Conformally natural extension of homeomorphisms of the circle

Douady, Adrien; Earle, Clifford J.

doi:10.1007/bf02392590

Cited by 331 publications

(355 citation statements)

References 12 publications

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“…Let µ 0 be the probability measure defined on B 0 by π * µ 0 = µ. There is a uniquê f -invariant probabilityμ on (M ,B) such that π * μ = µ: it is characterized by (20) E μ |f n (B 0 ) =f n * µ 0 for every n ≥ 0. To any smooth cocycle F : E → E over f , defined on a fiber bundle P : E → M , we may associate the smooth cocycleF :Ê →Ê overf defined byÊx = E π(x) and Fx = F π(x) .…”

Section: Examples a Few Simple Examples Illustrate The Contents Of Tmentioning

confidence: 99%

Extremal Lyapunov exponents: an invariance principle and applications

2010

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Abstract. We propose a new approach to analyzing dynamical systems that combine hyperbolic and non-hyperbolic ("center") behavior, e.g. partially hyperbolic diffeomorphisms. A number of applications illustrate its power.We find that any ergodic automorphism of the 4-torus with two eigenvalues in the unit circle is stably Bernoulli among symplectic maps. Indeed, any nearby symplectic map has no zero Lyapunov exponents, unless it is volume preserving conjugate to the automorphism itself. Another main application is to accessible skew-product maps preserving area on the fibers. We prove, in particular, that if the genus of the fiber is at least 2 then the Lyapunov exponents must be different from zero and vary continuously with the map.These, and other dynamical conclusions, originate from a general Invariance Principle we prove in here. It is formulated in terms of smooth cocycles, that is, fiber bundle morphisms acting by diffeomorphisms on the fibers. The extremal Lyapunov exponents measure the smallest and largest exponential rates of growth of the derivative along the fibers. The Invariance Principle states that if these two numbers coincide then the fibers carry some amount of structure which is transversely invariant, that is, invariant under certain canonical families of homeomorphisms between fibers.

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Section: Examples a Few Simple Examples Illustrate The Contents Of Tmentioning

confidence: 99%

Extremal Lyapunov exponents: an invariance principle and applications

2010

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“…Now let us show the continuity ofσ. Sinceσ is continuous from (Bel(Γ), · ∞ ) into itself (see [6]), it is sufficient to proveσ(µ n ) converges toσ(µ) in hyperbolic L p -norm. For every ε ′ > 0, take r 1 ∈ (0, 1) with…”

Section: Continuity Of the Bers Embeddingmentioning

confidence: 99%

“…We prepare for the proof in Section 2. Specifically, we characterize each point of T p (Γ) by its Douady-Earle extension, which is a quasiconformal self-mapping on ∆ with conformal naturality (see [6]). Originally, Cui [5] proved this result in the case of T 2 (1) and Tang [20] extended it to T p (1) for p ≥ 2, where 1 = {id ∆ } is the trivial group.…”

Section: Introductionmentioning

confidence: 99%

Introduction of a complex structure on the p-integrable Teichmüller space

Yanagishita

2014

Ann. Acad. Sci. Fenn. Math.

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Abstract. For p ≥ 1, the p-integrable Teichmüller space is the metric subspace of the Teichmüller space composed of the Teichmüller equivalence classes with p-integrable Beltrami coefficient. In this paper, for p ≥ 2, we introduce a complex structure on the p-integrable Teichmüller space of an arbitrary Fuchsian group satisfying a certain geometric condition. As an application, we show the coincidence of two canonical distances on the metric subspace.

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“…After endowing the Riemann surfaces M and N with the corresponding hyperbolic metrics it is well known that a (1 + ǫ)-quasiconformal map F : M ǫ → N ǫ is isotopic to a (1 + δ)-biLipschitz homeomorphism such that δ → 0 when ǫ → 0 (the function δ = δ(ǫ) does not depend on the choice of surfaces M and N ). One such biLipschitz map is obtained by taking the barycentric extension [1] of the boundary map of the lift to the universal covering of f : M ǫ → N ǫ (this observation was made in [2]). …”

Section: The First Main Results In This Paper Ismentioning

confidence: 99%

“…We first show that the barycentric extensions [1] of the Whitehead homeomorphisms (which we have defined in the previous section) have dilatations essentially supported in a neighborhood of the diagonal exchange for the corresponding Whitehead move.…”

Section: Mapping Classes With Small Dilatationsmentioning

confidence: 91%

The Teichmüller distance between finite index subgroups of $${PSL_2(\mathbb{Z})}$$

Marković

Šarić

2008

Geom Dedicata

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Abstract. For a given ǫ > 0, we show that there exist two finite index subgroups of P SL 2 (Z) which are (1 + ǫ)-quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any ǫ > 0 there are two finite regular covers of the Modular once punctured torus T 0 (or just the Modular torus) and a (1 + ǫ)-quasiconformal between them that is not homotopic to a conformal map. As an application of the above results, we show that the orbit of the basepoint in the Teichmüller space T (S p ) of the punctured solenoid S p under the action of the corresponding Modular group (which is the mapping class group of S p [5], [6]) has the closure in T (S p ) strictly larger than the orbit and that the closure is necessarily uncountable.

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Conformally natural extension of homeomorphisms of the circle

Cited by 331 publications

References 12 publications

Extremal Lyapunov exponents: an invariance principle and applications

Extremal Lyapunov exponents: an invariance principle and applications

Introduction of a complex structure on the p-integrable Teichmüller space

The Teichmüller distance between finite index subgroups of $${PSL_2(\mathbb{Z})}$$

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