Generic properties of measure preserving homeomorphisms

Alpern, Steve

doi:10.1007/bfb0063279

Cited by 30 publications

(59 citation statements)

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“…It is interesting to note that the logical structure of this section, where the Coding Theorem is obtained as a corollary of the Finite Multitower Theorem, is exactly the reverse of that of [1]. In that paper, the Coding Theorem was proved …rst, by a detailed limiting argument, and then the Finite Multitower Theorem was obtained as an easy consequence.…”

Section: Representations By Aperiodic Automorphismsmentioning

confidence: 95%

“…If f is setwise h periodic then it follows from the de…nition that there is an h-tower for f: If f is pointwise h-periodic it is setwise periodic [10,Lemma 1], and the h-tower must be periodic. If f is aperiodic, the existence of a multitower is given by the following result of [1]. Short and elegant proofs of this …nite version of the multitower theorem are given in [18] and [8].…”

Section: Automorphisms and Multitowersmentioning

confidence: 99%

“…Since we can start the cycles at any state (e.g. [3,1] is the same as [1,3]) all we need is that the cycles in the decomposition (6) all contain some common state, which we can use to label the entire bottom B h of the tower.…”

Section: Representations By Setwise Periodic Automorphismsmentioning

confidence: 99%

“…The subsequent results of Alpern [3], Haigh [9], Rodrigues del Tio and Valsero Blanco [20], and Kalpazidou [11][12] [15], established and extended Cohen's conjecture in various ways. The purpose of this paper is to show that the cycle decomposition techniques used in [3] to establish Cohen's conjecture, together with the multitower constructions of [1] [3], can be modi…ed to give short elementary proofs of results on representations by automorphisms of various types, including the well known Coding Theorem (Theorem 9). The cycle decompositions we use (the Lemmas of this paper) all are established by very elementary algorithms.…”

Section: Introductionmentioning

confidence: 99%

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Rotational (and Other) Representations of Stochastic Matrices

Alpern

Prasad

2007

Stochastic Analysis and Applications

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Joel E. Cohen (1981) conjectured that any stochastic matrix P = fp i;j g could be represented by some circle rotation f in the following sense: For some partition fS i g of the circle into sets consisting of …nite unions of arcs, we have (*), where denotes arc length. In this paper we show how cycle decomposition techniques originally used (Alpern, 1983) to establish Cohen's conjecture can be extended to give a short simple proof of the Coding Theorem, that any mixing (that is, P N > 0 for some N ) stochastic matrix P can be represented (in the sense of * but with S i merely measurable) by any aperiodic measure preserving bijection (automorphism) of a Lesbesgue probability space. Representations by pointwise and setwise periodic automorphisms are also established. While this paper is largely expository, all the proofs, and some of the results, are new.

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Section: Representations By Aperiodic Automorphismsmentioning

confidence: 95%

Section: Automorphisms and Multitowersmentioning

confidence: 99%

Section: Representations By Setwise Periodic Automorphismsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rotational (and Other) Representations of Stochastic Matrices

Alpern

Prasad

2007

Stochastic Analysis and Applications

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“…Theorem 1.1. (The Z d Alpern Multi-tower Theorem for rectangles) Let w 1 , w 2 , · · · ∈ N d , and p 1 , p 2 , · · · ∈ R + satisfy (1) for all i = 1, · · · , d gcd(w 1 i , w 2 i , · · · ) = 1 and…”

Section: Introductionunclassified

The ℤ^dAlpern multi-tower theorem for rectangles: a tiling approach

Şahіn

2009

Dynamical Systems

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We provide a proof of the Alpern multi-tower theorem for Z d actions. We reformulate the theorem as a problem of measurably tiling orbits of a Z d action by a collection of rectangles whose corresponding sides have no non-trivial common divisors. We associate to such a collection of rectangles a special family of generalized domino tilings. We then identify an intrinsic dynamic property of these tilings, viewed as symbolic dynamical systems, which allows for a multi-tower decomposition.2000 Mathematics Subject Classification. 37A15,37B50.

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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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Generic properties of measure preserving homeomorphisms

Cited by 30 publications

References 8 publications

Rotational (and Other) Representations of Stochastic Matrices

Rotational (and Other) Representations of Stochastic Matrices

The ℤ^dAlpern multi-tower theorem for rectangles: a tiling approach

A topological analog of Halmos' conjugacy lemma

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Generic properties of measure preserving homeomorphisms

Cited by 30 publications

References 8 publications

Rotational (and Other) Representations of Stochastic Matrices

Rotational (and Other) Representations of Stochastic Matrices

The ℤdAlpern multi-tower theorem for rectangles: a tiling approach

A topological analog of Halmos' conjugacy lemma

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Product

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The ℤ^dAlpern multi-tower theorem for rectangles: a tiling approach