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## Colloquium Mathematicum

2000 | 84/85 | 1 | 195-201
Tytuł artykułu

### Genericity of nonsingular transformations with infinite ergodic index

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EN
Abstrakty
EN
It is shown that in the group of invertible measurable nonsingular transformations on a Lebesgue probability space, endowed with the coarse topology, the transformations with infinite ergodic index are generic; they actually form a dense $G_δ$ set. (A transformation has infinite ergodic index if all its finite Cartesian powers are ergodic.) This answers a question asked by C. Silva. A similar result was proved by U. Sachdeva in 1971, for the group of transformations preserving an infinite measure. Exploring other possible (more restrictive) definitions of infinite ergodic index, we find, somewhat surprisingly, that if a nonsingular transformation on a Lebesgue probability space has an infinite} Cartesian power which is nonsingular with respect to the power measure, then it has to be measure preservingit.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
195-201
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-07-01
poprawiono
1999-08-17
Twórcy
autor
• Department of Mathematics and Statistics, Burnside Hall, McGill University, Montreal, Quebec, Canada H3A 2K6
autor
• Department of Mathematics, University of Mumbai, Vidyanagari, Kalina, Mumbai 400 098, India
Bibliografia
• [C-K] J. R. Choksi and S. Kakutani, Residuality of ergodic measurable transformations and of ergodic transformations which preserve an infinite measure, Indiana Univ. Math. J. 28 (1979), 453-469.
• [I] A. Iwanik, Approximation theorems for stochastic operators, ibid. 29 (1980), 415-425.
• [K] S. Kakutani, On equivalence of infinite product measures, Ann. of Math. 49 (1948), 214-224.
• [K-P] S. Kakutani and W. Parry, Infinite measure preserving transformations with 'mixing', Bull. Amer. Math. Soc. 69 (1963), 752-756.
• [S] U. Sachdeva, On category of mixing in infinite measure spaces, Math. Systems Theory 5 (1971), 319-330.
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Bibliografia
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