On the construction of minimal foliations by hyperbolic surfaces on 3-manifolds

Cuesta, Fernando Alcalde; Dal’Bo, Françoise; Martínez, Miguel; Verjovsky, Alberto

doi:10.1007/s13398-019-00677-6

Cited by 5 publications

(12 citation statements)

References 29 publications

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“…We shall define in Appendix what is a foliation by hyperbolic surfaces. Many examples may be found in [1]. Let us emphasize that every two-dimensional foliation without transverse invariant measure must be a foliation by hyperbolic surfaces: see Proposition 6.3.…”

Section: Finiteness Of Srb Measures -mentioning

confidence: 99%

“…Moreover its densities along unstable plaques, denoted by ψ u w , are uniformly log-bounded and satisfy the following for z 1…”

Section: Existence -Define the Unstable Jacobian Atmentioning

confidence: 99%

Section: Basins Of Gibbs U-states -We Now Prove That the Intersectionmentioning

confidence: 99%

“…Since v ∈ X we have v −n ∈ X 1 for every n and Leb cu -almost every point of W u 1 (G [0,1] (v −n )) belongs to B(µ). Using that B(µ) is G t -invariant this property also holds for G n+m (W u 1 (G [0,1] (v −n ))). Finally we deduce that Leb cu -almost every v ∈ W u (G [0,1] (v m )) belongs to B(µ) and the claim, as well as the lemma, follows.…”

Section: Basins Of Gibbs U-states -We Now Prove That the Intersectionmentioning

confidence: 99%

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Physical measures for the geodesic flow tangent to a transversally conformal foliation

Alvarez¹,

Yang²

2019

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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We consider a transversally conformal foliation F of a closed manifold M endowed with a smooth Riemannian metric whose restriction to each leaf is negatively curved. We prove that it satisfies the following dichotomy. Either there is a transverse holonomy-invariant measure for F , or the foliated geodesic flow admits a finite number of physical measures, which have negative transverse Lyapunov exponents and whose basin cover a set full for the Lebesgue measure. We also give necessary and sufficient conditions for the foliated geodesic flow to be partially hyperbolic in the case where the foliation is transverse to a projective circle bundle over a closed hyperbolic surface.

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Section: Finiteness Of Srb Measures -mentioning

confidence: 99%

“…Moreover its densities along unstable plaques, denoted by ψ u w , are uniformly log-bounded and satisfy the following for z 1…”

Section: Existence -Define the Unstable Jacobian Atmentioning

confidence: 99%

Section: Basins Of Gibbs U-states -We Now Prove That the Intersectionmentioning

confidence: 99%

Section: Basins Of Gibbs U-states -We Now Prove That the Intersectionmentioning

confidence: 99%

See 2 more Smart Citations

Physical measures for the geodesic flow tangent to a transversally conformal foliation

Alvarez¹,

Yang²

2019

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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“…The relationship between foliation theory and other topics of 3-manifolds is still being explored (e.g. [2,7,10,24]). Various analytic, geometric, and topological complications arise when singularities are allowed to exist in a foliation.…”

Section: Introductionmentioning

confidence: 99%

On Bott-Morse Foliations and their Poisson structures in dimension three

Evangelista-Alvarado¹,

Suárez–Serrato²,

Orozco³

et al. 2019

jsing

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We show that a Bott-Morse foliation in dimension 3 admits a linear, singular, Poisson structure of rank 2 with Bott-Morse singularities. We provide the Poisson bivectors for each type of singular component, and compute the symplectic forms of the characteristic distribution.This section follows notations used in [20] and [21], where Bott-Morse foliations on dimension 3 were described.Let M m be a closed, orientable, smooth manifold of dimension m, for m ≥ 3. Let F be a codimension-one smooth foliation with singularities on M . Denote by Sing(F) the set of singular points of F.

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Leaf topology of minimal hyperbolic foliations with non simply-connected generic leaf

Gusmão,

Meniño Cotón

2024

Proc. Amer. Math. Soc.

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A noncompact (oriented) surface satisfies the condition ( ⋆ ) (\star ) if their isolated ends are accumulated by genus. We show that every surface satisfying this condition is homeomorphic to the leaf of a minimal codimension one foliation on a closed 3 3 -manifold whose generic leaf is not simply connected. Moreover, the above result is also true for any prescription of a countable family of noncompact surfaces (satisfying ( ⋆ ) (\star ) ): they can coexist in the same minimal codimension one foliation as above. All the given examples are hyperbolic foliations, meaning that they admit a leafwise Riemannian metric of constant negative curvature.

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On the construction of minimal foliations by hyperbolic surfaces on 3-manifolds

Cited by 5 publications

References 29 publications

Physical measures for the geodesic flow tangent to a transversally conformal foliation

Physical measures for the geodesic flow tangent to a transversally conformal foliation

On Bott-Morse Foliations and their Poisson structures in dimension three

Leaf topology of minimal hyperbolic foliations with non simply-connected generic leaf

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