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 Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be an off-diagonal joining of a transformation . We construct a non-typical 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.**Daniel J. Rudolph,*An example of a measure preserving map with minimal self-joinings, and applications*, J. Analyse Math.**35**(1979), 97–122. MR**555301**, 10.1007/BF02791063**2.**Marina Ratner,*Horocycle flows, joinings and rigidity of products*, Ann. of Math. (2)**118**(1983), no. 2, 277–313. MR**717825**, 10.2307/2007030**3.**A. del Junco, M. Rahe, and L. Swanson,*Chacon’s automorphism has minimal self-joinings*, J. Analyse Math.**37**(1980), 276–284. MR**583640**, 10.1007/BF02797688**4.**Jonathan King,*The commutant is the weak closure of the powers, for rank-1 transformations*, Ergodic Theory Dynam. Systems**6**(1986), no. 3, 363–384. MR**863200**, 10.1017/S0143385700003552**5.**Bernard Host,*Mixing of all orders and pairwise independent joinings of systems with singular spectrum*, Israel J. Math.**76**(1991), no. 3, 289–298. MR**1177346**, 10.1007/BF02773866**6.**A. del Junco and D. Rudolph,*On ergodic actions whose self-joinings are graphs*, Ergodic Theory Dynam. Systems**7**(1987), no. 4, 531–557. MR**922364**, 10.1017/S0143385700004193**7.**A. M. Vershik,*Dynamic theory of growth in groups: entropy, boundaries, examples*, Uspekhi Mat. Nauk**55**(2000), no. 4(334), 59–128 (Russian, with Russian summary); English transl., Russian Math. Surveys**55**(2000), no. 4, 667–733. MR**1786730**, 10.1070/rm2000v055n04ABEH000314**8.**O. N. Ageev,*On ergodic transformations with homogeneous spectrum*, J. Dynam. Control Systems**5**(1999), no. 1, 149–152. MR**1680995**, 10.1023/A:1021701019156**9.**V. V. Ryzhikov,*Transformations having homogeneous spectra*, J. Dynam. Control Systems**5**(1999), no. 1, 145–148. MR**1680999**, 10.1023/A:1021748902318**10.**A. B. Katok, Ja. G. Sinaĭ, and A. M. Stepin,*The theory of dynamical systems and general transformation groups with invariant measure*, Mathematical analysis, Vol. 13 (Russian), Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow, 1975, pp. 129–262. (errata insert) (Russian). MR**0584389****11.**G. R. Goodson,*A survey of recent results in the spectral theory of ergodic dynamical systems*, J. Dynam. Control Systems**5**(1999), no. 2, 173–226. MR**1693318**, 10.1023/A:1021726902801**12.**E. A. Robinson Jr.,*Ergodic measure preserving transformations with arbitrary finite spectral multiplicities*, Invent. Math.**72**(1983), no. 2, 299–314. MR**700773**, 10.1007/BF01389325**13.**O. N. Ageev,*The spectral multiplicity function and geometric representations of interval exchange transformations*, Mat. Sb.**190**(1999), no. 1, 3–28 (Russian, with Russian summary); English transl., Sb. Math.**190**(1999), no. 1-2, 1–28. MR**1701878**, 10.1070/SM1999v190n01ABEH000376**14.**Daniel J. Rudolph,*Fundamentals of measurable dynamics*, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1990. Ergodic theory on Lebesgue spaces. MR**1086631****15.**O. N. Ageev,*On the spectrum of Cartesian powers of classical automorphisms*, Mat. Zametki**68**(2000), no. 5, 643–647 (Russian, with Russian summary); English transl., Math. Notes**68**(2000), no. 5-6, 547–551. MR**1835446**, 10.1023/A:1026698921311**16.**O. Ageev, C. Silva,*Genericity of rigidity and multiple recurrence for infinite measure preserving and nonsingular transformations*, preprint.**17.**M. Lemańczyk, F. Parreau, and J.-P. Thouvenot,*Gaussian automorphisms whose ergodic self-joinings are Gaussian*, Fund. Math.**164**(2000), no. 3, 253–293. MR**1784644**

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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/S0002-9939-03-06796-0

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