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Keywords:
aperiodic endomorphism; 1-sided generator
aperiodic endomorphism; 1-sided generator
Summary:
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.
References:
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[2] Kowalski Z.S.: Minimal generators for ergodic endomorphisms. Studia Mathematica 16 (1988), 85-88. MR 0985076 | Zbl 0676.28009
[3] Parry W.: Entropy and Generators in Ergodic Theory. Benjamin, 1969. MR 0262464 | Zbl 0175.34001
[4] Rohlin V.A.: On the fundamental ideas of measure theory. Amer. Math. Soc. Transl. Ser. 1 10 (1962), 1-54 Mat. Sb. 25 (1949), 107-150. MR 0030584
[5] Walters P.: Some results on the classification of non-invertible measure preserving transformations. in: Recent Advances in Topological Dynamics, Lecture Notes in Math. 318, Springer, 1972, pp. 266-276. MR 0393424 | Zbl 0257.28011
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