Ergodic properties of equilibrium measures for smooth three dimensional flows

Ledrappier, François; Lima, Yuri; Sarig, Omri

doi:10.4171/cmh/378

Cited by 29 publications

(45 citation statements)

References 35 publications

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“…And in [12], we show that µ max is an ergodic measure with full mass on the rank 1 set (also called the regular set). Then by Theorem 1.2 of [10], the geodesic flow is Bernoulli with respect to the unique maximal entropy measure. This wraps up the proof of Theorem 2.4.…”

Section: The Bernoulli Propertymentioning

confidence: 99%

The transitivity of geodesic flows on rank 1 manifolds without focal points

Liu

2018

Differential Geometry and its Applications

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Section: The Bernoulli Propertymentioning

confidence: 99%

The transitivity of geodesic flows on rank 1 manifolds without focal points

Liu

2018

Differential Geometry and its Applications

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“…The next lemma is used in [LLS16]. Recall from Lemma 2.6 that there is a set Λ * χ of full µ Λ -measure s.t.…”

Section: Symbolic Markov Propertymentioning

confidence: 99%

Symbolic dynamics for three-dimensional flows with positive topological entropy

Lima

Sarig

2018

J. Eur. Math. Soc.

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We construct symbolic dynamics on sets of full measure (with respect to an ergodic measure of positive entropy) for C 1+ε flows on closed smooth three dimensional manifolds. One consequence is that the geodesic flow on the unit tangent bundle of a closed C ∞ surface has at least const ×(e hT /T ) simple closed orbits of period less than T , whenever the topological entropy h is positive -and without further assumptions on the curvature. Kat82]. It extends to flows of lesser regularity, under the additional assumption that they possess a measure of maximal entropy (Theorem 8.1). The lower bound Ce ht /T is sharp in many special cases [Hub59, Mar69, PP83, Kni97], but not in the general setup of this paper. For more on this, see §8. The theorem strengthens Katok's bound lim infWe obtain Theorem 1.1 by constructing a symbolic model that is a finite-to-one extension of ϕ. The orbits of this model are easier to understand than those of the original flow. This technique, called "symbolic dynamics", can be traced back to the work of Hadamard, Morse, Artin, and Hedlund.We proceed to describe the symbolic models used in this work. Let G be a directed graph with a countable set of vertices V . We write v → w if there is an edge from v to w, and we assume throughout that for every v there are u, w s.t.Birkhoff cocycle: Suppose r : Σ → R is a function. The Birkhoff sums of r are r n := r + r • σ + · · · + r • σ n−1 (n ≥ 1). There is a unique way to extend the definition to n ≤ 0 in such a way that the cocycle identity r m+n = r n + r m • σ n holds for all m, n ∈ Z: r 0 := 0 and r n := −r |n| • σ −|n| (n < 0). Topological Markov flow: Suppose r : Σ → R + is Hölder continuous and bounded away from zero and infinity. The topological Markov flow with roof function r and base map σ : Σ → Σ is the flow σUniform Poincaré section: The Poincaré section Λ is called uniform if its roof function is bounded away from zero and infinity. If Λ is uniform, then µ Λ is finite and it can be normalized. With this normalization, for every Borel subset E ⊂ Λ and 0All the Poincaré sections considered in this paper will be uniform, and each of them will be the disjoint union of finitely many embedded smooth two dimensional discs. Let ∂Λ denote the union of the boundaries of these discs. The set ∂Λ will introduce discontinuities to the Poincaré map of Λ. Singular set: The singular set of a Poincaré section Λ is S(Λ) :=    p ∈ Λ : p does not have a relative neighborhood V ⊂ Λ \ ∂Λ s.t. V is diffeomorphic to an open disc, and f Λ : V → f Λ (V ) and f −1 Λ : V → f −1 Λ (V ) are diffeomorphisms    . Regular set: Λ := Λ \ S(Λ). SYMBOLIC DYNAMICS FOR FLOWS 5 Basic constructions. Let ϕ be a flow satisfying our standing assumptions.Canonical transverse disc:The following lemmas are standard, see the appendix for proofs.Lemma 2.1. There is a constant r s > 0 which only depends on M and ϕ s.t. for every p ∈ M and 0 < r < r s , S := S r (p) is a C ∞ embedded closed disc, | (X q , T q S)| ≥ 1 2 radians for all q ∈ S, and dist M (·, ·) ≤ dist S (·, ·) ≤ 2 d...

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“…It was using this type of approach that M. Ratner [36] proved that Anosov flows with u-Gibbs measures are Bernoulli. Also using a symbolic approach, F. Ledrappier, Y. Lima and O. Sarig [24] proved that, with respect to an ergodic measure of maximum entropy, smooth flows with positive speed and positive topological entropy on a compact smooth three dimensional manifold are either Bernoulli or isomorphic to the product of a Bernoulli flow and a rotational flow.…”

Section: Technical Considerationsmentioning

confidence: 99%

“…If we denote µ P R i := m P R i (·/R i ) we can rewrite the previous equality as, For a detailed proof of this Lemma see [11] Pg. 24 Proof. See [11], Pg.26.…”

Section: Second Bad Setmentioning

confidence: 99%

On the Bernoulli property for certain partially hyperbolic diffeomorphisms

Ponce

Tahzibi²,

Varão

2018

Advances in Mathematics

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We address the classical problem of equivalence between Kolmogorov and Bernoulli property of smooth dynamical systems. In a natural class of volume preserving partially hyperbolic diffeomorphisms homotopic to Anosov ("derived from Anosov") on 3-torus, we prove that Kolmogorov and Bernoulli properties are equivalent.In our approach, we propose to study the conditional measures of volume along central foliation to recover fine ergodic properties for partially hyperbolic diffeomorphisms. As an important consequence we obtain that there exists an almost everywhere conjugacy between any volume preserving derived from Anosov diffeomorphism of 3-torus and its linearization.Our results also hold in higher dimensional case when central bundle is one dimensional and stable and unstable foliations are quasi-isometric.

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Ergodic properties of equilibrium measures for smooth three dimensional flows

Cited by 29 publications

References 35 publications

The transitivity of geodesic flows on rank 1 manifolds without focal points

The transitivity of geodesic flows on rank 1 manifolds without focal points

Symbolic dynamics for three-dimensional flows with positive topological entropy

On the Bernoulli property for certain partially hyperbolic diffeomorphisms

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