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

Alvarez, Sébastien; Yang, Jiagang

doi:10.1016/j.anihpc.2018.03.009

Cited by 8 publications

(21 citation statements)

References 44 publications

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“…The dichotomy has also been established in the case where the foliation is transverse to a projective CP 1 -bundle over a closed manifold with negative sectional curvature: see [2,10,12] where the authors prove the uniqueness of the SRB measure in the absence of transverse invariant measure. Recently, in joint work with Yang (see [5]) and using the main result of the present paper, we were able to prove the dichotomy for transversally conformal foliations with negatively curved leaves.…”

Section: Introductionmentioning

confidence: 68%

“…The purpose of this paper is twofold. Firstly, we wish to give new sufficient conditions for a foliation with negatively curved leaves to admit a transverse invariant measure that are decisive for establishing the aforementioned dichotomy (see [5]). Secondly, this paper serves as a companion to [3].…”

Section: Introductionmentioning

confidence: 99%

“…Note that this theorem implies that foliated geodesic flows of most smooth foliations with negatively curved leaves don't preserve any smooth measure. Indeed, assume that M is equipped with a smooth Riemannian metric whose restriction to the leaves has negative sectional curvature everywhere (see [5] for a discussion on the existence of such metrics). Suppose the foliated geodesic flow G t preserves a smooth measure.…”

Section: Introductionmentioning

confidence: 99%

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Gibbs u-states for the foliated geodesic flow and transverse invariant measures

Alvarez

2017

Isr. J. Math.

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This paper is devoted to the study of Gibbs u-states for the geodesic flow tangent to a foliation F of a manifold M having negatively curved leaves. By definition, they are the probability measures on the unit tangent bundle to the foliation that are invariant under the foliated geodesic flow and have Lebesgue disintegration in the unstable manifolds of this flow.On the one hand we give sufficient conditions for the existence of transverse invariant measures. In particular we prove that when the foliated geodesic flow has a Gibbs su-state, i.e. an invariant measure with Lebesgue disintegration both in the stable and unstable manifolds, then this measure has to be obtained by combining a transverse invariant measure and the Liouville measure on the leaves.On the other hand we exhibit a bijective correspondence between the set of Gibbs u-states and a set of probability measure on M that we call φ u -harmonic. Such measures have Lebesgue disintegration in the leaves and their local densities have a very specific form: they possess an integral representation analogue to the Poisson representation of harmonic functions. IntroductionThis is the second of a series of three papers in which we study a notion of Gibbs measure for the geodesic flow tangent to the leaves of a closed foliated manifold with negatively curved leaves [2,3].Let F be a foliation of a closed manifold M (we say that (M , F ) is a closed foliated manifold) whose leaves L are endowed with smooth (i.e. of class C ∞ ) Riemannian metrics g L which vary continuously transversally in the smooth topology (we refer to the assignment L → g L as a leafwise metric). The unit tangent bundle of F is the set M of unit vectors tangent to F . An element of M shall be denoted by v or, when we want to specify its basepoint, by (x, v) with x ∈ M and v ∈ T 1 x F , the unit sphere of the tangent space to F at x. The set M is naturally a manifold endowed with a foliation F whose leaves are the unit tangent bundles of the leaves of F (see §3.1). It also carries a flow G t , the foliated geodesic flow, which preserves the leaves of F and whose restriction to a leaf T 1 L is precisely its geodesic flow. We will be mostly interested in negatively curved leafwise metrics i.e. in the case where all metrics g L have negative sectional curvature. In that case the foliated geodesic flow possesses a weak form of hyperbolicity that resembles the classical notion of partial hyperbolicity and that we shall analyze in detail in §3.2. This new notion has been introduced by Bonatti, Gómez-Mont and Martínez in a recent preprint [11]. They called it the foliated hyperbolicity. This means that F admits two continuous subfoliations, called stable and unstable foliations and denoted by W s and W u , which are G t -invariant and respectively uniformly contracted and dilated by G t . One would like to know the extent to which the classical results concerning partial hyperbolicity can be applied to that context.The motivating problem of this paper is the research of SRB measures (or phy...

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Section: Introductionmentioning

confidence: 68%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Gibbs u-states for the foliated geodesic flow and transverse invariant measures

Alvarez

2017

Isr. J. Math.

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“…Note that in [Thu,§4] it is claimed that one can choose a smooth metric in M which makes every leaf of F ε to have curvature arbitrarily close to ´1. and For smooth foliations this is proved in [AY, Theorem B] and attributed to Ghys (see also [AY,Remark 6.2]). In our case, leaves of F may be just C 1 , so it is more delicate to talk about curvature but still we only look at coarse geometric properties, so our statement suffices.…”

Section: Appendix a Branching Foliations And Prefoliations Revisitedmentioning

confidence: 84%

Collapsed Anosov flows and self orbit equivalences

Barthelmé¹,

Fenley²,

Potrie³

2020

Preprint

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We propose a generalization of the concept of discretized Anosov flows that covers a wide class of partially hyperbolic diffeomorphisms in 3manifolds, and that we call collapsed Anosov flows. They are related with Anosov flows via a self orbit equivalence of the flow. We show that all the examples from [BGHP] belong to this class, and that it is an open and closed class among partially hyperbolic diffeomorphisms. We provide some equivalent definitions which may be more amenable to analysis and are useful in different situations. Conversely we describe the class of partially hyperbolic diffeomorphisms that are collapsed Anosov flows associated with certain types of Anosov flows.

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“…To see this, notice first that in this context there cannot be a transverse invariant measure: if a minimal foliation has a transverse invariant measure and one leaf is not a disk, then infinitely many leaves must have non-trivial fundamental group, notice that one can lift a non-trivial loop to nearby leaves, and these cannot become homotopically trivial in their leaves because of Novikov's theorem (recall that a minimal foliation cannot have a Reeb-component). Therefore Candel's theorem applies and there is a smooth Riemannian metric on M such that leaves have negative curvature everywhere (see for example [4,Theorem B]). Thus, one can apply [4, Theorem A] to get a hyperbolic measure for the foliated geodesic flow which produces an infinite number of periodic orbits and each corresponds to a non-trivial closed geodesic in some leaves.…”

Section: Realizing a Cylinder As A Leafmentioning

confidence: 99%

Topology of leaves for minimal laminations by hyperbolic surfaces

Alvarez¹,

Brum²,

Martínez³

et al. 2019

Preprint

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We construct minimal laminations by hyperbolic surfaces whose generic leaf is a disk and contain any prescribed family of surfaces and with a precise control of the topologies of the surfaces that appear. The laminations are constructed via towers of finite coverings of surfaces for which we need to develop a relative version of residual finiteness which may be of independent interest. The main step in establishing this relative version of residual finiteness is to obtain finite covers with control on the second systole of the surface, which is done in the appendix. In a companion paper, the case of other generic leaves is treated.

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

Cited by 8 publications

References 44 publications

Gibbs u-states for the foliated geodesic flow and transverse invariant measures

Gibbs u-states for the foliated geodesic flow and transverse invariant measures

Collapsed Anosov flows and self orbit equivalences

Topology of leaves for minimal laminations by hyperbolic surfaces

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