Related papers: Characterization of geodesic flows on T^2 with and…

We study the Bowen topological entropy of generic and irregular points for certain dynamical systems. We define the topological entropy of noncompact sets for flows, analogous to Bowen's definition. We show that this entropy coincides with…

Let X be an infinite Riemann surface with an upper-bounded geodesic pants decomposition. The vertices of the corresponding dual graph G are pairs of pants and edges are cuffs with conductances equal to their lengths. We prove that the…

Let $\Sigma$ be a compact quotient of $T_4$, the Lie group of $4 \times 4$ upper triangular matrices with unity along the diagonal. The Lie algebra $t_4$ of $T_4$ has the standard basis $\{X_{ij}\}$ of matrices with $0$ everywhere but in…

When a shallow layer of inviscid fluid flows over a substrate, the fluid particle trajectories are, to leading order in the layer thickness, geodesics on the two-dimensional curved space of the substrate. Since the two-dimensional geodesic…

Motivated by Gromov's geodesic flow problem on hyperbolic groups $G$, we develop in this paper an analog using random walks. This leads to a notion of a harmonic analog $\Theta$ of the Bowen-Margulis-Sullivan measure on $\partial^2 G$. We…

In this paper we study the behavior of geodesics on cones over arbitrary $C^3$-smooth closed Riemannian manifolds. We show that the geodesic flow on such cones admits first integrals whose values uniquely determine almost all geodesics…

We introduce a notion of topological entropy for continuous actions of compactly generated topological groups on compact Hausdorff spaces. It is shown that any continuous action of a compactly generated topological group on a compact…

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…

We study expansive properties for the geodesic and horocycle flows on compact Riemann surfaces of constant negative curvature. It is well-known that the geodesic flow is expansive in the sense of Bowen-Walters and the horocycle flow is…

The paper is a study of geodesic in two-dimensional pseudo-Riemannian metrics. Firstly, the local properties of geodesics in a neighborhood of generic parabolic points are investigated. The equation of the geodesic flow has singularities at…

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 study topological aspects of the dynamics of the foliated horocycle flow on flat projective bundles over hyperbolic surfaces and we derive ergodic consequences. If $\rho : \Gamma \to {\rm PSL}(n+1,\mathbb{R})$ is a…

By studying completely integrable torus actions on contact manifolds we prove a conjecture of Toth and Zelditch that toric integrable geodesic flows on tori must have flat metrics.

Given any $K>0$, we construct two equivalent $C^2$ flows, one of which has positive topological entropy larger than $K$ and admits zero as the exponential growth of periodic orbits, in contrast, the other has zero topological entropy and…

We introduce a notion of stability for non-autonomous Hamiltonian flows on two-dimensional annular surfaces. This notion of stability is designed to capture the sustained twisting of particle trajectories. The main Theorem is applied to…

We consider the geodesic flow on a complete connected negatively curved manifold. We show that the set of invariant borel probability measures contains a dense $G_\delta$-subset consisting of ergodic measures fully supported on the…

The Euler equation of an ideal (i.e. inviscid incompressible) fluid can be regarded, following V.Arnold, as the geodesic flow of the right-invariant $L^2$-metric on the group of volume-preserving diffeomorphisms of the flow domain. In this…

If X is a proper CAT(-1)-space and $\Gamma$ a non-elementary discrete group of isometries acting properly discontinuously on X, it is shown that the geodesic flow on the quotient space Y=X/$\Gamma$ is topologically mixing, provided that the…

In this article we write the equations of barotropic compressible fluid mechanics as a geodesic equation on an infinite-dimensional manifold. The equations are given by \begin{align} u_t + \nabla_uu = -\frac{1}{\rho} \grad p \\ \rho_t +…

Let $Y$ be a topological Markov chain with finite leading and follower sets. Special flow over $Y$ whose height function depends on the time zero of elements of $Y$ is constructed. Then a formula for computing the entropy of this flow will…