Every aperiodic endomorphism $f$ of a nonatomic Lebesgue space which possesses a finite 1-sided generator has a 1-sided generator $\beta $ such that $k_f\leq \operatorname{card}\, \beta \leq k_f+1$. This is the best estimate for the minimal cardinality of a 1-sided generator. The above result is the generalization of the analogous one for ergodic case.
@article{118799,
author = {Zbigniew S. Kowalski},
title = {Minimal generators for aperiodic endomorphisms},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {36},
year = {1995},
pages = {721-725},
zbl = {0840.28006},
mrnumber = {1378693},
language = {en},
url = {http://dml.mathdoc.fr/item/118799}
}
Kowalski, Zbigniew S. Minimal generators for aperiodic endomorphisms. Commentationes Mathematicae Universitatis Carolinae, Tome 36 (1995) pp. 721-725. http://gdmltest.u-ga.fr/item/118799/
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