scholarly journals Maximally highly proximal flows

2020 ◽  
pp. 1-21 ◽  
Author(s):  
ANDY ZUCKER
Keyword(s):  
Structure Theorem ◽  
Dynamical Behavior ◽  
Minimal Flow ◽  
Complete Structure ◽  
Polish Groups ◽  
Polish Group

For $G$ a Polish group, we consider $G$ -flows which either contain a comeager orbit or have all orbits meager. We single out a class of flows, the maximally highly proximal (MHP) flows, for which this analysis is particularly nice. In the former case, we provide a complete structure theorem for flows containing comeager orbits, generalizing theorems of Melleray, Nguyen Van Thé, and Tsankov and of Ben Yaacov, Melleray, and Tsankov. In the latter, we show that any minimal MHP flow with all orbits meager has a metrizable factor with all orbits meager, thus ‘reflecting’ complicated dynamical behavior to metrizable flows. We then apply this to obtain a structure theorem for Polish groups whose universal minimal flow is distal.

A Metamathematical Condition Equivalent to the Existence of a Complete Left Invariant Metric for a Polish Group

2006 ◽  
Vol 71 (4) ◽  
pp. 1108-1124 ◽  
Author(s):  
Alex Thompson
Keyword(s):  
Equivalence Relation ◽  
Polish Space ◽  
Invariant Metrics ◽  
Orbit Equivalence ◽  
Invariant Metric ◽  
Continuous Action ◽  
Polish Groups ◽  
Polish Group ◽  
Left Invariant Metrics

AbstractStrengthening a theorem of Hjorth this paper gives a new characterization of which Polish groups admit compatible complete left invariant metrics. As a corollary it is proved that any Polish group without a complete left invariant metric has a continuous action on a Polish space whose associated orbit equivalence relation is not essentially countable.

Actions of non-compact and non-locally compact Polish groups

2000 ◽  
Vol 65 (4) ◽  
pp. 1881-1894 ◽  
Author(s):  
Sławomir Solecki
Keyword(s):  
Equivalence Relation ◽  
Polish Space ◽  
Equivalence Relations ◽  
Orbit Equivalence ◽  
Locally Compact ◽  
Continuous Action ◽  
Borel Equivalence Relations ◽  
Polish Groups ◽  
Local Compactness ◽  
Polish Group

AbstractWe show that each non-compact Polish group admits a continuous action on a Polish space with non-smooth orbit equivalence relation. We actually construct a free such action. Thus for a Polish group compactness is equivalent to all continuous free actions of this group being smooth. This answers a question of Kechris. We also establish results relating local compactness of the group with its inability to induce orbit equivalence relations not reducible to countable Borel equivalence relations. Generalizing a result of Hjorth, we prove that each non-locally compact, that is, infinite dimensional, separable Banach space has a continuous action on a Polish space with non-Borel orbit equivalence relation, thus showing that this property characterizes non-local compactness among Banach spaces.

scholarly journals REPRESENTATIONS OF IDEALS IN POLISH GROUPS AND IN BANACH SPACES

2015 ◽  
Vol 80 (4) ◽  
pp. 1268-1289 ◽  
Author(s):  
PIOTR BORODULIN–NADZIEJA ◽  
BARNABÁS FARKAS ◽  
GRZEGORZ PLEBANEK
Keyword(s):  
Banach Space ◽  
Abelian Group ◽  
Banach Spaces ◽  
Open Problems ◽  
Polish Groups ◽  
Polish Group ◽  
Focus On The Family ◽  
The Family ◽  
Null Ideal ◽  
Defining Sequence

AbstractWe investigate ideals of the form {A⊆ω: Σn∈Axnis unconditionally convergent} where (xn)n∈ωis a sequence in a Polish group or in a Banach space. If an ideal onωcan be seen in this form for some sequence inX, then we say that it is representable inX.After numerous examples we show the following theorems: (1) An ideal is representable in a Polish Abelian group iff it is an analytic P-ideal. (2) An ideal is representable in a Banach space iff it is a nonpathological analytic P-ideal.We focus on the family of ideals representable inc0. We characterize this property via the defining sequence of measures. We prove that the trace of the null ideal, Farah’s ideal, and Tsirelson ideals are not representable inc0, and that a tallFσP-ideal is representable inc0iff it is a summable ideal. Also, we provide an example of a peculiar ideal which is representable inℓ1but not in ℝ.Finally, we summarize some open problems of this topic.

scholarly journals Isometry groups of separable metric spaces

2009 ◽  
Vol 146 (1) ◽  
pp. 67-81 ◽  
Author(s):  
MACIEJ MALICKI ◽  
SŁAWOMIR SOLECKI
Keyword(s):  
Metric Space ◽  
Isometry Group ◽  
Metric Spaces ◽  
Structure Theorem ◽  
Ultrametric Space ◽  
Isometry Groups ◽  
Locally Compact ◽  
Natural Action ◽  
Polish Group ◽  
Separable Metric Space

AbstractWe show that every locally compact Polish group is isomorphic to the isometry group of a proper separable metric space. This answers a question of Gao and Kechris. We also analyze the natural action of the isometry group of a separable ultrametric space on the space. This leads us to a structure theorem representing an arbitrary separable ultrametric space as a bundle with an ultrametric base and with ultrahomogeneous fibers which are invariant under the action of the isometry group.

REMARKS ON THE IMPRIMITIVITY THEOREM FOR NONLOCALLY COMPACT POLISH GROUPS

2000 ◽  
Vol 03 (02) ◽  
pp. 247-262
Author(s):  
FRANCESCO FIDALEO
Keyword(s):  
Invariant Measure ◽  
Probability Measure ◽  
Invariant Probability Measure ◽  
Equivalence Classes ◽  
Fréchet Spaces ◽  
Locally Compact ◽  
Polish Groups ◽  
Polish Group ◽  
One To One

In this paper we analyze the possibility of establishing a Theorem of Imprimitivity in the case of nonlocally compact Polish groups. We prove that systems of imprimitivity for a Polish group G based on a locally compact homogeneous G-space M ≡ G/H equipped with a quasi-invariant probability measure μ, are in one-to-one correspondence with elements of the space [Formula: see text] of the first cohomology of the group G of equivalence classes of continuous cocycles. As a corollary, we have the complete Imprimitivity Theorem [Formula: see text] in the case of discrete countable homogeneous G-spaces equipped with a quasi-invariant measure. Finally, we outline the possibility of establishing the complete Imprimitivity Theorem for particular classes of Polish groups. These examples cover the case of (separable) Fréchet spaces, for which it is shown that the complete Imprimitivity Theorem holds as well.

scholarly journals Cardinal invariants of Haar null and Haar meager sets

2020 ◽  
pp. 1-27
Author(s):  
Márton Elekes ◽  
Márk Poór
Keyword(s):  
Borel Probability Measure ◽  
Invariant Metric ◽  
Locally Compact ◽  
Compact Metric Space ◽  
Borel Set ◽  
Polish Groups ◽  
Polish Group ◽  
Cardinal Invariants ◽  
Borel Probability ◽  
Meager Sets

A subset X of a Polish group G is Haar null if there exists a Borel probability measure μ and a Borel set B containing X such that μ(gBh) = 0 for every g, h ∈ G. A set X is Haar meager if there exists a compact metric space K, a continuous function f : K → G and a Borel set B containing X such that f−1(gBh) is meager in K for every g, h ∈ G. We calculate (in ZFC) the four cardinal invariants (add, cov, non, cof) of these two σ-ideals for the simplest non-locally compact Polish group, namely in the case $G = \mathbb {Z}^\omega$ . In fact, most results work for separable Banach spaces as well, and many results work for Polish groups admitting a two-sided invariant metric. This answers a question of the first named author and Vidnyánszky.

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

On Lattice-Ordered Rees Matrix Γ-Semigroups

2013 ◽  
Vol 59 (1) ◽  
pp. 209-218 ◽  
Author(s):  
Kostaq Hila ◽  
Edmond Pisha
Keyword(s):  
Structure Theorem ◽  
Lattice Of Congruences

Abstract The purpose of this paper is to introduce and give some properties of l-Rees matrix Γ-semigroups. Generalizing the results given by Guowei and Ping, concerning the congruences and lattice of congruences on regular Rees matrix Γ-semigroups, the structure theorem of l-congruences lattice of l - Γ-semigroup M = μº(G : I; L; Γe) is given, from which it follows that this l-congruences lattice is distributive.

Dynamical behavior of a nonlinear rotorcraft model

1989 ◽  
Author(s):  
GEORGE FLOWERS ◽  
BENSONH. TONGUE
Keyword(s):  
Dynamical Behavior
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