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Local rigidity of certain actions of solvable groups on the boundaries of rank-one symmetric spaces
The orbital equivalence of Bernoulli actions and their Sinai factors
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 |
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.
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 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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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