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 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.
@article{bwmeta1.element.bwnjournal-article-cmv84i1p195bwm,
author = {J. Choksi and M. Nadkarni},
title = {Genericity of nonsingular transformations with infinite ergodic index},
journal = {Colloquium Mathematicae},
volume = {84/85},
year = {2000},
pages = {195-201},
zbl = {1007.28011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv84i1p195bwm}
}
Choksi, J.; Nadkarni, M. Genericity of nonsingular transformations with infinite ergodic index. Colloquium Mathematicae, Tome 84/85 (2000) pp. 195-201. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv84i1p195bwm/
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