Gerhard Knieper

Differentiability and analyticity of topological entropy for Anosov and geodesic flows

SummaryIn this paper we investigate the regularity of the topological entropyhtop forCk perturbations of Anosov flows. We show that the topological entropy varies (almost) as smoothly as the perturbation. The results in this paper, along with several related results, have been announced in [KKPW].

Spherical means on compact Riemannian manifolds of negative curvature

Abstract We show that the spherical mean of functions on the unit tangent bundle of a compact manifold of negative curvature converges to a measure containing a vast amount of information about the asymptotic geometry of those manifolds. This measure is related to the unique invariant measure for the strong unstable foliation, as well as the Patterson-Sullivan measure at infinity. It turns out to be invariant under the geodesic flow if and only if the mean curvature of the horospheres is constant. We use this measure in the study of rigidity problems.

Formulas for the derivative and critical points of topological entropy for Anosov and geodesic flows

This paper represents part of a program to understand the behavior of topological entropy for Anosov and geodesic flows. In this paper, we have two goals. First we obtain some regularity results forC1 perturbations. Second, and more importantly, we obtain explicit formulas for the derivative of topological entropy. These formulas allow us to characterize the critical points of topological entropy on the space of negatively curved metrics.

Chapter 6 Hyperbolic dynamics and riemannian geometry

Abstract In this survey we will focus on recent results on the dynamics of geodesic flows on compact manifolds of nonpositive curvature with hyperbolic or at least weakly hyperbolic behavior. Furthermore, we will discuss rigidity theorems on compact negatively curved manifolds like entropy rigidity, the minimal entropy problem, rigidity of stable and unstable foliations and the infinitesimal spectral rigidity theorem. In many cases we will provide complete proofs and not only statements of theorems. Some of the results are new and did not appear somewhere else. For instance, we give a new upper bound for the number of closed geodesics in rank 1 manifolds and a new entropy comparison result. Moreover, we will give an intrinsic proof of a formula of Weitzenbock type on the tangent bundle (Pestovs identity). However, we will not discuss geodesic flows on manifolds with positive curvature. The reason is that until now there is no example of such a flow with a hyperbolic set of positive Liouville measure. Since we do not leave the geometric setting, we do not discuss the interesting subject of general Euler-Lagrange flows. Many of those topics are covered in the recent book by G. Paternain [76] which is in some sense complementary to this survey. Furthermore, we do not cover geodesic metric spaces as for instance Gromov hyperbolic spaces or the Alexandrov geometry of nonpositively curved metric spaces. This theory is treated in many text books and surveys as for instance [6,16,42].

MINIMAL GEODESIC FOLIATION ON T2 IN CASE OF VANISHING TOPOLOGICAL ENTROPY

On a Riemannian 2-torus

Asymptotically harmonic spaces in dimension 3

(T^2,g)

The uniqueness of the maximal measure for geodesic flows on symmetric spaces of higher rank

we study the geodesic flow in the case of low complexity described by zero topological entropy. We show that this assumption implies a nearly integrable behavior. In our previous paper \cite{GK} we already obtained that the asymptotic direction and therefore also the rotation number exists for all geodesics. In this paper we show that for all

A second derivative formula of the Liouville entropy at spaces of constant negative curvature

r \in \mathbb{R} \cup \{\infty\}

On the Asymptotic Geometry of Nonpositively Curved Manifolds

the universal cover

Hyperbolic dynamics and Riemannian Geometry

\Br^2