. Akad. Nauk SSSR (N.S.), 119 (1958), 861-864.   Google Scholar

[36]

Z. Kosloff, On a type $\rm III_1$ Bernoulli shift, Ergodic Theory Dynam. Systems, 31 (2011), 1727-1743.  doi: 10.1017/S0143385710000647.  Google Scholar

[37]

Z. Kosloff, The zero-type property and mixing of Bernoulli shifts, Ergodic Theory Dynam. Systems, 33 (2013), 549-559.  doi: 10.1017/S0143385711001052.  Google Scholar

[38]

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

[39]

Z. Kosloff, On manifolds admitting stable type $\rm III_1$ Anosov diffeomorphisms, J. Mod. Dyn., 13 (2018), 251-270.  doi: 10.3934/jmd.2018020.  Google Scholar

[40]

Z. Kosloff, Conservative Anosov diffeomorphisms of the two torus without an absolutely continuous invariant measure, Ann. Sci. Éc. Norm. Supér. (4), to appear, arXiv: 1410.7707. Google Scholar

[41]

Z. Kosloff and T. Soo, Some factors of nonsingular Bernoulli shifts, arXiv: 2010.04636. Google Scholar

[42]

U. Krengel, Transformations without finite invariant measure have finite strong generators, in Contributions to Ergodic Theory and Probability (Proc. Conf., Ohio State Univ., Columbus, Ohio, 1970), Springer, Berlin, 1970,133–157.  Google Scholar

[43]

W. Krieger, On entropy and generators of measure-preserving transformations, Trans. Amer. Math. Soc., 149 (1970), 453-464.  doi: 10.1090/S0002-9947-1970-0259068-3.  Google Scholar

[44]

W. Krieger, On the Araki-Woods asymptotic ratio set and non-singular transformations of a measure space, in Contributions to Ergodic Theory and Probability (Proc. Conf., Ohio State Univ., Columbus, Ohio, 1970), Lecture Notes in Math., 160, 1970,158–177.  Google Scholar

[45]

W. Krieger, On ergodic flows and the isomorphism of factors, Math. Ann., 223 (1976), 19-70.  doi: 10.1007/BF01360278.  Google Scholar

[46]

D. Maharam, Incompressible transformations, Fund. Math., 56 (1964), 35-50.  doi: 10.4064/fm-56-1-35-50.  Google Scholar

[47]

L. D. Mešalkin, A case of isomorphism of Bernoulli schemes, Dokl. Akad. Nauk SSSR, 128 (1959), 41-44.   Google Scholar

[48]

D. Ornstein, Newton's laws and coin tossing, Notices Amer. Math. Soc., 60 (2013), 450-459.  doi: 10.1090/noti974.  Google Scholar

[49]

D. S. Ornstein, On invariant measures, Bull. Amer. Math. Soc., 66 (1960), 297-300.  doi: 10.1090/S0002-9904-1960-10478-8.  Google Scholar

[50]

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

[51]

D. J. Rudolph and C. E. Silva, Minimal self-joinings for nonsingular transformations, Ergodic Theory Dynam. Systems, 9 (1989), 759-800.  doi: 10.1017/S0143385700005320.  Google Scholar

[52]

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

[53]

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

[54]

J. Sinaĭ, On the concept of entropy for a dynamic system, Dokl. Akad. Nauk SSSR, 124 (1959), 768-771.   Google Scholar

[55]

J. G. Sinaĭ, On a weak isomorphism of transformations with invariant measure, Mat. Sb. (N.S.), 63 (1964), 23-42.   Google Scholar

[56]

T. Soo, Translation-equivariant matchings of coin flips on $\mathbb{Z}^d$, Adv. in Appl. Probab., 42 (2010), 69-82.  doi: 10.1239/aap/1269611144.  Google Scholar

[57]

S. Vaes and J. Wahl, Bernoulli actions of type Ⅲ1 and $L^2$-cohomology, Geom. Funct. Anal., 28 (2018), 518-562.  doi: 10.1007/s00039-018-0438-y.  Google Scholar

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