Sankhya: The Indian Journal of Statistics

2000, Volume 62, Series A, Pt. 3, pp.307--317

JOINING PROPERTIES OF ERGODIC DYNAMICAL SYSTEMS HAVING SIMPLE SPECTRUM

By

GEOFFREY R. GOODSON, *Towson University, MD, USA*

*SUMMARY. *It was shown in Goodson (1995) that
if *T* is an ergodic automorphism having simple spectrum and
defined on a standard Borel probability space for which *T* and
*T*^{ -1} are isomorphic, i.e., there is an automorphism *S*
satisfying *ST*=*T*^{-1}*S*, then *S*^{ 2}=*I*, the identity automorphism.
If *T* and *T*^{-1} are not isomorphic, we can no longer consider
conjugations but rather joinings of *T* and *T*^{-1}. It is shown
that if *P* is a joining (Markov intertwining) between *T* and
*T*^{-1}, then *P* is self--adjoint. Applications of the methods
are given to provide new proofs of some well know results.

*AMS (1991) subject classification.* 28D05, 37A30,
47A35.

*Key words and phrases. * Simple spectrum, joining,
Markov intertwining.