Independence in topological and C*-dynamics

Kerr, David; Li, Hanfeng

doi:10.1007/s00208-007-0097-z

Cited by 207 publications

(319 citation statements)

References 75 publications

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“…However, if we replace the trace norm with the operator norm then we can no longer obtain a handle on asymptotic commutation relations using the same measure-theoretic devices. Nevertheless, we are able to show that a type of norm asymptotic Abelianness is generic for G-actions on the CAR algebra when G is discrete and admits a nontorsion Abelian quotient, generalizing a result for single * -automorphisms from [24].…”

Section: Introductionsupporting

confidence: 67%

“…For dynamical systems on highly noncommutative operator algebras, mixing properties are closely related to asymptotic Abelianness (see Sections 9 and 10 of [24] and Example 4.3.24 in [6]). We prove that G-actions on the hyperfinite II 1 factor with a certain asymptotic Abelianness property relative to the trace norm are generic precisely when G does not have property T, and as a corollary obtain the parallel statement for weak mixing in accord with the commutative case.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic Abelianness, weak mixing, and property T

Kerr¹,

Pichot²

2008

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. Let G be a second countable locally compact group and H a closed subgroup. We characterize the lack of Kazhdan's property T for the pair (G, H) by the genericity of G-actions on the hyperfinite II1 factor with a certain asymptotic Abelianness property relative to H, as well as by the genericity of measure-preserving G-actions on a nonatomic standard probability space that are weakly mixing for H. The latter furnishes a definitive generalization of a classical theorem of Halmos for single automorphisms and strengthens a recent result of Glasner, Thouvenot, and Weiss on generic ergodicity. We also establish a weak mixing version of Glasner and Weiss's characterization of property T for discrete G in terms of the invariant state space of a Bernoulli shift and show that on the CAR algebra a type of norm asymptotic Abelianness is generic for G-actions when G is discrete and admits a nontorsion Abelian quotient.

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Section: Introductionsupporting

confidence: 67%

Section: Introductionmentioning

confidence: 99%

Asymptotic Abelianness, weak mixing, and property T

Kerr¹,

Pichot²

2008

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…Such families play a major role in the theory of tame dynamical systems. See, for example, [12,11,14,15,16].…”

Section: Fragmented Mapsmentioning

confidence: 99%

A note on tameness of families having bounded variation

Megrelishvili¹

2017

Topology and its Applications

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Abstract. We show that for arbitrary linearly ordered set (X, ≤) any bounded family of (not necessarily, continuous) real valued functions on X with bounded total variation does not contain independent sequences. We obtain generalized Helly's sequential compactness type theorems. One of the theorems asserts that for every compact metric space (Y, d) the compact space BVr (X, Y ) of all functions X → Y with variation ≤ r is sequentially compact in the pointwise topology. Another Helly type theorem shows that the compact space M + (X, Y ) of all order preserving maps X → Y is sequentially compact where Y is a compact metrizable partially ordered space in the sense of Nachbin.

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“…It is much easier to detect if a set is an entropy set, if we use the concept IE-tuples (the terminology comes from [35], but the concept appeared earlier in the literature, e.g. [29,Theorem 8.2]).…”

Section: Entropymentioning

confidence: 99%

“…It is known (see [13,29,35]) that a set K , which is not a singleton, is an entropy set iff every finite selection of its points x 1 , . .…”

Section: Entropymentioning

confidence: 99%

On Entropy of Graph Maps That Give Hereditarily Indecomposable Inverse Limits

Boroński

Oprocha

2015

J Dyn Diff Equat

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We prove that if f : G → G is a map on a topological graph G such that the inverse limit lim ← − (G, f ) is hereditarily indecomposable, and entropy of f is positive, then there exists an entropy set with infinite topological entropy. When G is the circle and the degree of f is positive then the entropy is always infinite and the rotation set of f is nondegenerate. This shows that the Anosov-Katok type constructions of the pseudo-circle as a minimal set in volume-preserving smooth dynamical systems, or in complex dynamics, obtained previously by Handel, Herman and Chéritat cannot be modeled on inverse limits. This also extends a previous result of Mouron who proved that if G = [0, 1], then h( f ) ∈ {0, ∞}, and combined with a result of Ito shows that certain dynamical systems on compact finitedimensional Riemannian manifolds must either have zero entropy on their invariant sets or be non-differentiable.

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Independence in topological and C*-dynamics

Cited by 207 publications

References 75 publications

Asymptotic Abelianness, weak mixing, and property T

Asymptotic Abelianness, weak mixing, and property T

A note on tameness of families having bounded variation

On Entropy of Graph Maps That Give Hereditarily Indecomposable Inverse Limits

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