Une version feuilletée équivariante du théorème de translation de Brouwer

The Brouwer’s plane translation theorem asserts that for a fixed point free orientation preserving homeomorphism f of the plane, every point belongs to a Brouwer line: a proper topological embedding C of R, disjoint from its image and separating f(C) and f^–1(C). Suppose that f commutes with the elements of a discrete group G of orientation preserving homeomorphisms acting freely and properly on the plane. We will construct a G-invariant topological foliation of the plane by Brouwer lines. We apply this result to give simple proofs of previous results about area-preserving homeomorphisms of surfaces and to prove the following theorem: any Hamiltonian homeomorphism of a closed surface of genus g ≥ 1 has infinitely many contractible periodic points.      

Geodesics Closed On The Projective Plane

Given a C ^∞ Riemannian metric g on P ² we prove that (, g) has constant curvature iff all geodesics are closed. Therefore is the first non-trivial example of a manifold such that the smooth Riemannian metrics which involve that all geodesics are closed are unique up to isometries and scaling. This remarkable phenomenon is not true on the 2-sphere, since there is a large set of C ^∞ metrics whose geodesics are all closed and have the same period 2π (called Zoll metrics), but no metric of this set can be obtained from another metric of this set via an isometry and scaling. As a corollary we conclude that all two-dimensional P-manifolds are SC-manifolds. Received: April 2007; Revision: September 2007; Accepted: September 2007     

On the average indices of closed geodesics on positively curved Finsler spheres

In this paper, we prove that on every Finsler n-sphere (S ⁿ , F) for n ≥  6 with reversibility λ and flag curvature K satisfying ${(\frac{\lambda}{\lambda+1})^2 \, < \, K \, \le \, 1}$ , either there exist infinitely many prime closed geodesics or there exist ${[\frac{n}{2}]-2}$ closed geodesics possessing irrational average indices. If in addition the metric is bumpy, then there exist n?3 closed geodesics possessing irrational average indices provided the number of prime closed geodesics is finite.     

Multiplicity of closed geodesics on Finsler spheres with irrationally elliptic closed geodesics

If all prime closed geodesics on (Sⁿ, F) with an irreversible Finsler metric F are irrationally elliptic, there exist either exactly 2 \(\left[ {\frac{{n + 1}}{2}} \right]\) or infinitely many distinct closed geodesics. As an application, we show the existence of three distinct closed geodesics on bumpy Finsler (S³, F) if any prime closed geodesic has non-zero Morse index.     

On the curvature of the diffeomorphisms group

An expression for the sectional curvature ofSDIFF(M) (the group of diffeomorphism preserving Riemannian density on a closed manifoldM) is obtained. In the case of a locally Euclidean manifoldM, the negativeness of curvature that implies the instability of solutions of Euler equations of ideal incompressible fluids onM is established.     

A simple proof of P. Carter's theorem

We give a simple proof of the following result of P. Carter: Given a twist homeomorphism of an annulus with at most one fixed point in the interior of the annulus, then there exists an essential simple closed curve inside this annulus meeting its image in at most the (possible) interior fixed point.

     

Closed geodesics on positively curved Finsler spheres

In this paper, we prove that for every Finsler n-sphere (Sn,F) for n?3 with reversibility λ and flag curvature K satisfying , either there exist infinitely many prime closed geodesics or there exists one elliptic closed geodesic whose linearized Poincaré map has at least one eigenvalue which is of the form exp(πiμ) with an irrational μ. Furthermore, there always exist three prime closed geodesics on any (S³,F) satisfying the above pinching condition.     

Behavior of Distant Maximal Geodesics in Finitely Connected Complete Two-Dimensional Riemannian Manifolds II

We study the behavior of maximal geodesics in a finitely connected complete two-dimensional Riemannian manifold M admitting curvature at infinity. In the case where M is homeomorphic to ² the Cohn–Vossen theorem states that the total curvature of M, say c(M), is 2. We already studied the case c(M)<2 in our previous paper. So we study the behavior of geodesics in M with total curvature 2 in this paper. Next we consider the case where M has nonempty boundary. In order to know the behavior of distant geodesics in M with boundary, it is useful to investigate the 'visual image' of the boundary of M. The latter half of this paper will be spent to study the asymptotic behavior of the visual image of a subset of M with located point tending to infinity.     

Local product structure for expansive homeomorphisms

Let be an expansive homeomorphism with dense topologically hyperbolic periodic points, M a closed manifold. We prove that there is a local product structure in an open and dense subset of M. Moreover, if some topologically hyperbolic periodic point has codimension one, then this local product structure is uniform. In particular, we conclude that the homeomorphism is conjugated to a linear Anosov diffeomorphism of a torus.     

On the volume of certain manifolds with closed geodesics

It is proved that the ratio of the volume of a Riemannian manifold of cohomology typeCP ² orCP ³ for which all geodesics are closed and of length 2 to the volume of the standard sphere assumes a fixed value.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 102–106     

Translated points and Rabinowitz Floer homology

We prove that if a contact manifold admits an exact filling, then every local contactomorphism isotopic to the identity admits a translated point in the interior of its support, in the sense of Sandon [Internat. J. Math. 23 (2012), 1250042]. In addition, we prove that if the Rabinowitz Floer homology of the filling is nonzero, then every contactomorphism isotopic to the identity admits a translated point, and if the Rabinowitz Floer homology of the filling is infinite dimensional, then every contactomorphism isotopic to the identity has either infinitely many translated points, or a translated point on a closed leaf. Moreover, if the contact manifold has dimension greater than or equal to 3, the latter option generically does not happen. Finally, we prove that a generic compactly supported contactomorphism on ${\mathbb{R}^{2n+1}}$ has infinitely many geometrically distinct iterated translated points all of which lie in the interior of its support.     

Periodic points and rotation numbers for area preserving diffeomorphisms of the plane

Letf be an orientation preserving diffeomorphism ofR ² which preserves area. We prove the existence of infinitely many periodic points with distinct rotation numbers around a fixed point off, provided only thatf has two fixed points whose infinitesimal rotation numbers are not both 0. We also show that if a fixed pointz off is enclosed in a “simple heteroclinic cycle” and has a non-zero infinitesimal rotation numberr, then for every non-zero rational numberp/q in an interval with endpoints 0 andr, there is a periodic orbit inside the heteroclinic cycle with rotation numberp/q aroundz.     

Closed geodesics and volume growth of open manifolds with sectional curvature bounded from below

In this paper, the relationship between the existence of closed geodesics and the volume growth of complete noncompact Riemannian manifolds is studied. First the authors prove a diffeomorphic result of such an n-m2nifold with nonnegative sectional curvature, which improves Marenich-Toponogov＇s theorem. As an application, a rigidity theorem is obtained for nonnegatively curved open manifold which contains a clesed geodesic. Next the authors prove a theorem about the nonexistence of closed geodesics for Riemannian manifolds with sectional curvature bounded from below by a negative constant.     

Grassmann manifold of aJH-algebra

We study the set of rankp idempotents in a topologically simple Hilbert Jordan algebra (JH-algebra for short). To produce the differential geometric structure on, we establish Jordan algebraic results concerning the structure of some two-generator subalgebras. We identify geodesics, the Riemannian distance and the sectional curvature of by using the Jordan algebraic structure.     

A symmetry of sphere map implies its chaos*

A well-known example, given by Shub, shows that for any |d| ≥ 2 there is a self-map of the sphere Sⁿ, n ≥ 2, of degree d for which the set of non-wandering points consists of two points. It is natural to ask which additional assumptions guarantee an infinite number of periodic points of such a map. In this paper we show that if a continuous map f : Sⁿ → Sⁿ commutes with a free homeomorphism g : Sⁿ → Sⁿ of a finite order, then f has infinitely many minimal periods, and consequently infinitely many periodic points. In other words the assumption of the symmetry of f originates a kind of chaos. We also give an estimate of the number of periodic points. *Research supported by KBN grant nr 2 P03A 045 22.     

Closed geodesics on orbifolds

In this paper, we try to generalize to the case of compact Riemannian orbifolds Q some classical results about the existence of closed geodesics of positive length on compact Riemannian manifolds M. We shall also consider the problem of the existence of infinitely many geometrically distinct closed geodesics.In the classical case the solution of those problems involve the consideration of the homotopy groups of M and the homology properties of the free loop space on M (Morse theory). Those notions have their analogue in the case of orbifolds. The main part of this paper will be to recall those notions and to show how the classical techniques can be adapted to the case of orbifolds.     

On the Existence of Focal Points near Closed Geodesics on Surfaces

We establish some criteria for the existence or nonexistence of focal points near closed geodesics on surfaces. These criteria are in terms of the curvature of the manifold along the closed geodesic and the average values of the partial derivatives of the curvature in the direction perpendicular to the geodesic. Our criteria lead to a new family of examples of surfaces with no focal points. We also show that if S is a compact surface with no focal points and an inequality relating the curvature of the surface to the curvature of the horocycles holds, then the horocycles (considered as curves in S) are uniformly C ^2+Lipschitz.     

Flatness of Gaussian curvature and area of ideal triangles

Let M be aC ^k ,k 4, compact surface of genus greater than two whose curvature is negative in all points but along a simple closed geodesic (t) where the curvature is zero at every point. We show that the area of ideal triangles having a lifting of as an edge is infinite. This provides a family of surfaces having ideal triangles of infinite area whose geodesic flows are equivalent to Anosov flows, in contrast with the well-known examples of surfaces with flat strips which also have ideal triangles of infinite area. By the CAT-comparison theory we can deduce, using these surfaces as models, that aC compact surface of non-positive curvature having one geodesic along which the curvature is zero has ideal triangles of infinite area.Partially supported by CNPq of Brazilian Government     

The curvature of the Sasaki metric of tangent sphere bundles

We study the sectional curvaturesK of the Sasaki metric of tangent sphere bundles over spaces of constant curvatureK(T ₁(M ⁿ, K)). We give precise bounds on the variation of the Ricci curvature and a bound on the scalar curvature ofT ₁ (M ⁿ, K) that is uniform onK. In an appendix we calculate and give lower bounds for the lengths of closed geodesics onT ₁ S ⁿ. titles.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 132–145.     

A topological version of the Poincaré-Birkhoff theorem with two fixed points

The main result of this paper gives a topological property satisfied by any homeomorphism of the annulus \mathbb A = \mathbb S¹ ×[-1, 1]{\mathbb {A} = \mathbb {S}^1 \times [-1, 1]} isotopic to the identity and with at most one fixed point. This generalizes the classical Poincaré-Birkhoff theorem because this property certainly does not hold for an area preserving homeomorphism h of \mathbb A{\mathbb {A}} with the usual boundary twist condition. We also have two corollaries of this result. The first one shows in particular that the boundary twist assumption may be weakened by demanding that the homeomorphism h has a lift H to the strip [(\mathbbA)\tilde] = \mathbbR ×[-1, 1]{\tilde{\mathbb{A}} = \mathbb{R} \times [-1, 1]} possessing both a forward orbit unbounded on the right and a forward orbit unbounded on the left. As a second corollary we get a new proof of a version of the Conley–Zehnder theorem in \mathbb A{\mathbb {A}} : if a homeomorphism of \mathbb A{\mathbb {A}} isotopic to the identity preserves the area and has mean rotation zero, then it possesses two fixed points.