Topological characteristic factors along cubes of minimal systems

Cai, Fangzhou; Shao, Song

doi:10.3934/dcds.2019216

Cited by 8 publications

(5 citation statements)

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“…It is a long open question whether for a minimal distal system in Glasner's theorem in [17] one can replace the largest class d distal factor by the maximal pro-nilfactor of order d. Indeed, this is the case where we consider characteristic factors along cubes of minimal systems. In [6], the topological characteristic factors along cubes of minimal systems are studied. It is shown that up to proximal extensions the pro-nilfactors are the topological characteristic factors along cubes of minimal systems.…”

Section: Introductionmentioning

confidence: 99%

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Regionally proximal relation of order d along arithmetic progressions and nilsystems

et al. 2020

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The regionally proximal relation of order d along arithmetic progressions, namely AP [d] for d ∈ N, is introduced and investigated. It turns out that if (X, T ) is a topological dynamical system with AP [d] = ∆, then each ergodic measure of (X, T ) is isomorphic to a d-step pro-nilsystem, and thus (X, T ) has zero entropy. Moreover, it is shown that if (X, T ) is a strictly ergodic distal system with the property that the maximal topological and measurable d-step pro-nilsystems are isomorphic, then AP [d] = RP [d] for each d ∈ N. It follows that for a minimal ∞-pro-nilsystem, AP [d] = RP [d] for each d ∈ N. An example which is a strictly ergodic distal system with discrete spectrum whose maximal equicontinuous factor is not isomorphic to the Kronecker factor is constructed.

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Section: Introductionmentioning

confidence: 99%

“…It is shown that up to proximal extensions the pro-nilfactors are the topological characteristic factors along cubes of minimal systems. In particular, for a distal minimal system, the maximal (d − 1)-step pro-nilfactor is the topological cubic characteristic factor of order d [6].…”

Section: Introductionmentioning

confidence: 99%

Regionally proximal relation of order d along arithmetic progressions and nilsystems

et al. 2020

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“…Up to an almost 1-1 modification, they showed that the topological characteristic factors are the pro-nilfactors (see [12, Theorem A]), which are the analogies in the ergodic situation. For other study on topological characteristic factors, see [4,5].…”

Section: Introductionmentioning

confidence: 99%

On structure theorems and non-saturated examples

Wu¹,

Xu²,

Ye³

2022

Preprint

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For any minimal system (X, T ) and d ≥ 1 there is an associated minimal systemIn this paper, we further study the structure of N d (X). We show that the maximal distal factor of N d (X) is N d (X dis ) with X dis being the maximal distal factor of X, and prove that as minimal systems (N d (X), G d (T )) has the same structure theorem as (X, T ). In addition, a non-saturated metric example (X, T ) is constructed, which is not T × T 2 -saturated and is a Toeplitz minimal system.

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“…It is a long open question whether for a minimal distal system in Glasner's theorem in [19] one can replace the largest class d distal factor by the maximal pro-nilfactor of order d. Indeed, this is the case when we consider characteristic factors along cubes of minimal systems. In [7], the topological characteristic factors along cubes of minimal systems are studied. It is shown that up to proximal extensions the pro-nilfactors are the topological characteristic factors along cubes of minimal systems.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Regionally proximal relation of order $d$ along arithmetic progressions and nilsystems

Glasner

Huang

Shao

et al. 2019

Preprint

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The regionally proximal relation of order d along arithmetic progressions, namely AP [d] for d ∈ N, is introduced and investigated. It turns out that if (X, T ) is a topological dynamical system with AP [d] = ∆, then each ergodic measure of (X, T ) is isomorphic to a d-step pro-nilsystem, and thus (X, T ) has zero entropy.Moreover, it is shown that if (X, T ) is a strictly ergodic distal system with the property that the maximal topological and measurable d-step pro-nilsystems are isomorphic, then AP [d] = RP [d] for each d ∈ N. It follows that for a minimal ∞-pro-nilsystem, AP [d] = RP [d] for each d ∈ N. An example which is a strictly ergodic distal system with discrete spectrum whose maximal equicontinuous factor is not isomorphic to the Kronecker factor is constructed.

show abstract

Topological characteristic factors along cubes of minimal systems

Cited by 8 publications

References 12 publications

Regionally proximal relation of order d along arithmetic progressions and nilsystems

Regionally proximal relation of order d along arithmetic progressions and nilsystems

On structure theorems and non-saturated examples

Regionally proximal relation of order $d$ along arithmetic progressions and nilsystems

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