Genericity of nonsingular transformations with infinite ergodic index

• Autorzy: J. R. Choksi , M. G. Nadkarni
• Języki publikacji: 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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2000
• Tom: 84/85
• Numer: 1
• Strony: 195-201

Daty

wydano

2000

otrzymano

1999-07-01

poprawiono

1999-08-17

Twórcy

autor

J. R. Choksi

• Department of Mathematics and Statistics, Burnside Hall, McGill University, Montreal, Quebec, Canada H3A 2K6

autor

M. G. Nadkarni

• 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.