The ℤ<sup><i>d</i></sup>Alpern multi-tower theorem for rectangles: a tiling approach

Şahіn, Ayşe A.

doi:10.1080/14689360903127188

Cited by 3 publications

(7 citation statements)

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“…We prove Theorem 1.7 in Section 10. This relates to the ‘

Z^{d}

‐Alpern Lemma’ [48, 55]. In particular, from the case when

F

consists of rectangular shapes of size two, it follows that the graph associated with a free Borel

Z^{d}

dynamical system admits a Borel perfect matching, after removing a null set.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Borel subsystems and ergodic universality for compact Zd‐systems via specification and beyond

Chandgotia

Meyerovitch²

2021

Proc. London Math. Soc.

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A Borel Zd dynamical system (X,S) is ‘almost Borel universal’ if any free Borel Zd dynamical system (Y,T) of strictly lower entropy is isomorphic to a Borel subsystem of (X,S), after removing a null set. We obtain and exploit a new sufficient condition for a topological Zd dynamical system to be almost Borel universal. We use our main result to deduce various conclusions and answer a number of questions. Along with additional results, we prove that a ‘generic’ homeomorphism of a compact manifold of topological dimension at least two can model any ergodic transformation, that non‐uniform specification implies almost Borel universality, and that 3‐colorings in Zd and dimers in Z2 are almost Borel universal.

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“…We prove Theorem 1.7 in Section 10. This relates to the ‘

Z^{d}

‐Alpern Lemma’ [48, 55]. In particular, from the case when

F

consists of rectangular shapes of size two, it follows that the graph associated with a free Borel

Z^{d}

dynamical system admits a Borel perfect matching, after removing a null set.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Borel subsystems and ergodic universality for compact Zd‐systems via specification and beyond

Chandgotia

Meyerovitch²

2021

Proc. London Math. Soc.

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“…Proof. Since the areas are bounded below in the definition (15), it follows that Ψ 1 (ν) is finite. To show that Ψ 1 (ν) is invariant, let C × D be a cylinder set in Z, where C and D are cylinder sets specifying at least the 0th and 1st coordinates of W 1 and W 2 .…”

Section: 5mentioning

confidence: 99%

“…Alpern's result was generalized to Z d actions by Prikhodko [8] and Şahin [15] for rectangular towers, but with no restriction on the number of tiles as a function of dimension. Notably, two tiles suffice in any dimension.…”

Section: Introductionmentioning

confidence: 99%

“…Notably, two tiles suffice in any dimension. Further, the proof in [15] is a generalization of ideas introduced in [12], suggesting that the continuity of the group imposes some restrictions on the number of tiles necessary for a general representation theorem. Motivated by the Z d Alpern Theorem, Rudolph asked whether the number of tiles in his R d result, d > 1, was sharp.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Rudolph’s two step coding theorem and Alpern’s lemma for ℝ^{𝕕} actions

Kra

Quas

Şahіn

2014

Trans. Amer. Math. Soc.

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Rudolph showed that the orbits of any measurable, measure preserving R d action can be measurably tiled by 2 d rectangles and asked if this number of tiles is optimal for d > 1. In this paper, using a tiling of R d by notched cubes, we show that d + 1 tiles suffice. Furthermore, using a detailed analysis of the set of invariant measures on tilings of R 2 by two rectangles, we show that while for R 2 actions with completely positive entropy this bound is optimal, there exist mixing R 2 actions whose orbits can be tiled by 2 tiles. download from IP 130.74.92.202.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 4254 BRYNA KRA, ANTHONY QUAS, AND AYŞE ŞAHİN Licensed to Univ of Mississippi. Prepared on Tue Jul 14 02:20:22 EDT 2015 for download from IP 130.74.92.202.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-usethere exists a factor mapWe show:The collection of tiles, called the basic tiles, that satisfy the conclusion of Theorem 1.1 is obtained by sub-dividing a single basic tile given by a d-dimensional rectangle with a smaller one removed from one corner. We call this the supertile and it tiles R d periodically. The supertile in dimension two is an amalgamation of tiles used by Rudolph in [12]. The periodic tiling using the higher-dimensional supertile was previously constructed by Stein [16] and also studied by Kolountzakis [5]. In these papers, a tiling that is essentially the periodic tiling that we use (up to scaling in one coordinate) is called the "notched cube" tiling, but the papers do not address the decomposition of the notched cube into the smaller tiles. In keeping with the terminology of the earlier references, we also refer to the periodic tiling in this work as the notched cube tiling.Section 2 contains the definitions of the basic tiles, the supertile and the notched cube tiling. In Section 2.2 we study the topological mixing properties of the notched cube tiling and we prove Theorem 1.1 in Section 2.3. Structure of 2-tiling measures.In Section 3 we study T -tilings of R 2 where T consists of two rectangular tiles. The main result of the section is: Theorem 1.2. Let T = {τ 1 , τ 2 }, where the tiles τ i are rectangles whose corresponding dimensions are irrationally related. Then (Y T , {S v } v∈R 2 ) has zero topological entropy.

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“…In the process of proving this theorem we prove a very strong version of the Rokhlin tower theorem. Alpern towers were generalized to two dimensions in both [13] and [17] and to flows by Rudolph [15]. Definition 6.…”

Section: Introductionmentioning

confidence: 99%

Infinite partitions and Rokhlin towers

Kalikow

2011

Ergod. Th. Dynam. Sys.

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We find a countable partition P on a Lebesgue space, labeled {1, 2, 3, . . .}, for any non-periodic measure-preserving transformation T such that P generates T and, for the T, P process, if you see an n on time −1 then you only have to look at times −n, 1 − n, . . . −1 to know the positive integer i to put at time 0. We alter that proof to extend every non-periodic T to a uniform martingale (i.e. continuous g function) on an infinite alphabet. If T has positive entropy and the weak Pinsker property, this extension can be made to be an isomorphism. We pose remaining questions on uniform martingales. In the process of proving the uniform martingale result we make a complete analysis of Rokhlin towers which is of interest in and of itself. We also give an example that looks something like an independent identically distributed process on Z 2 when you read from right to left but where each column determines the next if you read left to right.Please see the dedication to Dan Rudolph that we included in the paper, 'Non-intersecting splitting σ -algebras in a non-Bernoulli transformation', also in this edition.Countable and uncountable partitions can be strange. For processes generated by a finite alphabet, one definition of zero entropy is that a process has zero entropy if and only if the past determines the future. Parry [11, Theorem 8.2, p. 90] showed that every transformation has a countable generator such that the past determines the future. (For the present paper, generator means that the whole process (past and future) generates but Parry uses the word generator to mean that the past generates.) Here we improve on that result with Theorem 1.1 below.Definition 1. Throughout this paper T is a one-to-one measure-preserving transformation on a Lebesgue space with probability measure P such that P({ω : T i (ω) = ω for any i}) = 0. This is what we mean by a non-periodic transformation.Comment 2. If T is one-to-one and measure-preserving on a Lebesgue space then T −1 so is (measurability of T −1 is standard theory but not obvious) and if T is non-periodic then so is T −1 .Definition 2. When T is a measure-preserving transformation and P is a partition, the P, T process is the process in which every ω ∈ is assigned to a doubly infinite word . . . X −2 , X −1 , X 0 , X 1 , X 2 , . . . where X i is the element of P containing T i (ω). The word it maps ω to is called its T, P name. The T, P process is the process of T, P names endowed with the measure induced by the map from points to T, P names. Since T is measure preserving it is a stationary process.Comment 3. Let T be a measure-preserving transformation P a partition and . . . X −2 , X −1 , X 0 , X 1 , X 2 , . . . be the P, T process, then if the P, T name of ω is . . . a −2 , a −1 , a 0 , a 1 , a 2 , . . . then the P, T name of T (ω) is . . . b −2 , b −1 , b 0 , b 1 , b 2 , . . . defined by b i := a i+1 for all i, so the map from ω to its doubly infinite name is a homomorphism onto the doubly infinite sequences of elements of P with induced measure where the transfor...

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

Cited by 3 publications

References 14 publications

Borel subsystems and ergodic universality for compact Zd‐systems via specification and beyond

Borel subsystems and ergodic universality for compact Zd‐systems via specification and beyond

Rudolph’s two step coding theorem and Alpern’s lemma for ℝ^{𝕕} actions

Infinite partitions and Rokhlin towers

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

Cited by 3 publications

References 14 publications

Borel subsystems and ergodic universality for compact Zd‐systems via specification and beyond

Borel subsystems and ergodic universality for compact Zd‐systems via specification and beyond

Rudolph’s two step coding theorem and Alpern’s lemma for ℝ^{𝕕} actions

Infinite partitions and Rokhlin towers

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