Non-stationary smooth geometric structures for contracting measurable cocycles

Melnick, Karin

doi:10.1017/etds.2017.38

Cited by 6 publications

(7 citation statements)

References 14 publications

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“…Instead, we replace the generalized Kanai connection with a similar one assembled from the flow invariant system of measurable affine connections on unstable manifolds constructed by Melnick in [Mel16]. The connections are defined on whole unstable manifolds but they are only defined for unstable manifolds W u (v) for v in a set of full measure.…”

Section: Non-strictly 1 4 -Pinched Casementioning

confidence: 99%

“…Following the notation in [Mel16], let her E be the smooth tautological bundle over SM whose fiber at v is W u (v). We consider the cocycle F t v which is g t restricted to W u (v).…”

Section: Non-strictlymentioning

confidence: 99%

“…We consider the cocycle F t v which is g t restricted to W u (v). The ratio of maximal to minimal positive Lyapunov exponents lies in [1, 2), and hence the integer r appearing in Theorem 3.12 of [Mel16] is 1. This theorem then reads in our notation as: Lemma 6.1.…”

Section: Non-strictlymentioning

confidence: 99%

“…By a result of Connell [Con03] relying on Theorem 1.2, if M is not already a locally symmetric space, then there is no uniform 2 : 1 resonance in the Lyapunov spectrum. Now we can use recent work of Melnick [Mel16] on normal forms to obtain a suitably invariant connection (cf. also ).…”

Section: Introductionmentioning

confidence: 99%

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Hyperbolic rank rigidity for manifolds of -pinched negative curvature

Connell¹,

Nguyen²,

Spatzier³

2018

Ergod. Th. Dynam. Sys.

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A Riemannian manifold M has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature -1 with the geodesic. If in addition, the sectional curvatures of M lie in the interval [−1, − 1 4 ], and M is closed, we show that M is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial converse to Hamenstädt's hyperbolic rank rigidity result for sectional curvatures ≤ −1, and complements well-known results on Euclidean and spherical rank rigidity.is clear that the general case will be much more difficult, even if we assume that the metric has negative or at least non-positive curvature.Finally let us note a consequence of Theorem 1.1 in terms of dynamics. Consider the geodesic flow g t on the unit tangent bundle of a closed manifold M. For a geodesic c ⊂ M, the maximal Lyapunov exponent λ max (c), for c is the biggest exponential growth rate of the norm of a Jacobi field J(t) along c:Note that λ max (c) ≤ 1 if the sectional curvatures of M are bounded below by −1, by Rauch's comparison theorem.Given an ergodic g t -invariant measure µ on the unit tangent bundle SM, λ max (c) is constant µ-a.e.. In fact, it is just the maximal Lyapunov exponent in the sense of dynamical systems for g t and µ (cf. Section 4).Corollary 1.4. Let M be a closed Riemannian manifold with sectional curvatures K between −1 ≤ K ≤ − 1 4 . Let µ be a probability measure of full support on the unit tangent bundle SM which is invariant and ergodic under the geodesic flow g t . Suppose that the maximal Lyapunov exponent for g t and µ is 1. Then M is a rank one locally symmetric space.We supply a proof in Section 6. In fact, the reduction to Theorem 1.1 is identical to Constantine's in [Con08, Section 6] which in turn adapts an argument of Connell for upper curvature bounds [Con03]. Definitions, Semicontinuity and Invariance on Stable ManifoldsLet M be compact manifold of negative sectional curvature, and denote its unit tangent bundle by SM. We let g t : SM → SM be the geodesic flow, and denote by pt : SM → M the footpoint map, i.e. v ∈ T pt(v) M. For v ∈ SM, let c v be the geodesic determined by v and let v ⊥ denote the perpendicular complement of v in T pt(v) M. Recall that rk h (v) is the dimension of the subspace of v ⊥ which are the initial vectors of Jacobi fields that make curvature −1 with g t v for all t ≥ 0, and rk h (M) is the minimum of rk h (v) for v ∈ SM.Lemma 2.1. Let v be a unit vector recurrent under the geodesic flow. Suppose that rk h (v) > 0. Then there is also an unstable or stable Jacobi field making curvature -1 with g t v for all t ∈ R.Proof. Since rk h (v) > 0, there is a Jacobi field J(t) making curvature -1 with g t v for all t ≥ 0. First assume that J(t) is not stable. Decompose J(t) into its stable and unstable components J(t) = J s (t) + J u (t). Suppose g tn v → v with t n → ∞. Then, for a suitable subsequence of t n , J(t+tn) J(tn) will converge to a Jacobi field Y (t) along c v (t). Note then that...

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Section: Non-strictly 1 4 -Pinched Casementioning

confidence: 99%

“…Following the notation in [Mel16], let her E be the smooth tautological bundle over SM whose fiber at v is W u (v). We consider the cocycle F t v which is g t restricted to W u (v).…”

Section: Non-strictlymentioning

confidence: 99%

Section: Non-strictlymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hyperbolic rank rigidity for manifolds of -pinched negative curvature

Connell¹,

Nguyen²,

Spatzier³

2018

Ergod. Th. Dynam. Sys.

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“…In the case Q u = E u /V u , with V u = {0}, that we consider in this paper, we point also to work of Melnick [44] in which smooth invariant connections on quotient bundles are constructed in the case of nonuniformly contracting foliations. These results resemble ours, however we are able to derive much stronger properties of these connections under weaker hypotheses in our setting because we assume the existence of a dominated splitting instead of a measurable invariant splitting.…”

Section: Quotient Normal Formsmentioning

confidence: 99%

Characterizing symmetric spaces by their Lyapunov spectra

Butler¹

2017

Preprint

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We prove that closed negatively curved locally symmetric spaces are characterized up to isometry among all homotopy equivalent negatively curved manifolds by the Lyapunov spectra of the periodic orbits of their geodesic flows. This is done by constructing a new invariant measure for the geodesic flow that we refer to as the horizontal measure. We show that the Lyapunov spectrum of the horizontal measure alone suffices to locally characterize these locally symmetric spaces up to isometry. We associate to the horizontal measure a new invariant, the horizontal dimension. We tie this invariant to extensions of curvature pinching rigidity theorems for complex hyperbolic manifolds to pinching rigidity theorems for the Lyapunov spectrum. Our methods extend to give a rigidity theorem for smooth Anosov flows f t orbit equivalent to the geodesic flow g t X of a closed negatively curved locally symmetric space X: f t is smoothly orbit equivalent to g t X if and only if its Lyapunov spectra on all periodic orbits are a multiple of the corresponding Lyapunov spectra for g t X .

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On classification of higher rank Anosov actions on compact manifold

Damjanović

2020

Isr. J. Math.

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Non-stationary smooth geometric structures for contracting measurable cocycles

Cited by 6 publications

References 14 publications

Hyperbolic rank rigidity for manifolds of -pinched negative curvature

Hyperbolic rank rigidity for manifolds of -pinched negative curvature

Characterizing symmetric spaces by their Lyapunov spectra

On classification of higher rank Anosov actions on compact manifold

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