Quasi-disjointness in ergodic theory

Author:
Kenneth Berg

Journal:
Trans. Amer. Math. Soc. **162** (1971), 71-87

MSC:
Primary 28.70

DOI:
https://doi.org/10.1090/S0002-9947-1971-0284563-1

MathSciNet review:
0284563

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Abstract: We define and study a relationship, quasi-disjointness, between ergodic processes. A process is a measure-preserving transformation of a measure space onto itself, and ergodicity means that the space cannot be written as a disjoint union of invariant pieces, unless one of the pieces is of zero measure. We restrict our attention to spaces of total measure one which also satisfy additional regularity properties. In particular, the associated Hilbert space of square-summable functions is separable. A simple class of examples is given by translation by a fixed element on a compact Abelian metrizable group, such processes being known as Kronecker processes. We introduce the notion of a maximal common Kronecker factor (or quotient) process for two processes. Quasi-disjointness is a notion tied to the homomorphisms from two processes into their maximal common Kronecker factor, and reduces to a previous notion, disjointness, when that factor is trivial. We show that a substantial class of processes, the Weyl processes, are quasi-disjoint from every ergodic process. As a corollary, we show that a Weyl process and an ergodic process are disjoint if and only if they have no nontrivial Kronecker factor in common, or, equivalently, if they form an ergodic product. We give an example which suggests an analogous theory could be constructed in topological dynamics.

**[1]**S. K. Berberian,*Notes on spectral theory*, Van Nostrand Math. Studies, no. 5, Van Nostrand, Princeton, N. J., 1966. MR**32**#8170. MR**0190760 (32:8170)****[2]**J. R. Blum and D. L. Hanson,*On invariant probability measures*. I, Pacific J. Math.**10**(1960), 1125-1129. MR**24**#A3260.**[3]**N. Bourbaki,*Les structures fondamentales de l'analyse*. Livre VI:*Intégration*, Actualités Sci. Indust., no. 1175, Hermann, Paris, 1952. MR**14**, 960. MR**0054691 (14:960h)****[4]**H. Furstenberg,*Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation*, Math. Systems Theory**1**(1967), 1-49. MR**35**#4369. MR**0213508 (35:4369)****[5]**F. J. Hahn,*On affine transformations of compact abelian groups*, Amer. J. Math.**85**(1963), 428-446. MR**27**#5889. MR**0155956 (27:5889)****[6]**-,*Minimal dynamical systems with quasi-discrete spectrum*, J. London Math. Soc.**40**(1965), 309-323. MR**30**#5292. MR**0175107 (30:5292)****[7]**-,*Some characteristic properties of dynamical systems with quasi-discrete spectra*, Math. Systems Theory**2**(1968), 179-190. MR**37**#6435. MR**0230877 (37:6435)****[8]**P. R. Halmos,*Lectures on ergodic theory*, Publ. Math. Soc. Japan, no. 3, Math. Soc. Japan, Tokyo, 1956; reprint, Chelsea, New York, 1960. MR**20**#3958. MR**0111817 (22:2677)****[9]**-,*Introduction to Hilbert space and the theory of spectral multiplicity*, Chelsea, New York, 1951. MR**13**, 563.**[10]**-,*Measure theory*, Van Nostrand, Princeton, N. J., 1950. MR**11**, 504. MR**0033869 (11:504d)****[11]**G. W. Mackey,*Borel structure in groups and their duals*, Trans. Amer. Math. Soc.**85**(1957), 134-165. MR**19**, 752. MR**0089999 (19:752b)****[12]**P. A. Meyer,*Probability and potentials*, Blaisdell, Waltham, Mass., 1966. MR**34**#5119. MR**0205288 (34:5119)****[13]**H. L. Royden,*Real analysis*, 2nd ed., Macmillan, New York, 1968. MR**0151555 (27:1540)****[14]**V. S. Varadarajan,*Groups of automorphisms of Borel spaces*, Trans. Amer. Math. Soc.**109**(1963), 191-220. MR**28**#3139. MR**0159923 (28:3139)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0284563-1

Keywords:
Ergodic process,
spectral measure,
Weyl process,
ergodic group extension,
disjoint processes,
ergodic decomposition,
disintegration of a measure,
Kronecker process

Article copyright:
© Copyright 1971
American Mathematical Society