The maximal abelian subextension determines weak mixing for group extensions

Author:
E. Arthur Robinson

Journal:
Proc. Amer. Math. Soc. **114** (1992), 443-450

MSC:
Primary 28D05; Secondary 22D40

MathSciNet review:
1062835

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Weak mixing is an important property for group extensions because its absence is the principal obstruction to lifting a large number of stronger mixing properties. Whether or not an extension is weakly mixing can be determined by studying the sub-extension corresponding to the quotient by the commutator subgroup. Several conditions equivalent to weak mixing are given. In particular, an extension by any group with no abelian factors (for example any nonabelian simple group) is automatically weak mixing if it is ergodic. The proof uses spectral multiplicity theory.

**[JP]**Roger Jones and William Parry,*Compact abelian group extensions of dynamical systems. II*, Compositio Math.**25**(1972), 135–147. MR**0338318****[KN]**H. B. Keynes and D. Newton,*Ergodicity in**extensions*, Global Theory of Dynamical Systems- Proceedings, Northwestern 1979, Lecture Notes in Math., vol. 819, Springer, Berlin.**[M]**Sidney A. Morris,*Pontryagin duality and the structure of locally compact abelian groups*, Cambridge University Press, Cambridge-New York-Melbourne, 1977. London Mathematical Society Lecture Note Series, No. 29. MR**0442141****[R1]**E. Arthur Robinson Jr.,*Ergodic properties that lift to compact group extensions*, Proc. Amer. Math. Soc.**102**(1988), no. 1, 61–67. MR**915717**, 10.1090/S0002-9939-1988-0915717-4**[R2]**E. Arthur Robinson Jr.,*Nonabelian extensions have nonsimple spectrum*, Compositio Math.**65**(1988), no. 2, 155–170. MR**932641****[Ru1]**Daniel J. Rudolph,*𝑘-fold mixing lifts to weakly mixing isometric extensions*, Ergodic Theory Dynam. Systems**5**(1985), no. 3, 445–447. MR**805841**, 10.1017/S0143385700003060**[Ru2]**Daniel J. Rudolph,*Classifying the isometric extensions of a Bernoulli shift*, J. Analyse Math.**34**(1978), 36–60 (1979). MR**531270**, 10.1007/BF02790007**[Z1]**Robert J. Zimmer,*Extensions of ergodic group actions*, Illinois J. Math.**20**(1976), no. 3, 373–409. MR**0409770****[Z2]**Robert J. Zimmer,*Compact nilmanifold extensions of ergodic actions*, Trans. Amer. Math. Soc.**223**(1976), 397–406. MR**0422584**, 10.1090/S0002-9947-1976-0422584-0

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
28D05,
22D40

Retrieve articles in all journals with MSC: 28D05, 22D40

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1992-1062835-X

Article copyright:
© Copyright 1992
American Mathematical Society