Related papers: Counting closed geodesics on rank one manifolds

We enumerate a necessary condition for the existence of infinitely many geometrically distinct, non-constant, prime closed geodesics on an arbitrary closed Riemannian manifold $M$. That is, we show that any Riemannian metric on $M$ admits…

The Lichnerowicz conjecture asserts that all harmonic manifolds are either flat or locally symmetric spaces of rank~1. This conjecture has been proved by Z. Szab\'{o} \cite{Sz} for harmonic manifolds with compact universal cover. E. Damek…

We extend the results of arXiv:2206.08295v2 by showing that any homothety in $\mathbb T^2$ is homotopic to a non-uniformly hyperbolic ergodic area preserving map, provided that its degree is at least $5^2$. We also address other small…

Every oriented closed geodesic on the modular surface has a canonically associated knot in its unit tangent bundle coming from the periodic orbit of the geodesic flow. We study the volume of the associated knot complement with respect to…

We construct geometrically infinite hyperbolic surfaces supporting horocycles with tailored recurrence properties. In particular, we obtain the first examples of non-trivial minimal horocyclic orbit closures and of infinite locally-finite…

Let $S_{g,n}$ be a surface of genus $g $ with $n$ marked points. Let $X$ be a complete hyperbolic metric on $S_{g,n}$ with $n$ cusps. Every isotopy class $[\gamma]$ of a closed curve $\gamma \in \pi_{1}(S_{g,n})$ contains a unique closed…

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 study the geodesic flow of a class of 3-manifolds introduced by Benoist which have some hyperbolicity but are non-Riemannian, not CAT(0), and with non-C^1 geodesic flow. The geometries are nonstrictly convex Hilbert geometries in…

We prove that if a geodesic flow on a closed orientable $C^\infty$ surface is transitive and has positive topological entropy, then it has a unique measure of maximal entropy. This covers all previous results of the literature on the…

Given a discrete group $\Gamma$ of isometries of a negatively curved manifold $\widetilde M$, a nontrivial conjugacy class $\mathfrak K$ in $\Gamma$ and $x_0\in\widetilde M$, we give asymptotic counting results, as $t\to +\infty$, on the…

We prove that for closed surfaces $M$ with Riemannian metrics without conjugate points and genus $\geq 2$ the geodesic flow on the unit tangent bundle $T^1M$ has a unique measure of maximal entropy. Furthermore, this measure is fully…

We show that the set of harmonic maps from the 2-dimensional stratified spheres with uniformly bounded energies contains only finitely many homotopy classes. We apply this result to construct infinitely many harmonic map flows and mean…

The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and…

In this article we analyze totally periodic pseudo-Anosov flows in graph three manifolds. This means that in each Seifert fibered piece of the torus decomposition, the free homotopy class of regular fibers has a finite power which is also a…

Harmonic manifolds of hypergeometric type form a class of non-compact harmonic manifolds that includes rank one symmetric spaces of non-compact type and Damek-Ricci spaces. When normalizing the metric of a harmonic manifold of…

In this article we obtain a simple topological and dynamical systems condition which is necessary and sufficient for an arbitrary pseudo-Anosov flow in a closed, hyperbolic three manifold to be quasigeodesic. Quasigeodesic means that orbits…

In this paper we study the equilibrium measures of geodesic flows of closed manifolds without conjugate points which have a visibility universal covering. Specifically, the uniqueness problem for Bowen potentials which are constants on some…

In this paper we prove that the homotopy class of non-homothety linear endomorphisms on $\mathbb{T}^2$ with determinant greater than 2 contains a $C^1$ open set of non-uniformly hyperbolic endomorphisms. Furthermore, we prove that the…

In this article, we give explicit conditions for compact group extensions of hyperbolic flows (including geodesic flows on negatively curved manifolds) to exhibit quantifiable rates of mixing (or decay of correlations) with respect to the…

We produce infinitely many examples of Anosov flows in closed 3-manifolds where the set of periodic orbits is partitioned into two infinite subsets. In one subset every closed orbit is freely homotopic to infinitely other closed orbits of…