Related papers: Cusp excursion in hyperbolic manifolds and singula…

Extending the earlier results for analytic curve segments, in this article we describe the asymptotic behaviour of evolution of a finite segment of a C^n-smooth curve under the geodesic flow on the unit tangent bundle of a finite volume…

Neumann and Reid described in their paper "Rigidity of cusps in deformations of hyperbolic 3-orbifolds" (Math Ann. 295 (1993) no. 2, 223--237) a 2-cusped hyperbolic 3-orbifold in which the cusps are geometrically isolated. Based on…

We consider dynamical systems generated by partially hyperbolic surface endomorphisms of class C^r with one-dimensional strongly unstable subbundle. As the main result, we prove that such a dynamical system generically admits finitely many…

We continue the study of the geometry of infinite geodesics in first passage percolation (FPP) on Gromov-hyperbolic groups G, initiated by Benjamini-Tessera and developed further by the authors. It was shown earlier by the authors that,…

For a fixed cusp neighborhood (determined by depth D) of the modular surface, we investigate the class of reciprocal geodesics that enter this neighborhood (called a cusp excursion) a fixed number of times.

For $0<k<1$, a finite-type $k$-surface in $3$-dimensional hyperbolic space is a complete, immersed surface of finite area and of constant extrinsic curvature equal to $k$. In [32], we showed that such surfaces have finite genus and finitely…

We show that a quasi-geodesic in an injective metric space is Morse if and only if it is strongly contracting. Since mapping class groups and, more generally, hierarchically hyperbolic groups act properly and coboundedly on injective metric…

For $C^{1+}$ maps, possibly non-invertible and with singularities, we prove that each homoclinic class of an ergodic adapted hyperbolic measure carries at most one adapted hyperbolic measure of maximal entropy. We then apply this to study…

We consider globally hyperbolic maximal anti de Sitter 3-manifolds $M$ with a closed Cauchy surface $S$ of genus greater than one and prove that any pair of hyperbolic metrics on $S$ can be realized as the boundary metrics of the convex…

We consider foliations of the whole three dimensional hyperbolic space H^3 by oriented geodesics. Let L be the space of all the oriented geodesics of H^3, which is a four dimensional manifold carrying two canonical pseudo-Riemannian metrics…

Let $\text{G}(n)$ be equal either to $\text{PO}(n,1),\text{PU}(n,1)$ or $\text{PSp}(n,1)$ and let $\Gamma \leq \text{G}(n)$ be a uniform lattice. Denote by $\mathbb{H}^n_K$ the hyperbolic space associated to $\text{G}(n)$, where $K$ is a…

We prove that the Oseledets splittings of an ergodic hyperbolic measure of a $C^{1+ r}$ diffeomorphism can be approximated by that of atomic measures on hyperbolic periodic orbits. This removes the assumption on simple spectrum in…

We present new methods to construct a Ruelle-Pollicott spectrum for the geodesic flow on manifolds with strictly negative curvature and a finite number of hyperbolic cusps.

As a final work to establish that frame flows for geometrically finite hyperbolic manifolds of arbitrary dimensions are exponentially mixing with respect to the Bowen-Margulis-Sullivan measure, this paper focuses on the case with cusps. To…

In this paper, we show that, for scalar reaction-diffusion equations on the circle S1, the property of hyperbolicity of all equilibria and periodic orbits is generic with respect to the non-linearity . In other words, we prove that in an…

We prove the upper semicontinuity of the measure theoretic entropy for the geodesic flow on complete Riemannian manifolds without focal points and bounded sectional curvature. We then study the relationship between the escape of mass…

For a non-compact hyperbolic 3-manifold with cusps we prove an explicit formula that relates the regularized analytic torsion associated to the even symmetric powers of the standard representation of SL_2(C) to the corresponding…

We discuss two generalizations of the collar lemma. The first is the stable neighborhood theorem which says that a (not necessarily simple) closed geodesic in a hyperbolic surface has a \lq\lq stable neighborhood\rq\rq whose width only…

We study closed geodesics on hyperbolic surfaces, and give bounds for their angles of intersection and self-intersection, and for the sides of the polygons that they form, depending only on the lengths of the geodesics

We prove that in any hyperbolic orbifold with one boundary component, the product of any hyperbolic fundamental group element with a sufficiently large multiple of the boundary is represented by a geodesic loop that virtually bounds an…