共查询到20条相似文献,搜索用时 621 毫秒
1.
Motoko Kotani 《Mathematische Annalen》2001,320(3):507-529
The counting function of prime closed geodesics in a fixed homology class is investigated for a closed negatively curved
manifold. The coefficients in its asymptotic expansion is described by means of dynamical properties of the geodesic flow.
Their dependency on the homology class is clarified by looking at the dominant terms.
Received June 22, 1999 / Published online March 12, 2001 相似文献
2.
Richard Sharp 《Transactions of the American Mathematical Society》2004,356(12):4897-4908
In this paper, we study the distribution of closed geodesics on a compact negatively curved manifold. We concentrate on geodesics lying in a prescribed homology class and, under certain conditions, obtain a local limit theorem to describe the asymptotic behaviour of the associated counting function as the homology class varies.
3.
The geodesic flow of any Riemannian metric on a geodesically convex surface of negative Euler characteristic is shown to be semi-equivalent to that of any hyperbolic metric on a homeomorphic surface for which the boundary (if any) is geodesic. This has interesting corollaries. For example, it implies chaotic dynamics for geodesic flows on a torus with a simple contractible closed geodesic, and for geodesic flows on a sphere with three simple closed geodesics bounding disjoint discs.
4.
Thirteen simple closed geodesics are found in the lemniscate. Among these are nine “mirrors”—geodesics of reflection symmetry—which generate the full octahedral group and determine a triangulation of the lemniscate as a disdyakis dodecahedron. New visualizations of the lemniscate are presented. 相似文献
5.
Given a smooth closed manifold M, the Morse–Witten complex associated to a Morse function f and a Riemannian metric g on M consists of chain groups generated by the critical points of f and a boundary operator counting isolated flow lines of the negative gradient flow. Its homology reproduces singular homology of M. The geometric approach presented here was developed in Weber [Der Morse–Witten Komplex, Diploma Thesis, TU Berlin, 1993] and is based on tools from hyperbolic dynamical systems. For instance, we apply the Grobman–Hartman theorem and the λ-lemma (Inclination Lemma) to analyze compactness and define gluing for the moduli space of flow lines. 相似文献
6.
In this paper, we use Chas–Sullivan theory on loop homology and Leray–Serre spectral sequence to investigate the topological structure of the non-contractible component of the free loop space on the real projective spaces with odd dimensions. Then we apply the result to get the resonance identity of non-contractible homologically visible prime closed geodesics on such spaces provided the total number of distinct prime closed geodesics is finite. 相似文献
7.
Mark Pollicott Richard Sharp 《Proceedings of the American Mathematical Society》2007,135(12):3887-3894
We describe how closed geodesics lying in a prescribed homology class on a negatively curved manifold split when lifted to a finite cover. This generalizes a result of Zelditch in the case of compact hyperbolic surfaces.
8.
Jan Philipp Schröder 《Israel Journal of Mathematics》2017,217(1):197-229
Given an arbitrary Riemannian metric on a closed surface, we consider length-minimizing geodesics in the universal cover. Morse and Hedlund proved that such minimal geodesics lie in bounded distance of geodesics of a Riemannian metric of constant curvature. Knieper asked when two minimal geodesics in bounded distance of a single constant-curvature geodesic can intersect. In this paper we prove an asymptotic property of minimal rays, showing in particular that intersecting minimal geodesics as above can only occur as heteroclinic connections between pairs of homotopic closed minimal geodesics. A further application characterizes the boundary at infinity of the universal cover defined by Busemann functions. A third application shows that flat strips in the universal cover of a nonpositively curved surface are foliated by lifts of closed geodesics of a single homotopy class. 相似文献
9.
In this paper we study the convergence behavior of grafting rays to the Thurston boundary of Teichmüller space. When the grafting is done along a weighted system of simple closed curves or along a maximal uniquely ergodic lamination this behavior is the same as for Teichmüller geodesics and lines of minima. We also show that the rays grafted along a weighted system of simple closed curves are at bounded distance from Teichmüller geodesics. 相似文献
10.
Paul Norbury 《Geometriae Dedicata》2008,134(1):153-176
We give an identity involving sums of functions of lengths of simple closed geodesics, known as a McShane identity, on any
non-orientable hyperbolic surface with boundary which generalises Mirzakhani’s identities on orientable hyperbolic surfaces
with boundary.
相似文献
11.
K. Guruprasad 《Topology》2006,45(3):611-641
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. 相似文献
12.
John Franks 《Inventiones Mathematicae》1992,108(1):403-418
Summary We show that an area preserving homeomorphism of the open or closed annulus which has at least one periodic point must in fact have infinitely many interior periodic points. A consequence is the theorem that every smooth Riemannian metric onS
2 with positive Gaussian curvature has infinitely many distinct closed geodesics.In this paper we investigate area preserving homeomorphisms of the annulus and their periodic points. The main result is that an area preserving homeomorphism of the annulus which has at least one periodic point (perhaps on the boundary) must in fact have infinitely many interior periodic points.The motivation and main application of this result is the furthering of a program begun by Birkhoff [B] in his book Dynamical Systems. There he shows that for many Riemannian metrics onS
2, including those with positive curvature, the problem of finding closed geodesics reduces to finding periodic points of a certain area preserving homeomorphism of the annulus. The annulus map in question can be shown to have a periodic point so our main result above can be applied to show the existence of infinitely many distinct closed geodesics whenever this annulus map exists. This is done in Sect. 4 Other quite different approaches to the problem of finding infinitely many geodesics have been successful in handling the cases which do not reduce to the investigation of an annulus homeomorphism (see [Ba]).Oblatum 20-III-1991 & 6-XI-1991 相似文献
13.
14.
We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka’s sutured instanton Floer homology theory. This is the first invariant of contact manifolds—with or without boundary—defined in the instanton Floer setting. We prove that our invariant vanishes for overtwisted contact structures and is nonzero for contact manifolds with boundary which embed into Stein fillable contact manifolds. Moreover, we propose a strategy by which our contact invariant might be used to relate the fundamental group of a closed contact 3-manifold to properties of its Stein fillings. Our construction is inspired by a reformulation of a similar invariant in the monopole Floer setting defined by Baldwin and Sivek (arXiv:1403.1930, 2014). 相似文献
15.
16.
Igor Rivin 《Geometriae Dedicata》2001,87(1-3):345-360
We study simple closed geodesics on a hyperbolic surface of genus g with b geodesic boundary components and c cusps. We show that the number of such geodesics of length at most L is of order L
6g+2b+2c–6. This answers a long-standing open question. 相似文献
17.
Claire M. Baribaud 《Israel Journal of Mathematics》1999,109(1):339-347
We study the non-simple closed geodesics of the Riemann surfaces of signature (0, 3). In the aim of classifying them, we define
one parameter: the number of strings. We show that for a given number of strings, a minimal geodesic exists; we then give
its representation and its length which depends on the boundary geodesics. 相似文献
18.
K. Bromberg 《Journal of the American Mathematical Society》2004,17(4):783-826
Given a geometrically finite hyperbolic cone-manifold, with the cone-singularity sufficiently short, we construct a one-parameter family of cone-manifolds decreasing the cone-angle to zero. We also control the geometry of this one-parameter family via the Schwarzian derivative of the projective boundary and the length of closed geodesics.
19.
Let be a convex co-compact, torsion-free, discrete group of isometries of real hyperbolic space H
n+1. We compute the asymptotics of the counting function for closed geodesics in homology classes for the quotient manifold X = \H
n+1, under the assumption that H
1(X, Z) is infinite. Our results imply asymptotic equipartition of geodesics in distinct homology classes. 相似文献
20.
In the recent paper [31] of Long and Duan (2009), we classified closed geodesics on Finsler manifolds into rational and irrational two families, and gave a complete understanding on the index growth properties of iterates of rational closed geodesics. This study yields that a rational closed geodesic cannot be the only closed geodesic on every irreversible or reversible (including Riemannian) Finsler sphere, and that there exist at least two distinct closed geodesics on every compact simply connected irreversible or reversible (including Riemannian) Finsler 3-dimensional manifold. In this paper, we study the index growth properties of irrational closed geodesics on Finsler manifolds. This study allows us to extend results in [31] of Long and Duan (2009) on rational, and in [12] of Duan and Long (2007), [39] of Rademacher (2010), and [40] of Rademacher (2008) on completely non-degenerate closed geodesics on spheres and CP2 to every compact simply connected Finsler manifold. Then we prove the existence of at least two distinct closed geodesics on every compact simply connected irreversible or reversible (including Riemannian) Finsler 4-dimensional manifold. 相似文献