Infinite ergodic index ℤ d -actions in infinite measure

E. Muehlegger; A. Raich; C. Silva; M. Touloumtzis; B. Narasimhan; W. Zhao

Colloquium Mathematicae (1999)

  • Volume: 82, Issue: 2, page 167-190
  • ISSN: 0010-1354

Abstract

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We construct infinite measure preserving and nonsingular rank one ℤ d -actions. The first example is ergodic infinite measure preserving but with nonergodic, infinite conservative index, basis transformations; in this case we exhibit sets of increasing finite and infinite measure which are properly exhaustive and weakly wandering. The next examples are staircase rank one infinite measure preserving ℤ d -actions; for these we show that the individual basis transformations have conservative ergodic Cartesian products of all orders, hence infinite ergodic index. We generalize this example to obtain a stronger condition called power weakly mixing. The last examples are nonsingular ℤ d -actions for each Krieger ratio set type with individual basis transformations with similar properties.

How to cite

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Muehlegger, E., et al. "Infinite ergodic index $ℤ^d$ -actions in infinite measure." Colloquium Mathematicae 82.2 (1999): 167-190. <http://eudml.org/doc/210755>.

@article{Muehlegger1999,
abstract = {We construct infinite measure preserving and nonsingular rank one $ℤ^d$-actions. The first example is ergodic infinite measure preserving but with nonergodic, infinite conservative index, basis transformations; in this case we exhibit sets of increasing finite and infinite measure which are properly exhaustive and weakly wandering. The next examples are staircase rank one infinite measure preserving $ℤ^d$-actions; for these we show that the individual basis transformations have conservative ergodic Cartesian products of all orders, hence infinite ergodic index. We generalize this example to obtain a stronger condition called power weakly mixing. The last examples are nonsingular $ℤ^d$-actions for each Krieger ratio set type with individual basis transformations with similar properties.},
author = {Muehlegger, E., Raich, A., Silva, C., Touloumtzis, M., Narasimhan, B., Zhao, W.},
journal = {Colloquium Mathematicae},
keywords = {infinite measure preserving transformation; conservative index basis transformation},
language = {eng},
number = {2},
pages = {167-190},
title = {Infinite ergodic index $ℤ^d$ -actions in infinite measure},
url = {http://eudml.org/doc/210755},
volume = {82},
year = {1999},
}

TY - JOUR
AU - Muehlegger, E.
AU - Raich, A.
AU - Silva, C.
AU - Touloumtzis, M.
AU - Narasimhan, B.
AU - Zhao, W.
TI - Infinite ergodic index $ℤ^d$ -actions in infinite measure
JO - Colloquium Mathematicae
PY - 1999
VL - 82
IS - 2
SP - 167
EP - 190
AB - We construct infinite measure preserving and nonsingular rank one $ℤ^d$-actions. The first example is ergodic infinite measure preserving but with nonergodic, infinite conservative index, basis transformations; in this case we exhibit sets of increasing finite and infinite measure which are properly exhaustive and weakly wandering. The next examples are staircase rank one infinite measure preserving $ℤ^d$-actions; for these we show that the individual basis transformations have conservative ergodic Cartesian products of all orders, hence infinite ergodic index. We generalize this example to obtain a stronger condition called power weakly mixing. The last examples are nonsingular $ℤ^d$-actions for each Krieger ratio set type with individual basis transformations with similar properties.
LA - eng
KW - infinite measure preserving transformation; conservative index basis transformation
UR - http://eudml.org/doc/210755
ER -

References

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