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One of the key differences between Finsler metrics and Riemannian metrics is the non-reversibility, i.e. given two points p and q, the Finsler distance d(p, q) is not necessarily equal to d(q, p). In this paper, we build the main tools to investigate the non-reversibility in the context of large-scale geometry of uniform Finsler Cartan–Hadamard manifolds.
In the second part of this paper, we use the large-scale geometry to prove the following dynamical theorem: Let φ be the geodesic flow of a closed negatively curved Finsler manifold. If its Anosov splitting is C ² , then its cohomological pressure is equal to its Liouville metric entropy. This result generalizes a previous Riemannian result of U. Hamenstädt.

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... (−∞, 0]) in a Finsler universal covering space ( M , F ). Let S F ( M ) = {u ∈ T M | F (u) = 1}. According to [4], the lifted geometric flip σ F : ...

... By Proposition 3.14 in [4], there exists a unique positive geodesic ray γ such that γ(0) = x and the Hausdorff distance ...

... It is proved in [4] that σ F is a homeomorphism, which is obviously invariant under the natural action of the fundamental group π 1 (M ) over M . Therefore the quotient map σ F : S F M → S F M is well-defined and extends naturally to a map defined on T M which preserves F viewed as a function on T M . ...

A Finsler manifold is said to be geodesically reversible if the reversed curve of any geodesic remains a geometrical geodesic. Well-known examples of geodesically reversible Finsler metrics are Randers metrics with closed [Formula: see text]-forms. Another family of well-known examples are projectively flat Finsler metrics on the [Formula: see text]-sphere that have constant positive curvature. In this paper, we prove some geometrical and dynamical characterizations of geodesically reversible Finsler metrics, and we prove several rigidity results about a family of the so-called Randers-type Finsler metrics. One of our results is as follows: let [Formula: see text] be a Riemannian–Finsler metric on a closed surface [Formula: see text], and [Formula: see text] be a small antisymmetric potential on [Formula: see text] that is a natural generalization of [Formula: see text]-form (see Sec. 1). If the Randers-type Finsler metric [Formula: see text] is geodesically reversible, and the geodesic flow of [Formula: see text] is topologically transitive, then we prove that [Formula: see text] must be a closed [Formula: see text]-form. We also prove that this rigidity result is not true for the family of projectively flat Finsler metrics on the [Formula: see text]-sphere of constant positive curvature.

... Let (M, F ) be a closed C ∞ Finsler manifold of negative flag curvature (see [5]), it is well-known that its geodesic flow defined on S F M is Anosov ( [8]). See, for instance, [7] for a general study of Finsler geodesic flows in negative flag curvature. According to Theorem 2 and the comments concerning the Anosov hypothesis in [12], we have the following proposition: The proof of the following proposition is given only for the convenience of the reader, since its arguments are known. ...

Let (M, F) be a closed C∞ Finsler manifold and φ its geodesic flow. In the case that φ is Anosov, we extend to the Finsler setting a Riemannian vanishing result of M. Gromov about the L∞-cohomology.

... Later, he proved that a symmetric compact Finsler space with negative curvature would be isometric to a locally symmetric negatively curved Riemann space. See [10,8] for more details. Next, Z. Shen considers the case of the negative but not necessarily constant flag curvature by imposing that the S-curvature is constant and proves that the Akbar-Zadeh rigidity theorem still holds in this case, see [16]. ...

Here, using the projectively invariant pseudo-distance and Schwarzian derivative, it is shown that every connected complete Finsler space of the constant negative Ricci scalar is reversible. In particular, every complete Randers metric of constant negative Ricci (or flag) curvature is Riemannian.

... Later, he proved that a symmetric compact Finsler space with negative curvature would be isometric to a locally symmetric negatively curved Riemann space. See [10,8] for more details. Next, Z. Shen considers the case of the negative but not necessarily constant flag curvature by imposing that the S-curvature is constant and proves that the Akbar-Zadeh rigidity theorem still holds in this case, see [16]. ...

Here, using the projectively invariant pseudo-distance and Schwarzian derivative, it is shown that every connected complete Finsler space of the constant negative Ricci scalar is reversible. In particular, every complete Randers metric of constant negative Ricci (or flag) curvature is Riemannian.

... To finish proving our claim about the examples of Theorem 1 we also need the following result (which is not new, see for instance [12,13]) Lemma 13. LetF be a reversible Finsler metric on a manifold M , let β be a 1-form on M such that F * (β) < 1, and let F =F + β. ...

In this article, we show that a Finsler--Laplacian introduced previously can
detect changes in the Finsler metric that the marked length spectrum cannot. We
also construct examples of non-reversible Finsler metrics in negative curvature
such that $4\lambda_1 > h^2$, where $\lambda_1$ is the bottom of the
$L^2$-spectrum and $h$ the topological entropy of the flow.

We first describe the action of the fundamental group of a closed surface of variable negative curvature on the oriented geodesics in its universal covering in terms of a naturally defined flat connection whose holonomy lies in the group of Hamiltonian diffeomorphisms of . Consideration of the holonomy necessitates an extension from Riemannian to Finsler metrics. The second part of the paper follows the Higgs bundle approach to flat connections adapted to this infinite‐dimensional group and focuses on a family of metrics, relying on a construction of O. Biquard, which is parametrised by the infinite‐dimensional space of CR functions on the unit circle bundle of a hyperbolic surface. This generates an alternative approach to defining a connection and offers the possibility of this vector space representing a moduli space which generalizes and includes the classical Teichmüller space.

Let g be a negatively curved Riemannian metric of a closed C
∞ manifold M of dimension at least three. Let L
λ be a C
∞ one-parameter convex superlinear Lagrangian on TM such that \({L_0(v)= \frac{1}{2} g(v, v)}\) for any v ∈ TM. We denote by \({\varphi^\lambda}\) the restriction of the Euler-Lagrange flow of L
λ on the \({\frac{1}{2}}\) -energy level. If λ is small enough then the flow \({\varphi^\lambda}\) is Anosov. In this paper we study the geometric consequences of different assumptions about the regularity of the Anosov distributions of \({\varphi^\lambda}\) . For example, in the case that the initial Riemannian metric g is real hyperbolic, we prove that for λ small, \({\varphi^\lambda}\) has C
3 weak stable and weak unstable distributions if and only if \({\varphi^\lambda}\) is C
∞ orbit equivalent to the geodesic flow of g.

Let (M, F) be a closed C
∞ Finsler manifold. The lift of the Finsler metric F to the universal covering space defines an asymmetric distance [(d)\tilde]{\widetilde d} on [(M)\tilde]{\widetilde M}. It is well-known that the classical comparison theorem of Aleksandrov does not exist in the Finsler setting. Therefore,
it is necessary to introduce new Finsler tools for the study of the asymmetric metric space ([(M)\tilde], [(d)\tilde]){(\widetilde M, \widetilde d)}. In this paper, by using the geometric flip map and the unstable-stable angle introduced in [2], we prove that if (M, F) is a closed Finsler manifold of negative flag curvature, then ([(M)\tilde], [(d)\tilde]){(\widetilde M, \widetilde d)} is an asymmetric δ-hyperbolic space in the sense of Gromov.
Mathematics Subject Classification (2010)37A35–34D20–37D35

In the first part of this dissertation, we give a new definition of a Laplace
operator for Finsler metric as an average, with regard to an angle measure, of
the second directional derivatives. This operator is elliptic, symmetric with
respect to the Holmes-Thompson volume, and coincides with the usual
Laplace--Beltrami operator when the Finsler metric is Riemannian. We compute
explicit spectral data for some Katok-Ziller metrics. When the Finsler metric
is negatively curved, we show, thanks to a result of Ancona that the Martin
boundary is H\"older-homeomorphic to the visual boundary. This allow us to
deduce the existence of harmonic measures and some ergodic preoperties. In the
second part of this dissertation, we study Anosov flows in 3-manifolds, with
leaf-spaces homeomorphic to $\mathbb{R}$. When the manifold is hyperbolic,
Thurston showed that the (un)stable foliations induces a regulating flow. We
use this second flow to study isotopy class of periodic orbits of the Anosov
flow and existence of embedded cylinders.

We describe which Anosov flows on compact manifolds have C∞ stable and unstable distributions and a contact canonical 1-form: up to finite coverings and up to a C∞ change of parameters, each of them is isomorphic to the geodesic flow on (the unit tangent bundle of) a compact locally symmetric space of strictly negative curvature.

I - Differentiable Manifolds.- II - Foliations.- III - The Topology of the Leaves.- IV - Holonomy and the Stability Theorems.- V - Fiber Bundles and Foliations.- VI - Analytic Foliations of Codimension One.- VII - Novikov's Theorem.- VIII - Topological Aspects of the Theory of Group Actions.- Appendix - Frobenius' Theorem.- 1. Vector fields and the Lie bracket.- 2. Frobenius' theorem.- 3. Plane fields defined by differential forms.- Exercises.

By using a formula relating topological entropy and cohomological pressure, we obtain several rigidity results about contact Anosov flows. For example, we prove the following result: Let ϕ be a C1 contact Anosov flow. If its Anosov splitting is C2 and it is C0 orbit equivalent to the geodesic flow of a closed negatively curved Riemannian manifold, then the cohomological pressure and the metric entropy of ϕ coincide. This result generalizes a result of U. Hamenstadt for geodesic flows. AMS Mathematics subject classification: 37A35, 34D20, 37D35, 37D40

We consider transitive Anosov flows φ: M → M and give necessary and sufficient conditions for every homology class in H1(M,) to contain a closed φ-orbit. Under these conditions, we derive an asymptotic formula for the number of closed φ-orbits in a fixed homology class, generalizing a result of Katsuda and Sunada.

Let M denote a closed Riemannian manifold whose geodesic flow is Anosov. Given a real number λ and a smooth one form θ, consider the twisted geodesic flow obtained by twisting the canonical symplectic structure by the lift of λdθ to the tangent bundle of M . For λ in a certain open interval around the origin the twisted flow remains Anosov. We show that the Anosov splitting of the twisted geodesic flow is never of class C 1 unless λ = 0.

Let (M, F) be a closed C
∞ Finsler manifold. The lift of the Finsler metric F to the universal covering space defines an asymmetric distance [(d)\tilde]{\widetilde d} on [(M)\tilde]{\widetilde M}. It is well-known that the classical comparison theorem of Aleksandrov does not exist in the Finsler setting. Therefore,
it is necessary to introduce new Finsler tools for the study of the asymmetric metric space ([(M)\tilde], [(d)\tilde]){(\widetilde M, \widetilde d)}. In this paper, by using the geometric flip map and the unstable-stable angle introduced in [2], we prove that if (M, F) is a closed Finsler manifold of negative flag curvature, then ([(M)\tilde], [(d)\tilde]){(\widetilde M, \widetilde d)} is an asymmetric δ-hyperbolic space in the sense of Gromov.
Mathematics Subject Classification (2010)37A35–34D20–37D35

The geodesic flow of a compact Finsler manifold with negative flag curvature is an Anosov flow [23]. We use the structure of the stable and unstable foliation to equip the geodesic ray boundary of the universal covering with a Hölder structure. Gromov's geodesic rigidity and the Theorem of Dinaburg--Manning on the relation between the topological entropy and the volume entropy are generalized to the case of Finsler manifolds.

The text includes the most recent results, although many of the problems discussed are classical.

This book focuses on the elementary but essential items among these results.

Several of the basic features of automorphic function theory—notably the notion of limit set—can be extended to apply to the study of Riemannian manifolds M of nonpositive curvature. Under somewhat stronger curvature conditions (e.g. K ≦ c < 0) M is called a Visibility manifold. For such manifolds there results a classification into three types: parabolic, axial, and fuchsian. This trichotomy is closely related to many of the most basic topological and geometric properties of M, and such relationships will be examined in some detail. For example, the trichotomy may be expressed in terms of the number (suitably counted) of closed geodesics in M, namely: 0, 1, or ∞, As to methodology: the conventional analytic machinery of C∞ Riemannian geometry is used, at least initially; however, at many crucial points it will be the qualitative behavior of geodesics (ála Busemann) that is important.

Under suitable conditions it is shown how to change the velocity of aC
2 AxiomA attractor so that the Sinai-Ruelle-Bowen measure coincides with the measure of maximal entropy. These measures are obtained as limits of certain closed orbital measures.

. Let M be a closed connected C 1 Riemannian manifold whose geodesic ow is Anosov. Let be a smooth 1-form on M . Given 2 R small, let hEL () be the topological entropy of the Euler-Lagrange ow of the Lagrangian L (x; v) = 1 2 jvj 2 x x (v); and let hF () be the topological entropy of the geodesic ow of the Finsler metric, F (x; v) = jvj x x (v): We show that h 00 EL (0) + h 00 F (0) = h 2 Var(); where Var() is the variance of with respect to the measure of maximal entropy of and h is the topological entropy of . We derive various consequences from this formula. 1. Introduction Let M be a closed connected C 1 Riemannian manifold whose geodesic ow is Anosov. This happens for example, when all the sectional curvatures are negative. Let be a smooth 1-form on M . We think of as a function : TM ! R such that for each x 2 M , x is a linear functional of T x M . For 2 R consider the 1-parameter family of convex superlinear Lagrangians ...

Some new developments in Finsler geometry

- D Egloff

D. Egloff, Some new developments in Finsler geometry, PH.D thesis,
Universität Freiburg, (1995).

Feuilletages des sphères et dynamiques Nord-Sud

- P Foulon

P. Foulon, Feuilletages des sphères et dynamiques Nord-Sud, C. R.
Acad. Sci. Paris, Ser. I, 318 (1994), 1041-1042.