In this paper we study the set of bounded geodesics on a general, geometrically finite $(N+1)$-manifold of constant negative curvature. We obtain the result that the Hausdorff dimension of this set is equal to $2 \delta$, where $\delta$ denotes the exponent of convergence of the associated Kleinian group. The proof of this shows, in particular, that if the group has parabolic elements, then the set of limit points which are badly approximable with respect to the parabolic fixed points has Hausdorff dimension equal to $\delta$.

Green [5] conjectured that if $M$ is a closed Riemannian manifold of negative sectional curvature such that the mean curvatures of the horospheres through each point depend only on the point, then $V$ is a locally symmetric space of rank one. He proved this in dimension two. In this paper we prove that under Green's assumption, $M$ must be asymptotically harmonic and that the geodesic flow on $M$ is $C^{\infty}$ conjugate to that of a locally symmetric space of rank one. Combining this with the recent rigidity theorem of Besson–Courtois–Gallot [1], it follows that Green's conjecture is true for all dimensions.

Consider a long piece of a trajectory $x, T(x), T(T(x)), \ldots, T^{n-1}(x)$ of an interval exchange transformation $T$. A generic interval exchange transformation is uniquely ergodic. Hence, the ergodic theorem predicts that the number $\chi_i(x,n)$ of visits of our trajectory to the $i$th subinterval would be approximately $\lambda_i n$. Here $\lambda_i$ is the length of the corresponding subinterval of our unit interval $X$. In this paper we give an estimate for the deviation of the actual number of visits to the $i$th subinterval $X_i$ from one predicted by the ergodic theorem.

We prove that for almost all interval exchange transformations the following bound is valid: $$ \max_{\ssty x\in X \atop \ssty 1\le i\le m} \limsup_{n\to +\infty} \frac {\log | \chi_i(x,n) -\lambda_in|}{\log n} = \frac{\theta_2}{\theta_1} < 1. $$ Roughly speaking the error term is bounded by $n^{\theta_2/\theta_1}$. The numbers $0\le \theta_2 < \theta_1$ depend only on the permutation $\pi$ corresponding to the interval exchange transformation (actually, only on the Rauzy class of the permutation). In the case of interval exchange of two intervals we obviously have $\theta_2=0$. In the case of exchange of three and more intervals the numbers $\theta_1, \theta_2$ are the two top Lyapunov exponents related to the corresponding generalized Gauss map on the space of interval exchange transformations.

The limit above ‘converges to the bound’ uniformly for all $x\in X$ in the following sense. For any $\varepsilon >0$ the ratio of logarithms would be less than $\theta_2(\pi)/\theta_1(\pi)+\varepsilon $ for all $n\ge N(\varepsilon)$, where $N(\varepsilon)$ does not depend on the starting point $x\in X$.