Introduction to the Modern Theory of Dynamical Systems

Katok, Anatole; Hasselblatt, Boris

doi:10.1017/cbo9780511809187

Cited by 2,911 publications

(3,628 citation statements)

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“…Similarly u has a uniformly Hölder version on almost all local stable manifolds. By [KH,Proposition 19.1.1], this proves the result.…”

Section: Proof Let ρ Denote a Right-invariant Metric On Gsupporting

confidence: 63%

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Livsic theorems for connected Lie groups

Pollicott

Walkden

2001

Trans. Amer. Math. Soc.

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Abstract. Let φ be a hyperbolic diffeomorphism on a basic set Λ and let G be a connected Lie group. Let f : Λ → G be Hölder. Assuming that f satisfies a natural partial hyperbolicity assumption, we show that if u : Λ → G is a measurable solution to f = uφ · u −1 a.e., then u must in fact be Hölder. Under an additional centre bunching condition on f , we show that if f assigns 'weight' equal to the identity to each periodic orbit of φ, then f = uφ · u −1 for some Hölder u. These results extend well-known theorems due to Livšic when G is compact or abelian.

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“…Similarly u has a uniformly Hölder version on almost all local stable manifolds. By [KH,Proposition 19.1.1], this proves the result.…”

Section: Proof Let ρ Denote a Right-invariant Metric On Gsupporting

confidence: 63%

“…As u(φ t x) = F t (x)u(x) a.e., u has a Hölder version along orbits. By repeated use of [KH,Proposition 19.1.1], this is sufficient to conclude that u has a Hölder version.…”

Section: Flowsmentioning

confidence: 99%

Livsic theorems for connected Lie groups

Pollicott

Walkden

2001

Trans. Amer. Math. Soc.

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“…Thus Θ 1 contains a dense G δ subset of the circle. 7 We now suppose that there exists some θ 0 ∈ Θ 1 such that K θ 0 is not a single point. We choose a connected component (a, b) of T 1 \ K θ 0 ; note that a = b.…”

Section: Ergodic Measuresmentioning

confidence: 99%

“…[7]) holds, with invariant strips playing the role of periodic orbits in the unforced case: Theorem 1.1 (Theorems 3.1 and 4.1 in [1]). …”

Section: Introductionmentioning

confidence: 99%

Denjoy constructions for fibered homeomorphisms of the torus

Béguin¹,

Crovisier

Jäger³

et al. 2009

Trans. Amer. Math. Soc.

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Abstract. We construct different types of quasiperiodically forced circle homeomorphisms with transitive but non-minimal dynamics. Concerning the recent Poincaré-like classification by Jäger and Stark for this class of maps, we demonstrate that transitive but non-minimal behaviour can occur in each of the different cases. This closes one of the last gaps in the topological classification.Actually, we are able to get some transitive quasiperiodically forced circle homeomorphisms with rather complicated minimal sets. For example, we show that in some of the examples we construct, the unique minimal set is a Cantor set and its intersection with each vertical fibre is uncountable and nowhere dense (but may contain isolated points).We also prove that minimal sets of the latter kind cannot occur when the dynamics are given by the projective action of a quasiperiodic SL(2, R)-cocycle. More precisely, we show that for a quasiperiodic SL(2, R)-cocycle, any minimal proper subset of the torus either is a union of finitely many continuous curves or contains at most two points on generic fibres.

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“…First we will show that for .A-foliations there exists a global cross-section such that the corresponding return map does not admit interval exchange transformations (IET), see [21] for the definition. Next we will prove that under the restrictions, imposed on the divergence of the foliation in singular points, there exists an ergodic measure, invariant under the return map (or, what is the same, there are no 'wandering intervals').…”

Section: Proofmentioning

confidence: 99%

The Poincaré-Bendixson theorem and arational foliations on the sphere

Nikolaev¹

1996

Annales De L’institut Fourier

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Introduction to the Modern Theory of Dynamical Systems

Cited by 2,911 publications

References 0 publications

Livsic theorems for connected Lie groups

Livsic theorems for connected Lie groups

Denjoy constructions for fibered homeomorphisms of the torus

The Poincaré-Bendixson theorem and arational foliations on the sphere

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