The action of a semisimple Lie group on its maximal compact subgroup

Budak, Talin; Işık, Nilgün; Milnes, Paul; Pym, J. S.

doi:10.1090/s0002-9939-01-05984-6

Cited by 4 publications

(2 citation statements)

References 11 publications

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“…Note that this result is similar to one obtained in , where the authors consider the much smaller flow

(G (R), K (R))

rather than

(G (double-struckR), S_{G} (double-struckR))

. Corollary If

Z (G)

is finite, then it embeds abstractly in

H (G (double-struckR), S_{G} (double-struckR))

. Corollary Assume

G = K ⋉ H

.…”

Section: The Ellis Groupmentioning

confidence: 99%

Definable topological dynamics and real Lie groups

Jagiella

2015

Mathematical Logic Qtrly

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We investigate definable topological dynamics of groups definable in an o‐minimal expansion of the field of reals. Assuming that a definable group G admits a model‐theoretic analogue of Iwasawa decomposition, namely the compact‐torsion‐free decomposition G=KH, we give a description of minimal subflows and the Ellis group of its universal definable flow SG(double-struckR) in terms of this decomposition. In particular, the Ellis group of this flow is isomorphic to NG(H)∩K(double-struckR). This provides a range of counterexamples to a question by Newelski whether the Ellis group is isomorphic to G/G00. We further extend the results to universal topological covers of definable groups, interpreted in a two‐sorted structure containing the o‐minimal sort double-struckR and a sort for an abelian group.

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“…Note that this result is similar to one obtained in , where the authors consider the much smaller flow

(G (R), K (R))

rather than

(G (double-struckR), S_{G} (double-struckR))

. Corollary If

Z (G)

is finite, then it embeds abstractly in

H (G (double-struckR), S_{G} (double-struckR))

. Corollary Assume

G = K ⋉ H

.…”

Section: The Ellis Groupmentioning

confidence: 99%

Definable topological dynamics and real Lie groups

Jagiella

2015

Mathematical Logic Qtrly

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“…Some examples are to be found in Namioka [31] (1984), Milnes [29] (1986) and [30] (1989), Glasner [16] (1976) and [20] (1993), Berg, Gove & Hadad [4] (1998), Budak, Işik, Milnes & Pym. [9] (2001), and Glasner & Megrelishvili [23] (2004). Rarely is the enveloping semigroup metrizable (a notable exception is the case of weakly almost periodic metric systems; see Downarowicz [10] (1998) and Glasner [22] (2003), Theorem 1.48).…”

Section: Introductionmentioning

confidence: 99%

On tame dynamical systems

Glasner

2006

Colloq. Math.

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Abstract. A dynamical version of the Bourgain-Fremlin-Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of βN, or it is a "tame" topological space whose topology is determined by the convergence of sequences. In the latter case we say that the dynamical system is tame. We show that (i) a metric distal minimal system is tame iff it is equicontinuous, (ii) for an abelian acting group a tame metric minimal system is PI (hence a weakly mixing minimal system is never tame), and (iii) a tame minimal cascade has zero topological entropy. We also show that for minimal distal-but-not-equicontinuous systems the canonical map from the enveloping operator semigroup onto the Ellis semigroup is never an isomorphism. This answers a long standing open question. We give a complete characterization of minimal systems whose enveloping semigroup is metrizable. In particular it follows that for an abelian acting group such a system is equicontinuous.

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Enveloping semigroups in topological dynamics

Glasner

2007

Topology and its Applications

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The action of a semisimple Lie group on its maximal compact subgroup

Cited by 4 publications

References 11 publications

Definable topological dynamics and real Lie groups

Definable topological dynamics and real Lie groups

On tame dynamical systems

Enveloping semigroups in topological dynamics

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