On asymmetry of the future and the past for limit selfjoinings
Author:
Oleg N. Ageev
Journal:
Proc. Amer. Math. Soc. 131 (2003), 20532062
MSC (2000):
Primary 37Axx, 28D05, 28D15, 20M14, 47B65; Secondary 47A05, 47A15, 47Dxx, 60Gxx
Published electronically:
February 5, 2003
MathSciNet review:
1963750
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let be an offdiagonal joining of a transformation . We construct a nontypical transformation having asymmetry between limit sets of for positive and negative powers of . It follows from a correspondence between subpolymorphisms and positive operators, and from the structure of limit polynomial operators. We apply this technique to find all polynomial operators of degree in the weak closure (in the space of positive operators on ) of powers of Chacon's automorphism and its generalizations.
 1.
D. Rudolph, An example of a measure preserving map with minimal selfjoinings, and applications, J. Analyse Math. 35 (1979), 97122. MR 81e:28011
 2.
M. Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math.(2) 118 (1983), 277313. MR 85k:58063
 3.
A del Junco, M. Rahe, and L. Swanson, Chacon's automorphism has minimal selfjoinings, J. Analyse Math. 37 (1980), 276284. MR 81j:28027
 4.
J. King, The commutant is the weak closure of the powers, for rank1 transformations, Erg. Theory and Dyn. Sys. 6 (1986), 363384. MR 88a:28021
 5.
B. Host, Mixing of all orders and pairwise independent joinings of systems with singular spectrum, Isr. J. Math. 76 (1991), 289298. MR 93k:28022
 6.
A del Junco and D. Rudolph, On ergodic actions whose selfjoinings are graphs, Erg. Theory and Dyn. Sys. 7 (1987), 531557. MR 89e:28029
 7.
A. Vershik, Dynamic theory of growth in groups: entropy, boundaries, examples, Uspekhi Mat. Nauk 55 (2000), 59128; English transl., Russian Math. Surv. 55 (2000), 667733. MR 2001m:37019
 8.
O. Ageev, On ergodic transformations with homogeneous spectrum, J. Dynam. Control Systems 5 (1999), 149152. MR 99m:28034
 9.
V. Ryzhikov, Transformations having homogeneous spectra, J. Dynam. Control Systems 5 (1999), 145148. MR 99m:28038
 10.
A. Katok, Ja. Sinai, and A. Stepin, The theory of dynamical systems and general transformation groups with invariant measure, Mathematical analysis, Vol. 13 (Russian), pp. 129262. (errata insert) Akad. Nauk SSSR VINITI, Moscow, 1975. MR 58:28430
 11.
G. Goodson, A survey of recent results in the spectral theory of ergodic dynamical systems, J. Dynam. Control Systems 5 (1999), 173226. MR 2000f:28021
 12.
E. Robinson, Jr., Ergodic measure preserving transformations with arbitrary finite spectral multiplicities, Invent. Math. 72 (1983), 299314. MR 85a:28014
 13.
O. Ageev, The spectral multiplicity function and geometric representations of interval exchange transformations, Math. Sb. 190 (1999), 328; English transl., Sb. Math. 190 (1999), 128. MR 2000m:28015
 14.
D. Rudolph, Fundamentals of measurable dynamics. Ergodic theory on Lebesgue spaces, The Clarendon Press, Oxford University Press, New York, 1990. MR 92e:28006
 15.
O. Ageev, The spectrum of Cartesian powers of classical automorphisms, Math. Notes 68 (2000), 547551. MR 2001m:37014
 16.
O. Ageev, C. Silva, Genericity of rigidity and multiple recurrence for infinite measure preserving and nonsingular transformations, preprint.
 17.
M. Lemanczyk, B. Host, J.P. Thouvenot, Gaussian automorphisms whose ergodic selfjoinings are Gaussian, Fund. Math. 164 (2000), 253293. MR 2001h:37009
 1.
 D. Rudolph, An example of a measure preserving map with minimal selfjoinings, and applications, J. Analyse Math. 35 (1979), 97122. MR 81e:28011
 2.
 M. Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math.(2) 118 (1983), 277313. MR 85k:58063
 3.
 A del Junco, M. Rahe, and L. Swanson, Chacon's automorphism has minimal selfjoinings, J. Analyse Math. 37 (1980), 276284. MR 81j:28027
 4.
 J. King, The commutant is the weak closure of the powers, for rank1 transformations, Erg. Theory and Dyn. Sys. 6 (1986), 363384. MR 88a:28021
 5.
 B. Host, Mixing of all orders and pairwise independent joinings of systems with singular spectrum, Isr. J. Math. 76 (1991), 289298. MR 93k:28022
 6.
 A del Junco and D. Rudolph, On ergodic actions whose selfjoinings are graphs, Erg. Theory and Dyn. Sys. 7 (1987), 531557. MR 89e:28029
 7.
 A. Vershik, Dynamic theory of growth in groups: entropy, boundaries, examples, Uspekhi Mat. Nauk 55 (2000), 59128; English transl., Russian Math. Surv. 55 (2000), 667733. MR 2001m:37019
 8.
 O. Ageev, On ergodic transformations with homogeneous spectrum, J. Dynam. Control Systems 5 (1999), 149152. MR 99m:28034
 9.
 V. Ryzhikov, Transformations having homogeneous spectra, J. Dynam. Control Systems 5 (1999), 145148. MR 99m:28038
 10.
 A. Katok, Ja. Sinai, and A. Stepin, The theory of dynamical systems and general transformation groups with invariant measure, Mathematical analysis, Vol. 13 (Russian), pp. 129262. (errata insert) Akad. Nauk SSSR VINITI, Moscow, 1975. MR 58:28430
 11.
 G. Goodson, A survey of recent results in the spectral theory of ergodic dynamical systems, J. Dynam. Control Systems 5 (1999), 173226. MR 2000f:28021
 12.
 E. Robinson, Jr., Ergodic measure preserving transformations with arbitrary finite spectral multiplicities, Invent. Math. 72 (1983), 299314. MR 85a:28014
 13.
 O. Ageev, The spectral multiplicity function and geometric representations of interval exchange transformations, Math. Sb. 190 (1999), 328; English transl., Sb. Math. 190 (1999), 128. MR 2000m:28015
 14.
 D. Rudolph, Fundamentals of measurable dynamics. Ergodic theory on Lebesgue spaces, The Clarendon Press, Oxford University Press, New York, 1990. MR 92e:28006
 15.
 O. Ageev, The spectrum of Cartesian powers of classical automorphisms, Math. Notes 68 (2000), 547551. MR 2001m:37014
 16.
 O. Ageev, C. Silva, Genericity of rigidity and multiple recurrence for infinite measure preserving and nonsingular transformations, preprint.
 17.
 M. Lemanczyk, B. Host, J.P. Thouvenot, Gaussian automorphisms whose ergodic selfjoinings are Gaussian, Fund. Math. 164 (2000), 253293. MR 2001h:37009
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
37Axx,
28D05,
28D15,
20M14,
47B65,
47A05,
47A15,
47Dxx,
60Gxx
Retrieve articles in all journals
with MSC (2000):
37Axx,
28D05,
28D15,
20M14,
47B65,
47A05,
47A15,
47Dxx,
60Gxx
Additional Information
Oleg N. Ageev
Affiliation:
Department of Mathematics, Moscow State Technical University, 2nd Baumanscaya St. 5, 105005 Moscow, Russia
Email:
ageev@mx.bmstu.ru
DOI:
http://dx.doi.org/10.1090/S0002993903067960
PII:
S 00029939(03)067960
Keywords:
Joinings,
Chacon's automorphism,
weak operator convergence
Received by editor(s):
April 19, 2001
Published electronically:
February 5, 2003
Additional Notes:
The author was supported in part by the Max Planck Institute of Mathematics, Bonn, and RFBR Grants #1001596107, #990101104
Communicated by:
Michael Handel
Article copyright:
© Copyright 2003
American Mathematical Society
