Genericity of nonsingular transformations with infinite ergodic index

J. Choksi; M. Nadkarni

Colloquium Mathematicae (2000)

  • Volume: 84/85, Issue: 1, page 195-201
  • ISSN: 0010-1354

Abstract

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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 infiniteCartesian power which is nonsingular with respect to the power measure, then it has to be measure preservingit.

How to cite

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Choksi, J., and Nadkarni, M.. "Genericity of nonsingular transformations with infinite ergodic index." Colloquium Mathematicae 84/85.1 (2000): 195-201. <http://eudml.org/doc/210797>.

@article{Choksi2000,
abstract = {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 infiniteCartesian power which is nonsingular with respect to the power measure, then it has to be measure preservingit. },
author = {Choksi, J., Nadkarni, M.},
journal = {Colloquium Mathematicae},
keywords = {nonsingular transformation; ergodic theorem; ergodic index},
language = {eng},
number = {1},
pages = {195-201},
title = {Genericity of nonsingular transformations with infinite ergodic index},
url = {http://eudml.org/doc/210797},
volume = {84/85},
year = {2000},
}

TY - JOUR
AU - Choksi, J.
AU - Nadkarni, M.
TI - Genericity of nonsingular transformations with infinite ergodic index
JO - Colloquium Mathematicae
PY - 2000
VL - 84/85
IS - 1
SP - 195
EP - 201
AB - 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 infiniteCartesian power which is nonsingular with respect to the power measure, then it has to be measure preservingit.
LA - eng
KW - nonsingular transformation; ergodic theorem; ergodic index
UR - http://eudml.org/doc/210797
ER -

References

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  1. [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. Zbl0377.28012
  2. [I] A. Iwanik, Approximation theorems for stochastic operators, ibid. 29 (1980), 415-425. Zbl0474.47003
  3. [K] S. Kakutani, On equivalence of infinite product measures, Ann. of Math. 49 (1948), 214-224. Zbl0030.02303
  4. [K-P] S. Kakutani and W. Parry, Infinite measure preserving transformations with 'mixing', Bull. Amer. Math. Soc. 69 (1963), 752-756. Zbl0126.31801
  5. [S] U. Sachdeva, On category of mixing in infinite measure spaces, Math. Systems Theory 5 (1971), 319-330. Zbl0226.28008

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