Related papers: $C^0$-Contact Anosov flows

For Anosov flows on compact Riemann manifolds we study the rate of decay along the flow of diameters of balls $B^s(x,\ep)$ on local stable manifolds at Lyapunov regular points $x$. We prove that this decay rate is similar for all…

We employ the curve shortening flow to establish three new results on the dynamics of geodesic flows of closed Riemannian surfaces. The first one is the stability, under $C^0$-small perturbations of the Riemannian metric, of certain flat…

We prove Calegari's conjecture that every quasigeodesic flow on a closed hyperbolic 3-manifold can be deformed to a flow that is simultaneously quasigeodesic and pseudo-Anosov.

We prove polynomial decay of correlations for geodesic flows on a class of nonpositively curved surfaces where zero curvature only occurs along one closed geodesic. We also prove that various statistical limit laws, including the central…

We prove that contact homeomorphisms preserve characteristic foliations on surfaces in contact $3$-manifolds. More precisely, since the characteristic foliation is a singular $1$-dimensional foliation, we show that singular points are…

In this paper we study the phenomenon of phase transitions for the geodesic flow on some geometrically finite negatively curved manifolds. We define a class of potentials going slowly to zero through the cusps of $M$ for which the pressure…

We prove a classification theorem for transitive Anosov and pseudo-Anosov flows on closed 3-manifolds, up to orbit equivalence. In many cases, flows on a 3-manifold $M$ are completely determined by the set of free homotopy classes of their…

For a geometrically finite Kleinian group $\Gamma$, the Bowen-Margulis-Sullivan measure is finite and is the unique measure of maximal entropy for the geodesic flow, as shown by Sullivan and Otal-Peign\'e respectively. Moreover, it is…

In this paper, we study flows of hypersurfaces in hyperbolic space, and apply them to prove geometric inequalities. In the first part of the paper, we consider volume preserving flows by a family of curvature functions including positive…

We consider the Gauss curvature type flow for uniformly convex hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1}\ (n\geqslant 2)$. We prove that if the initial closed hypersurface is smooth and uniformly convex, then the smooth…

Many natural real-valued functions of closed curves are known to extend continuously to the larger space of geodesic currents. For instance, the extension of length with respect to a fixed hyperbolic metric was a motivating example for the…

In this article, we consider the weighted ergodic optimization problem Axiom A attractors of a $C^2$ flow on a compact smooth manifold. The main result obtained in this paper is that for a generic observable from function space $\mc…

We prove that an Anosov flow with $\mathcal{C}^{1}$ stable bundle mixes exponentially whenever the stable and unstable bundles are not jointly integrable. This allows us to show that if a flow is sufficiently close to a volume-preserving…

Strong hyperbolicity is a coarse notion of negative curvature, stronger than Gromov hyperbolicity, that includes all CAT(-k) metrics for k positive and allows the use of dynamical techniques available in negative curvature, such as…

We study the twisted Ruelle zeta function $\zeta_X(s)$ for smooth Anosov vector fields $X$ acting on flat vector bundles over smooth compact manifolds. In dimension $3$, we prove Fried conjecture, relating Reidemeister torsion and…

On a compact manifold of any dimension $d\geq 3$, we show that joint non-integrability of the stable and unstable foliation of a hyperbolic attractor with one-dimensional expanding direction, for a vector field of class $C^2$, implies…

In this paper, we describe the intersection between geodesic and conformal currents on closed hyperbolic three-manifolds. We use this to prove some sharp bounds which involve the Liouville entropy of a negatively curved metric, the minimal…

We show that every pseudo-Anosov flow on a graph manifold is almost equivalent, i.e. orbit equivalent in the complement of a finite collection of closed orbits, to a totally periodic pseudo-Anosov flow or a suspension Anosov flow. The proof…

We show that the geodesic flow and the exponential map of a $C^k$ submanifold of $\mathbb{R}^n$ with $k\geq 2$ are of class $C^{k-1}$.

An example of a real-analytic metric on a compact manifold whose geodesic flow is Liouville integrable by $C^\infty$ functions and has positive topological entropy is constructed.