Relatively minimal extensions of topological flows

Mieczysław Mentzen

Colloquium Mathematicae (2000)

  • Volume: 84/85, Issue: 1, page 51-65
  • ISSN: 0010-1354

Abstract

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The concept of relatively minimal (rel. min.) extensions of topological flows is introduced. Several generalizations of properties of minimal extensions are shown. In particular the following extensions are rel. min.: distal point transitive, inverse limits of rel. min., superpositions of rel. min. Any proximal extension of a flow Y with a dense set of almost periodic (a.p.) points contains a unique subflow which is a relatively minimal extension of Y. All proximal and distal factors of a point transitive flow with a dense set of a.p. points are rel. min. In the class of point transitive flows with a dense set of a.p. points, distal open extensions are disjoint from all proximal extensions. An example of a relatively minimal point transitive extension determined by a cocycle which is a coboundary in the measure-theoretic sense is given.

How to cite

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Mentzen, Mieczysław. "Relatively minimal extensions of topological flows." Colloquium Mathematicae 84/85.1 (2000): 51-65. <http://eudml.org/doc/210808>.

@article{Mentzen2000,
abstract = {The concept of relatively minimal (rel. min.) extensions of topological flows is introduced. Several generalizations of properties of minimal extensions are shown. In particular the following extensions are rel. min.: distal point transitive, inverse limits of rel. min., superpositions of rel. min. Any proximal extension of a flow Y with a dense set of almost periodic (a.p.) points contains a unique subflow which is a relatively minimal extension of Y. All proximal and distal factors of a point transitive flow with a dense set of a.p. points are rel. min. In the class of point transitive flows with a dense set of a.p. points, distal open extensions are disjoint from all proximal extensions. An example of a relatively minimal point transitive extension determined by a cocycle which is a coboundary in the measure-theoretic sense is given.},
author = {Mentzen, Mieczysław},
journal = {Colloquium Mathematicae},
keywords = {factors; flows; topological dynamics},
language = {eng},
number = {1},
pages = {51-65},
title = {Relatively minimal extensions of topological flows},
url = {http://eudml.org/doc/210808},
volume = {84/85},
year = {2000},
}

TY - JOUR
AU - Mentzen, Mieczysław
TI - Relatively minimal extensions of topological flows
JO - Colloquium Mathematicae
PY - 2000
VL - 84/85
IS - 1
SP - 51
EP - 65
AB - The concept of relatively minimal (rel. min.) extensions of topological flows is introduced. Several generalizations of properties of minimal extensions are shown. In particular the following extensions are rel. min.: distal point transitive, inverse limits of rel. min., superpositions of rel. min. Any proximal extension of a flow Y with a dense set of almost periodic (a.p.) points contains a unique subflow which is a relatively minimal extension of Y. All proximal and distal factors of a point transitive flow with a dense set of a.p. points are rel. min. In the class of point transitive flows with a dense set of a.p. points, distal open extensions are disjoint from all proximal extensions. An example of a relatively minimal point transitive extension determined by a cocycle which is a coboundary in the measure-theoretic sense is given.
LA - eng
KW - factors; flows; topological dynamics
UR - http://eudml.org/doc/210808
ER -

References

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  1. [1] J. Aaronson, M. Lemańczyk, C. Mauduit and H. Nakada, Koksma's inequality and group extensions of Kronecker transformations, in: Algorithms, Fractals, and Dynamics, Plenum Press, New York, 1995, 27-50. Zbl0878.28009
  2. [2] R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969. 
  3. [3] S. Glasner and B. Weiss, On the construction of minimal skew products, Israel J. Math. 34 (1979), 321-336. Zbl0434.54032
  4. [4] M. Lemańczyk and K. K. Mentzen, Topological ergodicity of real cocycles over minimal rotations, preprint. Zbl1002.54023
  5. [5] K. Schmidt, Cocycles of Ergodic Transformation Groups, Lecture Notes in Math. 1, MacMillan of India, 1977. Zbl0421.28017
  6. [6] J. de Vries, Elements of Topological Dynamics, Kluwer Acad. Publ., 1993. Zbl0783.54035

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