Some old and new Rokhlin towers

Kornfeld, Isaac

doi:10.1090/conm/356/06502

Cited by 7 publications

(13 citation statements)

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“…The literature related to this very useful result is rich (2) ; we refer the reader to [9,14] for more detailed expositions. In particular, the result has been generalized in several directions.…”

Section: Introductionmentioning

confidence: 95%

“…Let us mention here the generalization to Z d -actions proved by Conze [3] and independently by Katznelson and Weiss [8], and let us refer again to [9,14] for information on other such extensions.…”

Section: Introductionmentioning

confidence: 99%

“…This starts with the version of Theorem 1.1 in which T is just an endomorphism. Up to the early 2000s, this version was part of the folklore (some explicit mentions of the result outline ways to prove it by modifying some of the existing proofs of the versions for automorphisms; see for instance [9]). The first publication containing a full proof of a version for endomorphisms seems to be [6].…”

Section: Introductionmentioning

confidence: 99%

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Towers for commuting endomorphisms, and combinatorial applications

Ávila¹,

Candela²

2016

Annales De l'Institut Fourier

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Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Towers for commuting endomorphisms, and combinatorial applications

Ávila¹,

Candela²

2016

Annales De l'Institut Fourier

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“…MRT is a generalization of one of the basic constructions in ergodic theory, Rokhlin's Lemma (see for example Kornfeld's survey [18]). …”

Section: Introductionmentioning

confidence: 99%

“…In recent work on topological (and measurable) orbit equivalence, Kornfeld and Ormes [19] use MRT to prove strong orbit equivalence theorems and generalizations of the Jewett-Krieger theorem. A nice survey of the use of MRT and associated Bratteli diagrams to obtain Vershik adic-maps in topological orbit equivalence theory is in [18].…”

mentioning

confidence: 99%

MultiTowers, conjugacies and codes: Three theorems in Ergodic theory, one variation on Rokhlin’s Lemma

Alpern

Prasad

2008

Proc. Amer. Math. Soc.

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We dedicate this paper to the memory of Shizuo Kakutani. We miss his kind manner, gentle presence and keen insight.Abstract. We show that three theorems about the measurable dynamics of a fixed aperiodic measure preserving transformation τ of a Lebesgue probability space (X, A, µ) are equivalent. One theorem asserts that the conjugates of τ are dense in the uniform topology on the space of automorphisms. The other two results assert the existence of a partition of the space X with special properties. One partition result shows that given a mixing Markov chain, there is a code (i.e., a partition of the space) so that the itinerary process given by τ and the partition has the distribution of the given Markov Chain. The other partition result is a generalization of the Rokhlin Lemma, stating that the space can be partitioned into denumerably many columns and the measures of the columns can be prescribed in advance. Thus the first two results are equivalent to this strengthening of Rokhlin's Lemma.

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Existence and non-existence of solutions to the coboundary equation for measure-preserving systems

Adams

Rosenblatt²

2022

Ergod. Th. Dynam. Sys.

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A fundamental question in the field of cohomology of dynamical systems is to determine when there are solutions to the coboundary equation: $$ \begin{align*} f = g - g \circ T. \end{align*} $$ In many cases, T is given to be an ergodic invertible measure-preserving transformation on a standard probability space $(X, {\mathcal B}, \mu )$ and is contained in $L^p$ for $p \geq 0$ . We extend previous results by showing for any measurable f that is non-zero on a set of positive measure, the class of measure-preserving T with a measurable solution g is meager (including the case where $\int _X f\,d\mu = 0$ ). From this fact, a natural question arises: given f, does there always exist a solution pair T and g? In regards to this question, our main results are as follows. Given measurable f, there exist an ergodic invertible measure-preserving transformation T and measurable function g such that $f(x) = g(x) - g(Tx)$ for almost every (a.e.) $x\in X$ , if and only if $\int _{f> 0} f\,d\mu = - \int _{f < 0} f\,d\mu $ (whether finite or $\infty $ ). Given mean-zero $f \in L^p(\mu )$ for $p \geq 1$ , there exist an ergodic invertible measure-preserving T and $g \in L^{p-1}(\mu )$ such that $f(x) = g(x) - g( Tx )$ for a.e. $x \in X$ . In some sense, the previous existence result is the best possible. For $p \geq 1$ , there exists a dense $G_{\delta }$ set of mean-zero $f \in L^p(\mu )$ such that for any ergodic invertible measure-preserving T and any measurable g such that $f(x) = g(x) - g(Tx)$ almost everywhere, then $g \notin L^q(\mu )$ for $q> p - 1$ . Finally, it is shown that we cannot expect finite moments for solutions g, when $f \in L^1(\mu )$ . In particular, given any such that $\lim _{x\to \infty } \phi (x) = \infty $ , there exist mean-zero $f \in L^1(\mu )$ such that for any solutions T and g, the transfer function g satisfies: $$ \begin{align*} \int_{X} \phi ( | g(x) | )\,d\mu = \infty. \end{align*} $$

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Some old and new Rokhlin towers

Cited by 7 publications

References 0 publications

Towers for commuting endomorphisms, and combinatorial applications

Towers for commuting endomorphisms, and combinatorial applications

MultiTowers, conjugacies and codes: Three theorems in Ergodic theory, one variation on Rokhlin’s Lemma

Existence and non-existence of solutions to the coboundary equation for measure-preserving systems

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