Related papers: Geodesic flow, left-handedness, and templates

In this paper we study some generic properties of the geodesic flows on a convex sphere. We prove that, $C^r$ generically ($2\le r\le\infty$), every hyperbolic closed geodesic admits some transversal homoclinic orbits.

We give a formula for the topological pressure of the geodesic flow of a compact rank 1 manifold in terms of the growth of the number of closed hyperbolic (rank 1) geodesics. We derive an equidistribution result for these geodesics with…

We study the dynamics of unipotent flows on frame bundles of hyperbolic manifolds of infinite volume. We prove that they are topologi-cally transitive, and that the natural invariant measure, the so-called " Burger-Roblin measure ", is…

The periodic orbit conjecture states that, on closed manifolds, the set of lengths of the orbits of a non-vanishing vector field all whose orbits are closed admits an upper bound. This conjecture is known to be false in general due to a…

Any closed, oriented, hyperbolic three-manifold with nontrivial second homology has many quasigeodesic flows, where quasigeodesic means that flow lines are uniformly efficient in measuring distance in relative homotopy classes. The flows…

A Riemann surface equipped with its conformal hyperbolic metric is parabolic if and only if the geodesic flow on its unit tangent bundle is ergodic. Let X be a Cantor tree or a blooming Cantor tree Riemann surface. Fix a geodesic pants…

For every finite collection of curves on a surface, we define an associated (semi-)norm on the first homology group of the surface. The unit ball of the dual norm is the convex hull of its integer points. We give an interpretation of these…

In this article, we characterize two kinds of exceptional orbits of the geodesic flow associated with the Modular surface in terms of a two-parameter family of continued fraction expansion of endpoints of the lifts to the hyperbolic plane…

We consider the existence of simple closed geodesics or "geodesic knots" in finite volume orientable hyperbolic 3-manifolds. Previous results show that at least one geodesic knot always exists [Bull. London Math. Soc. 31(1) (1999) 81-86],…

We consider a transversally conformal foliation $\mathcal{F}$ of a closed manifold $M$ endowed with a smooth Riemannian metric whose restriction to each leaf is negatively curved. We prove that it satisfies the following dichotomy. Either…

In this paper we describe the topological behavior of the geodesic flow for a class of closed 3-manifolds realized as quotients of nonstrictly convex Hilbert geometries, constructed and described explicitly by Benoist. These manifolds are…

In this article we investigate the dynamical properties of the geodesic flow for a proper metric space endowed with a proper action by isometries of a group with a contracting element. We show that the existence of a contracting isometry is…

A Riemann surface $X$ is parabolic if and only if the geodesic flow (for the hyperbolic metric) on the unit tangent bundle of $X$ is ergodic. Consider a Riemann surface $X$ with a single topological end and a sequence $\alpha_n$ of pairwise…

The symmetrized bidisc \[ G \stackrel{\rm{def}}{=}\{(z+w,zw):|z|<1,\ |w|<1\}, \] under the Carath\'eodory metric, is a complex Finsler space of cohomogeneity $1$ in which the geodesics, both real and complex, enjoy a rich geometry. As a…

We prove that every homogeneous flow on a finite-volume homogeneous manifold has countably many independent invariant distributions unless it is conjugate to a linear flow on a torus. We also prove that the same conclusion holds for every…

In this paper, we establish the existence of periodic orbits of a twisted geodesic flow on all low energy levels and in all dimensions whenever the magnetic field form is symplectic and spherically rational. This is a consequence of a more…

We generalize the results of [AS], finding large classes of totally geodesic Seifert surfaces in hyperbolic knot and link complements, each the lift of a rigid 2-orbifold embedded in some hyperbolic 3-orbifold. In addition, we provide a…

We consider the billiard flow of elastically colliding hard balls on the flat $\nu$-torus ($\nu\ge 2$), and prove that no singularity manifold can even locally coincide with a manifold describing future non-hyperbolicity of the…

The author shows that equicontinuous geodesic flows on surfaces are periodic. A similar result for flows on 3-manifolds is also proven. The idea of the proof is to show that the return map is recurrent and therefore periodic.

We describe a class of completely integrable $G$-invariant magnetic geodesic flows on (co)adjoint orbits of a compact connected Lie group $G$ with magnetic field given by the Kirillov-Konstant 2-form.