action on a two torus

Author:
Kyewon Koh Park

Journal:
Proc. Amer. Math. Soc. **114** (1992), 955-963

MSC:
Primary 58F11; Secondary 28D15, 58F17

MathSciNet review:
1059631

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the group action on a two torus with Lebesgue measure. We show that any measure-preserving transformation that commutes with the group action is the action of a matrix of the form , where is an integer. We also show that all factors come from these commuting transformations. Finally we show that the set of self-joinings consists of the product measure and the measures sitting on the graphs , and . We provide an example whose self-joinings consist only of the product measure and the diagonal measure.

**[JRu]**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**[JRS]**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**[F]**Harry Furstenberg,*Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation*, Math. Systems Theory**1**(1967), 1–49. MR**0213508****[K]**Yitzhak Katznelson,*Ergodic automorphisms of 𝑇ⁿ are Bernoulli shifts*, Israel J. Math.**10**(1971), 186–195. MR**0294602****[PRo]**K. Park and A. Robinson,*The joinings within a class of**-actions*, J. Analyse Math. (to appear).**[R]**M. Ratner,*Joinings of horocycle flows*, preprint.**[Ru]**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**[V]**William A. Veech,*A criterion for a process to be prime*, Monatsh. Math.**94**(1982), no. 4, 335–341. MR**685378**, 10.1007/BF01667386**[Z]**Robert J. Zimmer,*Ergodic theory and semisimple groups*, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. MR**776417**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
58F11,
28D15,
58F17

Retrieve articles in all journals with MSC: 58F11, 28D15, 58F17

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1992-1059631-6

Article copyright:
© Copyright 1992
American Mathematical Society