Weakly Wandering Sequences in Ergodic Theory

Eigen, Stanley; Hajian, Arsen R.; 伊藤, 雄二; Prasad, V.

doi:10.1007/978-4-431-55108-9

Cited by 12 publications

(9 citation statements)

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“…It suffices to show that there exists a set A of positive measure for T such that C T (A) has long gaps. To guarantee this, we use the following characterization of a transformation with no finite invariant measure for which the reader can refer to [7]:…”

Section: 2mentioning

confidence: 99%

On conservative sequences and their application to ergodic multiplier problems

et al. 2018

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Abstract. The conservative sequence of a set A under a transformation T is the set of all n ∈ Z such that T n A ∩ A = ∅. By studying these sequences, we prove that given any countable collection of nonsingular transformations with no finite invariant measure {T i }, there exists a rank-one transformation S such that T i × S is not ergodic for all i. Moreover, S can be chosen to be rigid or have infinite ergodic index. We establish similar results for Z d actions and flows. Then, we find sufficient conditions on rankone transformations T that guarantee the existence of a rank-one transformation S such that T × S is ergodic, or, alternatively, conditions that guarantee that T × S is conservative but not ergodic. In particular, the infinite Chacón transformation satisfies both conditions. Finally, for a given ergodic transformation T , we study the Baire categories of the sets E(T ),ĒC(T ) andC(T ) of transformations S such that T × S is ergodic, ergodic but not conservative, and conservative, respectively.

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Section: 2mentioning

confidence: 99%

On conservative sequences and their application to ergodic multiplier problems

et al. 2018

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“…Similar statements apply to topological dynamical systems on the Cantor set. There are generalizations to actions by other groups, infinite measure systems, and nonsingular actions-see, for example, [10,9,7,13]. We are interested in finding conditions for representing a system that is presented in this way as a subshift on a (usually finite) alphabet.…”

Section: Introductionmentioning

confidence: 99%

Periodic codings of Bratteli-Vershik systems

Frick¹,

Petersen²,

Shields³

2018

Preprint

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We develop conditions for the coding of a Bratteli-Vershik system according to initial path segments to be periodic, equivalently for a constructive symbolic recursive scheme corresponding to a cutting and stacking process to produce a periodic sequence. This is a step toward understanding when a Bratteli-Vershik system can be essentially faithfully represented by means of a natural coding as a subshift on a finite alphabet.

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“…In [8] and [10], for an invertible transformation, one can see an equivalent condition for the existence of an equivalent finite invariant measure (I) as the conditions (IV) in the next Theorem 3.12. Further, they proved the equivalence of (I) and {m • T n } n∈Z m (equi-unif.)…”

mentioning

confidence: 96%

“…of a "strictly positive" fixed point of a Markov operator P over a probability space (X, F , m) as follows (cf. [3,8,10,16,27,35,37,38,40]):…”

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confidence: 99%

“…The last condition in the above list means that any weakly wandering set in the sense of Hajian and Kakutani has measure zero when P is the Perron-Frobenius operator corresponding to a nonsingular transformation T . We also know that, in ergodic measure preserving systems, if the invariant measure is σ-finite infinite then a weakly wandering set of positive measure always exists (see [8]). For the case of an absolutely continuous σ-finite (it might be infinite) invariant measure of a nonsingular transformation, many sufficient conditions are known (see [1,2,9,41,46]).…”

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confidence: 99%

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$\sigma$-finite invariant densities for eventually conservative Markov operators

Toyokawa

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We establish equivalent conditions for the existence of an integrable or locally integrable fixed point for a Markov operator with the maximal support. Maximal support means that almost all initial points will concentrate on the support of the invariant density under the iteration of the process. One of the equivalent conditions for the existence of a locally integrable fixed point is weak almost periodicity of the jump operator with respect to some sweep-out set. This result includes the case of the existence of an absolutely continuous σ-finite invariant measure when we consider a nonsingular transformation on a probability space. Weak almost periodicity implies the Jacobs-Deleeuw-Glicksberg splitting and we show that constrictive Markov operators which guarantee the spectral decomposition are typical weakly almost periodic operators.

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Weakly Wandering Sequences in Ergodic Theory

Cited by 12 publications

References 0 publications

On conservative sequences and their application to ergodic multiplier problems

On conservative sequences and their application to ergodic multiplier problems

Periodic codings of Bratteli-Vershik systems

$\sigma$-finite invariant densities for eventually conservative Markov operators

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