Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On asymmetry of the future and the past for limit self-joinings

Author: Oleg N. Ageev
Journal: Proc. Amer. Math. Soc. 131 (2003), 2053-2062
MSC (2000): Primary 37Axx, 28D05, 28D15, 20M14, 47B65; Secondary 47A05, 47A15, 47Dxx, 60Gxx
Published electronically: February 5, 2003
MathSciNet review: 1963750
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\triangle_T$ be an off-diagonal joining of a transformation $T$. We construct a non-typical transformation having asymmetry between limit sets of $\triangle_{T^n}$ for positive and negative powers of $T$. 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 $1$ in the weak closure (in the space of positive operators on $L_2$) of powers of Chacon's automorphism and its generalizations.

References [Enhancements On Off] (What's this?)

  • 1. D. Rudolph, An example of a measure preserving map with minimal self-joinings, and applications, J. Analyse Math. 35 (1979), 97-122. MR 81e:28011
  • 2. M. Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math.(2) 118 (1983), 277-313. MR 85k:58063
  • 3. A del Junco, M. Rahe, and L. Swanson, Chacon's automorphism has minimal self-joinings, J. Analyse Math. 37 (1980), 276-284. MR 81j:28027
  • 4. J. King, The commutant is the weak closure of the powers, for rank-1 transformations, Erg. Theory and Dyn. Sys. 6 (1986), 363-384. MR 88a:28021
  • 5. B. Host, Mixing of all orders and pairwise independent joinings of systems with singular spectrum, Isr. J. Math. 76 (1991), 289-298. MR 93k:28022
  • 6. A del Junco and D. Rudolph, On ergodic actions whose self-joinings are graphs, Erg. Theory and Dyn. Sys. 7 (1987), 531-557. MR 89e:28029
  • 7. A. Vershik, Dynamic theory of growth in groups: entropy, boundaries, examples, Uspekhi Mat. Nauk 55 (2000), 59-128; English transl., Russian Math. Surv. 55 (2000), 667-733. MR 2001m:37019
  • 8. O. Ageev, On ergodic transformations with homogeneous spectrum, J. Dynam. Control Systems 5 (1999), 149-152. MR 99m:28034
  • 9. V. Ryzhikov, Transformations having homogeneous spectra, J. Dynam. Control Systems 5 (1999), 145-148. 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. 129-262. (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), 173-226. MR 2000f:28021
  • 12. E. Robinson, Jr., Ergodic measure preserving transformations with arbitrary finite spectral multiplicities, Invent. Math. 72 (1983), 299-314. MR 85a:28014
  • 13. O. Ageev, The spectral multiplicity function and geometric representations of interval exchange transformations, Math. Sb. 190 (1999), 3-28; English transl., Sb. Math. 190 (1999), 1-28. 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), 547-551. 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 self-joinings are Gaussian, Fund. Math. 164 (2000), 253-293. 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

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 #100-15-96107, #99-01-01104
Communicated by: Michael Handel
Article copyright: © Copyright 2003 American Mathematical Society