Topological mixing of the geodesic flow on convex projective manifolds

Blayac, Pierre-Louis

doi:10.48550/arxiv.2009.05035

Cited by 3 publications

(8 citation statements)

References 8 publications

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“…All the convex projective manifolds studied in the work mentioned above, as well as in the present paper and in [9,12], satisfy the rank-one condition, which was introduced by M. Islam [42] and A. Zimmer [72].…”

Section: Theorem Cmentioning

confidence: 73%

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Ergodicity and equidistribution in Hilbert geometry

Blayac,

Zhu

2023

JMD

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We show that dynamical and counting results characteristic of negatively curved Riemannian geometry, or more generally CAT(−1) or rankone CAT(0) spaces, also hold for rank-one properly convex projective manifolds or orbifolds, equipped with their Hilbert metrics, admitting finite Sullivan measures built from appropriate conformal densities. In particular, this includes geometrically finite convex projective manifolds or orbifolds whose universal covers are strictly convex with C 1 boundary.More specifically, with respect to the Sullivan measure, the Hilbert geodesic flow is strongly mixing, and orbits and primitive closed geodesics equidistribute, allowing us to asymptotically enumerate these objects.

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Section: Theorem Cmentioning

confidence: 73%

“…In [9] and [12], the first author obtained good dynamical properties for the Hilbert geodesic flow in the setting of rank-one properly convex Hilbert geometries, including topological mixing and strong mixing of the geodesic flow with respect to a Sullivan measure (constructed via a Patterson-Sullivan density).…”

Section: Pierre-louis Blayac and Feng Zhumentioning

confidence: 99%

Ergodicity and equidistribution in Hilbert geometry

Blayac,

Zhu

2023

JMD

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“…4.19] established that Γ acts convex cocompactly on Ω if and only if C cor Ω (Γ) is non-empty and Λ con Ω (Γ) and Λ orb Ω (Γ) are equal and closed. In [Blab,Def. 1.1] we introduced a (φ t ) t∈R -invariant closed subset of T 1 M , called the biproximal unit tangent bundle and denoted by T 1 M bip ; it consists of those vectors v ∈ T 1 M such that φ ±∞ ṽ ∈ Λ prox for any lift ṽ ∈ T 1 Ω.…”

Section: The Hopf-tsuji-sullivan-roblin Dichotomymentioning

confidence: 99%

“…In this section we prove that the W ss and W su -invariant functions on T 1 M are essentially constant, and we derive as a corollary that the flow is ergodic and mixing. We will need the following result, about the local non-arithmeticity of the length spectrum; it is similar to [Blab,Prop. 4.1] but does not need the assumption of strong irreducibility.…”

Section: A Cross-ratio On the Boundarymentioning

confidence: 99%

Patterson--Sullivan densities in convex projective geometry

Blayac¹

2021

Preprint

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For any rank-one convex projective manifold with a compact convex core, we prove that there exists a unique probability measure of maximal entropy on the set of unit tangent vectors whose geodesic is contained in the convex core, and that it is mixing. We use this to establish asymptotics for the number of closed geodesics. In order to construct the measure of maximal entropy, we develop a theory of Patterson-Sullivan densities for general rankone convex projective manifolds. In particular, we establish a Hopf-Tsuji-Sullivan-Roblin dichotomy, and prove that, when it is finite, the measure on the unit tangent bundle induced by a Patterson-Sullivan density is mixing under the action of the geodesic flow.

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“…a rank-one symmetric divisible convex set, then Λ prox Ω = ∂Ω and Aut(Ω) acts transitively on it. If Ω is a higher-rank symmetric irreducible divisible convex set, then Λ prox Γ is an analytic submanifold of P(V ) of dimension less than dim(V ) − 2, and hence is a proper subset of ∂Ω (see [Blaa,§7]), on which Aut(Ω) acts transitively.…”

Section: Introductionmentioning

confidence: 99%

The boundary of rank-one divisible convex sets

Blayac¹

2021

Preprint

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We prove that for any non-symmetric irreducible divisible convex set, the proximal limit set is the full projective boundary.If Ω is symmetric, then it naturally identifies with the Riemannian symmetric space of Aut(Ω), and there is yet another natural dichotomy: namely, either Aut(Ω) has real rank 1, in which case Ω is an ellipsoid and Aut(Ω) is isomorphic to PO(n, 1) for n = dim(V ) − 1, or Aut(Ω) has real rank greater than one, it is isomorphic to PGL(n, K) for some n ≥ 3, and for K = R, C, or the classical division algebra of quaternions, or of octonions if n = 3 (see for instance [Ben08, §2.4]).Recently, A. Zimmer proved the following higher-rank rigidity result [Zim, Th. 1.4], analogous to a celebrated result in Riemannian geometry by Ballmann [Bal85] and Burns-Spatzier [BS87].If Ω is not symmetric, then it is rank-one in the following sense.Definition 1.1. A divisible convex set Ω ⊂ P(V ) is said to be rank-one if there exists in ∂Ω a strongly extremal point, namely a point ξ ∈ ∂Ω such that [ξ, η]∩Ω is non-empty for any η ∈ ∂Ω {ξ} (in other words, ξ is "visible" from any other point of the projective boundary).The notion of rank-one divisible convex sets (and more generally of rank-one geodesics, automorphisms, groups of automorphisms, quotients of properly convex open sets, which we do not define

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Topological mixing of the geodesic flow on convex projective manifolds

Cited by 3 publications

References 8 publications

Ergodicity and equidistribution in Hilbert geometry

Ergodicity and equidistribution in Hilbert geometry

Patterson--Sullivan densities in convex projective geometry

The boundary of rank-one divisible convex sets

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