Endomorphisms of a Lebesgue space. II. (English) Zbl 0305.28009

[For part I see Bull. Am. Math. Soc. 78, 272–276 (1972; Zbl 0232.28013).]
This paper introduces an equivalence relation, regular isomorphism, on measure preserving transformations of a Lebesgue space, motivated by codings of sequence spaces which anticipate only a finite amount of the future. \((X_i ,\mathcal L_i, m_i, T_i)\), \(i = 1,2\), are Lebesgue dynamical systems and \(\mathcal A_i\) \(T_i\)-invariant algebras such that \(T_i^n\mathcal A_i\nearrow \mathcal B_i\). The dynamical systems are regularly isomorphic if there is an isomorphism, \(\varphi\), such that \(\mathcal A_1\subset \varphi T_2^k \mathcal A_2\) and \(\mathcal A_2\subset \varphi^{-1} T_1^k\mathcal A_1\), for some \(k\). In this case the information functions \(I(\mathcal A_1/T_1^{-1}\mathcal A_1)\) and \(I(\varphi\mathcal A_2/\varphi T_2^{-1}\mathcal A_2)\) are cohomologous. This fact is used to give a numerical invariant of the relation. The Bernoulli shifts with weights \((1/4, 1/4, 1/4, 1/4)\) and \((1/2, 1/8, 1/8, 1/8, 1/8)\) are not regularly isomorphic and the natural extensions of \(\beta\)-transformations are not, in general, regularly isomorphic to Bernoulli shifts.
In a further paper by W. Parry [Endomorphisms of a Lebesgue space. III, Isr. J. Math. 21, 167–172 (1975; Zbl 0312.28018)] a new invariant is introduced and it is shown the Markov shifts \(\begin{pmatrix} p & q \\ p & q\end{pmatrix}\), \(\begin{pmatrix} p & q \\ q & p\end{pmatrix}\), and \(\begin{pmatrix} q & p \\ p & q\end{pmatrix}\) are in different equivalence classes, if \(p\ne q\). Ideas in a somewhat similar spirit appear in a paper by R. Bowen [Smooth partitions of Anosov diffeomorphisms are weak Bernoulli. Isr. J. Math. 21, 95–100 (1975; Zbl 0315.58020)].
Reviewer: William Parry


28D05 Measure-preserving transformations
37A05 Dynamical aspects of measure-preserving transformations
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