Metric properties of measure preserving homeomorphisms

Katok, Anatole; Stëpin, A. M.

doi:10.1070/rm1970v025n02abeh003793

Cited by 30 publications

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“…[By "typical" we mean dense G a in an appropriate space.) In this paper we explain the striking similarity of the "typicality" results which have been established separately in the above two contexts, in the following pairs: ergodicity [5,13], weak mixing [6,9], and other properties in [8,9] and [6,3]. We do this by proving the following result [Theorem 6) under the assumption I that the underlying manifold has the fixed point property: Any isomorphism-invariant measure theoretic property which is typical for automorphisms o/ a Lehesgue space is typical for measure preserving homeomorphisms.…”

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confidence: 61%

“…This result does not use the Conjugacy Lemma, and is valid without restriction (except the usual ones of [13] or [9]) on the manifold. We conclude the paper with a brief discussion of two alternative forms of a topological conjugacy lemma, each of which would have important consequences.…”

Section: Theorem 6 If S Is a Self-conjugate (Weak Topology) Dense G~mentioning

confidence: 73%

“…Then there is an ./in G such that g,=f-1 gf is in M and p(h,g')<~:. Conjecture 1 is due to Katok and Stepin [9], who seem to suggest that it would be of use in proving, for example, that weakly mixing homeomorphisms are typical. However, since the existence of a weakly mixing homeomorphism cannot apparently be established by example on every manifold, it is not clear that this conjecture could be used as they indicate.…”

Section: Theorem 6 If S Is a Self-conjugate (Weak Topology) Dense G~mentioning

confidence: 95%

“…We call our analog the Topological Conjugacy Lemma. Katok and Stepin [9] observed that it was precisely the absence of a suitable analog of Halmos' conjugacy lemma that necessitated a new technique to prove that weakly mixing homeomorphisms were typical. They conjectured a result similar to the one we prove, which we discuss along with a new conjecture at the end of the paper.…”

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A topological analog of Halmos' conjugacy lemma

Alpern

1978

Invent Math

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Abstract. Using a new lemma indicated by the title, and a recent measure preserving version of Lusin's Theorem, we prove the following theorem: Any isomorphism-invariant measure theoretic property which is "typical" for automorphisms of a kebesgue space is also "typical" for Lebesgue measure preserving homeomorphisms of the unit cube I n, n > 2. We also prove a partial converse of this theorem. Taken together, these results clarify the relationship between pairs of theorems proved by several authors, which established the "typicality" of specific properties (such as ergodicity and weak mixing) separately in the measurable and continuous cases.The determination of typical, or generic, properties of measure preserving transformations has proceeded at different paces but with eventually the same results in the measurable (i) and continuous (ii) contexts. For example, it is known for both (i) automorphisms of a Lebesgue space, and (ii) measure preserving homeomorphisms of a compact manifold, that ergodicity, weak mixing and zero entropy are typical. [By "typical" we mean dense G a in an appropriate space.) In this paper we explain the striking similarity of the "typicality" results which have been established separately in the above two contexts, in the following pairs: ergodicity [5,13], weak mixing [6,9], and other properties in [8,9] and [6,3]. We do this by proving the following result [Theorem 6) under the assumption I that the underlying manifold has the fixed point property: Any isomorphism-invariant measure theoretic property which is typical for automorphisms o/ a Lehesgue space is typical for measure preserving homeomorphisms. We also prove [Theorem 5) a partial converse of this result.As a basic tool in the comparison of the measurable and continuous contexts, we use a measure preserving version of Lusin's Theorem developed in a previous paper [3]. That result (Theorem 1 of this paper) is in turn based on a similar theorem due to J.C.

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confidence: 61%

Section: Theorem 6 If S Is a Self-conjugate (Weak Topology) Dense G~mentioning

confidence: 73%

Section: Theorem 6 If S Is a Self-conjugate (Weak Topology) Dense G~mentioning

confidence: 95%

mentioning

confidence: 99%

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A topological analog of Halmos' conjugacy lemma

Alpern

1978

Invent Math

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“…for all open Ad" (2) there exists a homeomorphism h of /" onto itself fixing the boundary pointwise such that for any A-measurable set S It is known that the above theorem remains valid if /" is replaced by any compact finite dimensional manifold [2], [4] or with /°°, the Hilbert cube, [8].…”

Section: ) N(a)>0 L(a)>0mentioning

confidence: 99%

A note an homeomorphic measures on topological groups

Morris

Peck

1983

Proceedings of the Edinburgh Mathematical Society

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The classical von Neumann–Oxtoby–Ulam Theorem states the following:Given non-atomic Borel probability measures μ, λ on In such thatthere exists a homeomorphism h of In onto itself fixing the boundary pointwise such that for any λ-measurable set SIt is known that the above theorem remains valid if In is replaced by any compact finite dimensional manifold [2], [4] or with I∞, the Hilbert cube, [8].

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