Related papers: On the geodesic flow on CAT(0) spaces

Almost nothing is known concerning the extension of $3$-dimensional Kronecker--Weyl equidistribution theorem on geodesic flow from the unit torus $[0,1)^3$ to non-integrable finite polycube translation $3$-manifolds. In the special case…

We consider geodesic flows between hypersurfaces in $\R^n$. However, rather than consider using geodesics in $\R^n$, which are straight lines, we consider an induced flow using geodesics between the tangent spaces of the hypersurfaces…

Since their introduction by Thurston, geodesic laminations on hyperbolic surfaces occur in many contexts. In this paper, we propose a generalization of geodesic laminations on locally CAT(0), complete, geodesic metric spaces, whose boundary…

In this work, using the method by He, we prove a short time existence for Ricci flow on a complete noncompact Riemannian manifold with the following properties: (i) there is $r_0>0$ such that the volume of any geodesic balls of radius $r\le…

We show the equivalences of several notions of entropy, like a version of the topological entropy of the geodesic flow and the Minkowski dimension of the boundary, in metric spaces with convex geodesic bicombings satisfying a uniform…

In this survey article we gather classical as well as recent results on minimal geodesics of Riemannian or Finsler metrics, giving special attention to the two-dimensional case. Moreover, we present open problems together with some first…

As we have proved in [L], the geodesic flows associated with the flat metrics on T^2 minimize the polynomial entropy. In this paper, we show that, among the geodesic flows that are Bott integrable and dynamically coherent, the geodesic…

We consider curvature flows in hyperbolic space with a monotone, symmetric, homogeneous of degree 1 curvature function F. Furthermore we assume F to be either concave and inverse concave or convex. For compact initial hypersurfaces, which…

Let $X$ be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank one axis. Assume $X$ is not homothetic to a metric graph with integer edge lengths. Let $P_t$ be the number of parallel classes of…

We consider the geodesic flow of reversible Finsler metrics on the 2-sphere and the 2-torus, whose geodesic flow has vanishing topological entropy. Following a construction of A. Katok, we discuss examples of Finsler metrics on both…

In this paper, we show that starting from a geodesic ball $\overline{B_{r_0}}(0)$ in $\mathbb{H}^n$, for $n\geq3$, with prescribed non-decreasing rotationally symmetric mean curvature and the fixed conformal class $[g_{\mathbb{S}^{n-1}}]$…

We study the influence of the existence of totally geodesic null hypersurface on the properties of a Lorentzian manifold. By coupling the rigging technique with the existence of a null foliation we prove the existence of a Riemann flow…

We develop different synthetic notions of Ricci flow in the setting of time-dependent metric measure spaces based on ideas from optimal transport. They are formulated in terms of dynamic convexity and local concavity of the entropy along…

It is shown that unlike the perfect fluid case, anisotropic fluids (principal stresses unequal) may be geodesic, without this implying the vanishing of (spatial) pressure gradients. Then the condition of vanishing four acceleration is…

We explain why semistability of a one-ended proper CAT(0) space can be determined by the geodesic rays. This is applied to boundaries of CAT(0) groups.

We investigate CAT(0) metric spaces whose associated Tits boundary is compact. Prominent examples of such spaces are of course the euclidean ones. However there exist non trivial geodesically complete CAT(0) spaces with compact Tits…

We describe an algorithm to compute the geodesics in an arbitrary CAT(0) cubical complex. A key tool is a correspondence between cubical complexes of global non-positive curvature and posets with inconsistent pairs. This correspondence also…

We study the generic invariant probability measures for the geodesic flow on connected complete nonpositively curved manifolds. Under a mild technical assumption, we prove that ergodicity is a generic property in the set of probability…

In this note we study the thermodynamic formalism for the positive geodesic flow on the modular surface. We define the pressure and prove the variational principle. We also establish conditions for the the pressure to be real analytic and…

Hamiltonian flows on compact surfaces are characterized, and the topological invariants of such flows with finitely many singular points are constructed from the viewpoints of integrable systems, fluid mechanics, and dynamical systems.…