The structure of ergodic measures for compact group extensions

Keynes, Harvey B.; Newton, D.

doi:10.1007/bf02760845

Cited by 27 publications

(21 citation statements)

References 9 publications

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“…For example, a problem related to the measurable Livsic theorem for non-hyperbolic systems occurs in the study of rational maps (that is, the invariant line field problem [8]). The measurable Livsic theorem is also known to have important consequences for the study of skew products [10,6,11]. We refer to these papers for more details.…”

Section: Local Results For Arbitrary Groupsmentioning

confidence: 99%

Measurable Cocycle Rigidity for Some Non-Compact Groups

Nicol

Pollicott

1999

Bulletin of the London Mathematical Society

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Section: Local Results For Arbitrary Groupsmentioning

confidence: 99%

Measurable Cocycle Rigidity for Some Non-Compact Groups

Nicol

Pollicott

1999

Bulletin of the London Mathematical Society

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“…It is immediate from Keynes and Newton [16] that F is not ergodic, and hence (by Livšic regularity) not transitive. Thus F is a non-transitive K m -extension C r -close to F , which is the desired contradiction.…”

Section: Stable Ergodicity 547mentioning

confidence: 99%

“…By Keynes and Newton [16] and Livšic regularity, a necessary and sufficient condition for the non-ergodicity of the skew extension f : X × G→X × G is that there exists a nontrivial irreducible unitary representation R of G on C d for some d ≥ 1 and a continuous function w : X→S 2d−1 such that Let µ be an equilibrium state on X.…”

Section: Transitivity and Ergodicity Of Compact Group Extensionsmentioning

confidence: 99%

“…Brin's original proof of stable transitivity of compact Lie group extensions over an Anosov diffeomorphism uses transitivity properties of the stable and unstable foliations, and the well-known 'quadrilateral construction'. A common theme of more recent work is the use of a result of Keynes and Newton [16] together with the Livšic regularity theorem [17], [20,Theorem 3.1]. While the existence of maximal transitivity components holds under the assumption that the base is a basic set (see [22, §5]), the quadrilateral argument appears to require at least some local path connectivity in the invariant foliations of the basic set.…”

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confidence: 99%

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Stable ergodicity for smooth compact Lie group extensions of hyperbolic basic sets

Field

Melbourne

Török

2005

Ergod. Th. Dynam. Sys.

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Link to this article: http://journals.cambridge.org/abstract_S0143385704000355How to cite this article: MICHAEL FIELD, IAN MELBOURNE and ANDREI TÖRÖK (2005). Stable ergodicity for smooth compact Lie group extensions of hyperbolic basic sets.Abstract. We obtain sharp results for the genericity and stability of transitivity, ergodicity and mixing for compact connected Lie group extensions over a hyperbolic basic set of a C 2 diffeomorphism. In contrast to previous work, our results hold for general hyperbolic basic sets and are valid in the C r -topology for all r > 0 (here r need not be an integer and C 1 is replaced by Lipschitz). Moreover, when r ≥ 2, we show that there is a C 2 -open and C r -dense subset of C r -extensions that are ergodic. We obtain similar results on stable transitivity for (non-compact) R m -extensions, thereby generalizing a result of Niţicȃ and Pollicott, and on stable mixing for suspension flows. THEOREM 1.7. For r > 0, there exists a C r -open and -dense subset W r of S r such that for all f ∈ W r , the skew extension f :Remark 1.8. If is a hyperbolic attractor, then we obtain the improved result that W r is C 0 -open in S r . Again, C α -openness, α ∈ [0, 1), fails for general hyperbolic basic sets.1.3. Suspension flows. We now describe our main result on the stability of weak mixing for suspensions flows over hyperbolic basic sets. Let R r denote the space of strictly positive functions (roof functions) in C r (M, R). Each roof function f ∈ R r defines a suspension flow f t : f → f on the suspension f . When = M is Anosov, it follows from the Anosov alternative [2] that f is mixing for all non-constant f ∈ R r .

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“…But what is more important here is that { ν g : g ∈ G} is the set of S ϕ -invariant ergodic measures, see e.g. [27].…”

Section: Sarnak's Conjecture For Continuous Extensions By Coboundariesmentioning

confidence: 99%

Möbius disjointness for models of an ergodic system and beyond

et al. 2018

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Given a topological dynamical system (X, T ) and an arithmetic function u : N → C, we study the strong MOMO property (relatively to u) which is a strong version of u-disjointness with all observable sequences in (X, T ). It is proved that, given an ergodic measure-preserving system (Z, D, κ, R), the strong MOMO property (relatively to u) of a uniquely ergodic model (X, T ) of R yields all other uniquely ergodic models of R to be u-disjoint. It follows that all uniquely ergodic models of: ergodic unipotent diffeomorphisms on nilmanifolds, discrete spectrum automorphisms, systems given by some substitutions of constant length (including the classical Thue-Morse and Rudin-Shapiro substitutions), systems determined by Kakutani sequences are Möbius (and Liouville) disjoint. The validity of Sarnak's conjecture implies the strong MOMO property relatively to µ in all zero entropy systems, in particular, it makes µ-disjointness uniform. The absence of strong MOMO property in positive entropy systems is discussed and, it is proved that, under the Chowla conjecture, a topological system has the strong MOMO property relatively to the Liouville function if and only if its topological entropy is zero.Measure-theoretic viewpoint While often one focuses on proving the Möbius disjointness for a particular class of zero entropy dynamical systems (see the bibliography in [40]), our approach is more abstract and concentrates on the measure-theoretic aspects. The starting point for us is the following:

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The structure of ergodic measures for compact group extensions

Cited by 27 publications

References 9 publications

Measurable Cocycle Rigidity for Some Non-Compact Groups

Measurable Cocycle Rigidity for Some Non-Compact Groups

Stable ergodicity for smooth compact Lie group extensions of hyperbolic basic sets

Möbius disjointness for models of an ergodic system and beyond

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