Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation.

*(English)*Zbl 0146.28502The approach to ergodic theory in this remarkable paper is complementary to the one developed, mainly by the Russian school, associated with numerical and group invariants. In fact the relationship investigated here between two measure preserving transformations (processes) and between two continuous maps (flows) is disjointness, an extreme form of non-isomorphism. (Most readers will be as upset as the reviewer with this use of the words ‘process’ and ‘flow’.) The concept seems rich enough to warrant quite a few papers on it and these papers will no doubt be largely stimulated by the author’s. An interesting aspect of the paper, apart from the new results it contains, is the entirely novel demonstration of a number of established theorems.

The paper is divided into four parts: I. Disjoint processes; II. Disjoint flows; III. Properties of minimal sets; IV. A problem in diophantine approximation.

Two processes \(X, Y\) are said to be disjoint \((X \perp Y)\) if whenever they are homomorphic images (factors) of the same process \(Z\) then there is a homomorphism of \(Z\) onto \(X\times Y\) which, when composed with the projections of \(X\times Y\) to \(X, Y\), yields the given homomorphisms. (The commutativity in the diagram of this definition is essential as a quick examination of a process \(X\) which is isomorphic to \(X\times X\) will reveal.) An equivalent definition insists that the inverse images of the two Borel fields are independent. The disjointness of two flows is defined similarly (but of course there is no analogous second definition). Two processes (flows) are co-prime if they have no non-trivial common factor. Disjointness implies co-primeness.

The converse is (or was) an open question which, the reviewer understands, has been solved negatively. Definitions are given of Bernoulli processes \(\mathcal B\), Pinsker processes \(\mathcal P\) (with completely positive entropy) deterministic processes \(\mathcal D\) (with zero entropy) without reference to entropy. A particularly interesting class are the Weyl processes \(\mathcal W\), which in view of the author’s structural theorem [Am. J. Math. 85, 477–515 (1963; Zbl 0199.27202)] is a measure theoretic analogue of the class of distal flows. In this connection the reviewer is a little puzzled by the omission of the condition that \(\mathcal W\) be closed under inverse limits, for it seems that such a definition would still yield the result “Mixing processes are disjoint from Weyl processes” and would provide yet another proof of L. M. Abramov’s result [Izv. Akad. Nauk SSSR, Ser. Mat. 26, 513–530 (1962; Zbl 0132.35902)] that processes with quasi-discrete spectrum have zero entropy. Disjointness relations are established between the various classes but M. S. Pinsker’s result [Sov. Math., Dokl. 1, 937–938 (1960); translation from Dokl. Akad. Nauk SSSR 133, 1025–1026 (1960; Zbl 0099.12302)] \(\mathcal P\perp \mathcal D\) is not proved. Two processes with positive entropy are not disjoint, in fact are not co-prime. This is a consequence of Sinai’s weak isomorphism theorem but it is good to see a proof which does not depend upon such a deep result. Part I ends with a discussion of the relationship between disjointness and a problem of filtering.

In Part II analogues of weak mixing \(\mathcal W\) and ergodicity \(\mathcal E\) are defined for flows. Distal flows \(\mathcal D\) are those such that \(T^{m_n} x\to z\), \(T^{m_n} y\to z\) implies \(x = y\). Flows \(T\) with a dense set of periodic points and such that \(T^n\) \((n\neq 0)\) is ergodic, are denoted by \(\mathcal T\). The main results: If two flows are disjoint, one must be minimal \((\mathcal M)\); \(\mathcal T\perp \mathcal M\); \(\mathcal W\times\mathcal M\subset \mathcal E\); \(\mathcal W\perp \mathcal D\cap \mathcal M\).

Part III is devoted to an analysis of the smallness of minimal sets for endomorphisms of compact abelian groups with special emphasis on the circle group and the endomorphism \(Tz = z^n\) \((n\neq 0)\). In fact this endomorphism is a \(\mathcal T\) flow and as such every minimal set is ‘restricted’ and therefore cannot be a basis for the group. If \(A\) is a \(T\) invariant (closed) subset of the circle then the topological entropy of \(T\) restricted to \(A\) is the Hausdorff dimension of \(A\) multiplied by \(\log n\). A reference to P. Billingsley [e. g. “Ergodic theory and information.” (New York etc.: John Wiley) (1965; Zbl 0141.16702)] and other authors would have been appropriate here. An example of a minimal set with positive topological entropy is given (cf. F. Hahn and Y. Katznelson, Trans. Am. Math. Soc. 126, 335–360 (1967; Zbl 0191.21502)], for a sharper result).

The main result of the final Part IV says that if \(\Sigma\) is a non-lacunary (multiplicative) semigroup of integers and if \(\alpha\) is irrational then \(\{n\alpha \bmod 1: n\in\Sigma\}\) is dense in the unit interval.

The paper is divided into four parts: I. Disjoint processes; II. Disjoint flows; III. Properties of minimal sets; IV. A problem in diophantine approximation.

Two processes \(X, Y\) are said to be disjoint \((X \perp Y)\) if whenever they are homomorphic images (factors) of the same process \(Z\) then there is a homomorphism of \(Z\) onto \(X\times Y\) which, when composed with the projections of \(X\times Y\) to \(X, Y\), yields the given homomorphisms. (The commutativity in the diagram of this definition is essential as a quick examination of a process \(X\) which is isomorphic to \(X\times X\) will reveal.) An equivalent definition insists that the inverse images of the two Borel fields are independent. The disjointness of two flows is defined similarly (but of course there is no analogous second definition). Two processes (flows) are co-prime if they have no non-trivial common factor. Disjointness implies co-primeness.

The converse is (or was) an open question which, the reviewer understands, has been solved negatively. Definitions are given of Bernoulli processes \(\mathcal B\), Pinsker processes \(\mathcal P\) (with completely positive entropy) deterministic processes \(\mathcal D\) (with zero entropy) without reference to entropy. A particularly interesting class are the Weyl processes \(\mathcal W\), which in view of the author’s structural theorem [Am. J. Math. 85, 477–515 (1963; Zbl 0199.27202)] is a measure theoretic analogue of the class of distal flows. In this connection the reviewer is a little puzzled by the omission of the condition that \(\mathcal W\) be closed under inverse limits, for it seems that such a definition would still yield the result “Mixing processes are disjoint from Weyl processes” and would provide yet another proof of L. M. Abramov’s result [Izv. Akad. Nauk SSSR, Ser. Mat. 26, 513–530 (1962; Zbl 0132.35902)] that processes with quasi-discrete spectrum have zero entropy. Disjointness relations are established between the various classes but M. S. Pinsker’s result [Sov. Math., Dokl. 1, 937–938 (1960); translation from Dokl. Akad. Nauk SSSR 133, 1025–1026 (1960; Zbl 0099.12302)] \(\mathcal P\perp \mathcal D\) is not proved. Two processes with positive entropy are not disjoint, in fact are not co-prime. This is a consequence of Sinai’s weak isomorphism theorem but it is good to see a proof which does not depend upon such a deep result. Part I ends with a discussion of the relationship between disjointness and a problem of filtering.

In Part II analogues of weak mixing \(\mathcal W\) and ergodicity \(\mathcal E\) are defined for flows. Distal flows \(\mathcal D\) are those such that \(T^{m_n} x\to z\), \(T^{m_n} y\to z\) implies \(x = y\). Flows \(T\) with a dense set of periodic points and such that \(T^n\) \((n\neq 0)\) is ergodic, are denoted by \(\mathcal T\). The main results: If two flows are disjoint, one must be minimal \((\mathcal M)\); \(\mathcal T\perp \mathcal M\); \(\mathcal W\times\mathcal M\subset \mathcal E\); \(\mathcal W\perp \mathcal D\cap \mathcal M\).

Part III is devoted to an analysis of the smallness of minimal sets for endomorphisms of compact abelian groups with special emphasis on the circle group and the endomorphism \(Tz = z^n\) \((n\neq 0)\). In fact this endomorphism is a \(\mathcal T\) flow and as such every minimal set is ‘restricted’ and therefore cannot be a basis for the group. If \(A\) is a \(T\) invariant (closed) subset of the circle then the topological entropy of \(T\) restricted to \(A\) is the Hausdorff dimension of \(A\) multiplied by \(\log n\). A reference to P. Billingsley [e. g. “Ergodic theory and information.” (New York etc.: John Wiley) (1965; Zbl 0141.16702)] and other authors would have been appropriate here. An example of a minimal set with positive topological entropy is given (cf. F. Hahn and Y. Katznelson, Trans. Am. Math. Soc. 126, 335–360 (1967; Zbl 0191.21502)], for a sharper result).

The main result of the final Part IV says that if \(\Sigma\) is a non-lacunary (multiplicative) semigroup of integers and if \(\alpha\) is irrational then \(\{n\alpha \bmod 1: n\in\Sigma\}\) is dense in the unit interval.

Reviewer: William Parry

##### MSC:

37A05 | Dynamical aspects of measure-preserving transformations |

37A25 | Ergodicity, mixing, rates of mixing |

37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |

37B40 | Topological entropy |

28D05 | Measure-preserving transformations |

28D20 | Entropy and other invariants |

11K06 | General theory of distribution modulo \(1\) |

##### Keywords:

disjointness of two flows; smallness of minimal sets; endomorphisms of compact abelian groups; circle group; problem in diophantine approximation
Full Text:
DOI

##### References:

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