Let $(M,g)$ be a compact Riemannian manifold of hyperbolic type, i.e$M$ is a manifold admitting another metric of strictly negativecurvature. In this paper we study the geodesic flow restricted to theset of geodesics which are minimal on the universal covering. Inparticular for surfaces we show that the topological entropy of theminimal geodesics coincides with the volume entropy of $(M,g)$generalizing work of Freire and Mañé.

We show that every group $G$ with no cyclic subgroup of finite indexthat acts properly and cocompactly by isometries on a proper geodesicGromov-hyperbolic space $X$ is growth-tight. In other words, theexponential growth rate of $G$ for the geometric (pseudo)-distanceinduced by $X$ is greater than the exponential growth rate of any ofits quotients by an infinite normal subgroup. This result unifies andextends previous works of Arzhantseva-Lysenok and Sambusetti using ageometric approach.

A closed geodesic on the modular surface gives rise to a knot on the3-sphere with a trefoil knot removed, and one can compute the linkingnumber of such a knot with the trefoil knot. We show that, whenordered by their length, the set of closed geodesics having aprescribed linking number become equidistributed on average withrespect to the Liouville measure. We show this by using thethermodynamic formalism to prove an equidistribution result for acorresponding set of quadratic irrationals on the unit interval.

A cyclic cover over $CP^1$ branched at four points inherits a natural flatstructure from the 'pillow' flat structure on the basic sphere. We give anexplicit formula for all individual Lyapunov exponents of the Hodge bundle overthe corresponding arithmetic Teichmüller curve. The key technical element isevaluation of degrees of line subbundles of the Hodge bundle, corresponding toeigenspaces of the induced action of deck transformations.

Let $M$ be a closed orientable irreducible $3$-dimensional manifold. Anembedded $2$-torus $\mathbb{T}$ is an Anosov torus if there exists adiffeomorphism $f$ over $M$ for which $\T$ is $f$-invariant and$f_\#|_\mathbb{T}:\pi_1(\mathbb{T})\to \pi_1(\mathbb{T})$ is hyperbolic.We prove that only few irreducible $3$-manifolds admit Anosov tori: (1) the$3$-torus $\mathbb{T}^3$; (2) the mapping torus of $-\Id$; and (3) the mappingtori of hyperbolic automorphisms of $\mathbb{T}^2$. This has consequences for instance in the context of partially hyperbolicdynamics of $3$-manifolds: if there is an invariant foliation$\mathcal{F}^{cu}$ tangent to the center-unstable bundle $E^c\oplus E^u$, then$\mathcal{F}^{cu}$ has no compact leaves [21]. This has led tothe first example of a non-dynamically coherent partially hyperbolicdiffeomorphism with one-dimensional center bundle [21].

Suppose $f\colon M\to M$ is a $C^{1+\alpha}$ $(\alpha>0)$ diffeomorphism on a compact smooth orientable manifold $M$ of dimension 2, and let $\mu_\Psi$ be an equilibrium measure for a Hölder-continuous potential $\Psi\colon M\to \mathbb R$. We show that if $\mu_\Psi$ has positive measure-theoretic entropy, then $f$ is measure-theoretically isomorphic mod $\mu_\Psi$ to the product of a Bernoulli scheme and a finite rotation.

We consider partially hyperbolic abelian algebraic high-rank actions on compacthomogeneous spaces obtained from simple indefinite orthogonal and unitarygroups. In the first part of the paper, we show local differentiable rigidityfor such actions. The conclusions are based on progress toward computations ofthe Schur multipliers of these non-split groups, which is the main aim of thesecond part.

In this paper, we show that if Rabinowitz Floer homology has infinitedimension, there exist infinitely many critical points of a Rabinowitz actionfunctional even though it could be non-Morse. This result is proved byexamining filtered Rabinowitz Floer homology.

We compute explicitly the action of the group of affine diffeomorphisms on therelative homology of two remarkable origamis discovered respectively by Forni(in genus $3$) and Forni and Matheus (in genus $4$). We show that, in bothcases, the action on the nontrivial part of the homology is through finitegroups. In particular, the action on some $4$-dimensional invariant subspace ofthe homology leaves invariant a root system of $D_4$ type. This provides as aby-product a new proof of (slightly stronger versions of) the results of Forniand Matheus: the nontrivial Lyapunov exponents of the Kontsevich-Zorichcocycle for the Teichmüller disks of these two origamis are equal to zero.

We consider a Fuchsian group Г and the factor surface H/Г, which has constant curvature $-1$ and maybe a few singularities. If we lift the surface continuously to $\H$ (except for a subset of a lower dimension), we obtain a fundamental domain $\D$ of Г. This can be done in different ways; ours is to restrict the choice to so-called Dirichlet domains, which always are convex polygonal subsets of $\H$. Given a generic geodesic on $\H$, one can produce a so-called geometric Morse code (or the cutting sequence) of the geodesic with respect to $\D$. We prove that the set of Morse codes of all generic geodesics on $\H$ with respect to $\D$ forms a topological Markov chain if and only if $\D$ is an ideal polygon.