The orbital equivalence of Bernoulli actions and their Sinai factors

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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.
[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.
[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.
[4]	N. Avraham-Re'em, On absolutely continuous invariant measures and Krieger-type of Markov subshifts, J. Analyse Math., to appear
[5]	T. Berendschot and S. Vaes, Nonsingular Bernoulli actions of arbitrary Krieger type, Anal. PDE, to appear
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	A. I. Danilenko and C. E. Silva, Ergodic theory: Non-singular transformations, preprint, arXiv: 0803.2424.
[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.
[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.
[18]	H. A. Dye, On groups of measure preserving transformations. Ⅰ, Amer. J. Math., 81 (1959), 119-159. doi: 10.2307/2372852.
[19]	H. A. Dye, On groups of measure preserving transformations. Ⅱ, Amer. J. Math., 85: 551–576, 1963. doi: 10.2307/2373108.
[20]	E. Fø lner, On groups with full Banach mean value, Math. Scand., 3 (1955), 243-254. doi: 10.7146/math.scand.a-10442.
[21]	P. R. Halmos, Invariant measures, Ann. of Math., 48 (1947), 735-754. doi: 10.2307/1969138.
[22]	T. Hamachi, On a Bernoulli shift with nonidentical factor measures, Ergodic Theory Dynam. Systems, 1 (1981), 273-283. doi: 10.1017/S0143385700001255.
[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.
[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.
[25]	A. E. Holroyd and Y. Peres, Extra heads and invariant allocations, Ann. Probab., 33 (2005), 31-52. doi: 10.1214/009117904000000603.
[26]	A. del Junco, Finitary codes between one-sided Bernoulli shifts, Ergodic Theory Dynam. Systems, 1 (1981), 285-301. doi: 10.1017/S0143385700001267.
[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.
[28]	S. Kakutani, On equivalence of infinite product measures, Ann. of Math., 49 (1948), 214-224. doi: 10.2307/1969123.
[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.
[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.
[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.
[32]	M. Keane and M. Smorodinsky, A class of finitary codes, Israel J. Math., 26 (1977), 352-371. doi: 10.1007/BF03007652.
[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.
[34]	A. N. Kolmogorov, Foundations of the Theory of Probability, Chelsea Publishing Co., New York, 1956.
[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.
[36]	Z. Kosloff, On a type $\rm III_1$ Bernoulli shift, Ergodic Theory Dynam. Systems, 31 (2011), 1727-1743. doi: 10.1017/S0143385710000647.
[37]	Z. Kosloff, The zero-type property and mixing of Bernoulli shifts, Ergodic Theory Dynam. Systems, 33 (2013), 549-559. doi: 10.1017/S0143385711001052.
[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.
[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.
[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.
[41]	Z. Kosloff and T. Soo, Some factors of nonsingular Bernoulli shifts, arXiv: 2010.04636.
[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.
[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.
[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.
[45]	W. Krieger, On ergodic flows and the isomorphism of factors, Math. Ann., 223 (1976), 19-70. doi: 10.1007/BF01360278.
[46]	D. Maharam, Incompressible transformations, Fund. Math., 56 (1964), 35-50. doi: 10.4064/fm-56-1-35-50.
[47]	L. D. Mešalkin, A case of isomorphism of Bernoulli schemes, Dokl. Akad. Nauk SSSR, 128 (1959), 41-44.
[48]	D. Ornstein, Newton's laws and coin tossing, Notices Amer. Math. Soc., 60 (2013), 450-459. doi: 10.1090/noti974.
[49]	D. S. Ornstein, On invariant measures, Bull. Amer. Math. Soc., 66 (1960), 297-300. doi: 10.1090/S0002-9904-1960-10478-8.
[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.
[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.
[52]	K. Schmidt, Cocycles on Ergodic Transformation Groups, Macmillan Lectures in Mathematics, 1, Macmillan Company of India, Ltd., Delhi, 1977.
[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.
[54]	J. Sinaĭ, On the concept of entropy for a dynamic system, Dokl. Akad. Nauk SSSR, 124 (1959), 768-771.
[55]	J. G. Sinaĭ, On a weak isomorphism of transformations with invariant measure, Mat. Sb. (N.S.), 63 (1964), 23-42.
[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.
[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.

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.
[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.
[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.
[4]	N. Avraham-Re'em, On absolutely continuous invariant measures and Krieger-type of Markov subshifts, J. Analyse Math., to appear
[5]	T. Berendschot and S. Vaes, Nonsingular Bernoulli actions of arbitrary Krieger type, Anal. PDE, to appear
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	A. I. Danilenko and C. E. Silva, Ergodic theory: Non-singular transformations, preprint, arXiv: 0803.2424.
[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.
[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.
[18]	H. A. Dye, On groups of measure preserving transformations. Ⅰ, Amer. J. Math., 81 (1959), 119-159. doi: 10.2307/2372852.
[19]	H. A. Dye, On groups of measure preserving transformations. Ⅱ, Amer. J. Math., 85: 551–576, 1963. doi: 10.2307/2373108.
[20]	E. Fø lner, On groups with full Banach mean value, Math. Scand., 3 (1955), 243-254. doi: 10.7146/math.scand.a-10442.
[21]	P. R. Halmos, Invariant measures, Ann. of Math., 48 (1947), 735-754. doi: 10.2307/1969138.
[22]	T. Hamachi, On a Bernoulli shift with nonidentical factor measures, Ergodic Theory Dynam. Systems, 1 (1981), 273-283. doi: 10.1017/S0143385700001255.
[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.
[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.
[25]	A. E. Holroyd and Y. Peres, Extra heads and invariant allocations, Ann. Probab., 33 (2005), 31-52. doi: 10.1214/009117904000000603.
[26]	A. del Junco, Finitary codes between one-sided Bernoulli shifts, Ergodic Theory Dynam. Systems, 1 (1981), 285-301. doi: 10.1017/S0143385700001267.
[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.
[28]	S. Kakutani, On equivalence of infinite product measures, Ann. of Math., 49 (1948), 214-224. doi: 10.2307/1969123.
[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.
[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.
[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.
[32]	M. Keane and M. Smorodinsky, A class of finitary codes, Israel J. Math., 26 (1977), 352-371. doi: 10.1007/BF03007652.
[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.
[34]	A. N. Kolmogorov, Foundations of the Theory of Probability, Chelsea Publishing Co., New York, 1956.
[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.
[36]	Z. Kosloff, On a type $\rm III_1$ Bernoulli shift, Ergodic Theory Dynam. Systems, 31 (2011), 1727-1743. doi: 10.1017/S0143385710000647.
[37]	Z. Kosloff, The zero-type property and mixing of Bernoulli shifts, Ergodic Theory Dynam. Systems, 33 (2013), 549-559. doi: 10.1017/S0143385711001052.
[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.
[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.
[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.
[41]	Z. Kosloff and T. Soo, Some factors of nonsingular Bernoulli shifts, arXiv: 2010.04636.
[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.
[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.
[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.
[45]	W. Krieger, On ergodic flows and the isomorphism of factors, Math. Ann., 223 (1976), 19-70. doi: 10.1007/BF01360278.
[46]	D. Maharam, Incompressible transformations, Fund. Math., 56 (1964), 35-50. doi: 10.4064/fm-56-1-35-50.
[47]	L. D. Mešalkin, A case of isomorphism of Bernoulli schemes, Dokl. Akad. Nauk SSSR, 128 (1959), 41-44.
[48]	D. Ornstein, Newton's laws and coin tossing, Notices Amer. Math. Soc., 60 (2013), 450-459. doi: 10.1090/noti974.
[49]	D. S. Ornstein, On invariant measures, Bull. Amer. Math. Soc., 66 (1960), 297-300. doi: 10.1090/S0002-9904-1960-10478-8.
[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.
[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.
[52]	K. Schmidt, Cocycles on Ergodic Transformation Groups, Macmillan Lectures in Mathematics, 1, Macmillan Company of India, Ltd., Delhi, 1977.
[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.
[54]	J. Sinaĭ, On the concept of entropy for a dynamic system, Dokl. Akad. Nauk SSSR, 124 (1959), 768-771.
[55]	J. G. Sinaĭ, On a weak isomorphism of transformations with invariant measure, Mat. Sb. (N.S.), 63 (1964), 23-42.
[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.
[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.

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