(with F. Ledrappier)

We present recent results about the asymptotic behavior of ergodic products of isometries of a metric space X. If we assume that the displacement is integrable, then either there is a sublinear diusion or there is, for almost every trajectory in X, a preferred direction at the boundary. We discuss the precise statement when X is a proper metric space ([KL1]) and compare it with classical ergodic theorems. Applications are given to ergodic theorems for nonintegrable functions, random walks on groups and Brownian motion on covering manifolds.

Outline of lectures given at the meeting Probabilistic and Dynamical Properties of (semi)-group Actions, held at Universidad de Santiago de Chile, January 2008, and at Wroclaw University as part of the EU Marie Curie Host Fellowship program Transfer of Knowledge, June 2008.

(with N. Monod)

It is observed that a wellnigh trivial application of the ergodic theorem from [KL1] yields a strong LLN for arbitrary concave moments. Not for publication: we found that it already is in J. Aaronson, An introduction to infinite ergodic theory (AMS Math. Surv. Mon. 50, 1997), pages 65-66.

(with F. Ledrappier)

We prove a general noncommutative law of large numbers. This applies in particular to random walks on any locally finite homogeneous graph, as well as to Brownian motion on Riemannian manifolds which admit a compact quotient. It also generalizes Oseledec’s multiplicative ergodic theorem. In addition, we show that epsilon-shadows of any ballistic random walk with finite moment on any group eventually intersect. Some related results concerning Coxeter groups and mapping class groups are recorded in the last section.

Final section of:

(with V. Metz and G. Noskov)

We obtain precise descriptions of all horoballs for Hilbert's geometry on simplices and for normed finite-dimensional vector spaces with norm given by some polyhedron. Certain geometrical consequences are deduced and several other applications are pointed out, which concern the dynamics of important classes of nonlinear self-maps of convex cones.

We provide an analysis of the dynamics of isometries and semicontractions of metric spaces. Certain subsets of the boundary at infinity play a fundamental role and are identified completely for the standard boundaries of CAT(0) spaces, Gromov hyperbolic spaces, Hilbert geometries, certain pseudoconvex domains, and partially for Thurston’s boundary of Teichmuller spaces. We present several rather general results concerning groups of isometries, as well as the proof of other more specific new theorems, for example concerning the existence of free nonabelian subgroups in CAT(0) geometry, iteration of holomorphic maps, a metric Furstenberg lemma, random walks on groups, noncompactness of automorphism groups of convex cones, and boundary behaviour of Kobayashi’s metric.

We prove that almost every path of a random walk on a finitely generated nonamenable group converges in the compactification of the group introduced by W. J. Floyd. In fact, we consider the more general setting of ergodic cocycles of some semigroup of one-Lipschitz maps of a complete metric space with a boundary constructed following Gromov. We obtain in addition that when the Floyd boundary of a finitely generated group is non-trivial, then it is in fact maximal in the sense that it can be identified with the Poisson boundary of the group with reasonable measures. The proof relies on works of Kaimanovich together with visibility properties of Floyd boundaries. Furthermore, we discuss mean proximality of 0[' and a conjecture of McMullen. Lastly, related statements about the convergence of certain sequences of points, for example quasigeodesic rays or orbits of one-Lipschitz maps, are obtained.

(Ed. by V.A. Kaimanovich, in collab. with K. Schmidt, W. Woess) de Gruyter, 2004.

This paper describes some situations when random walks (or related processes) of linear rate of escape converge in direction in various senses. We discuss random walks on isometry groups of fairly general metric spaces, and more specifically, random walks on isometry groups of nonpositive curvature, isometry groups of reflexive Banach spaces, and linear groups preserving a proper cone. We give an alternative proof of the main tool from subadditive ergodic theory and make a conjecture in this context involving Busemann functions.

Final section of:

(with G. Noskov)

We give some sufficient conditions for Hilbert's metric on convex domains D to be Gromov hyperbolic. The conditions involve an intersecting chords property, which we in turn relate to the Menger curvature of triples of boundary points and, in the case the boundary is smooth, to differential geometric curvature of dD. In particular, the intersecting chords property and hence Gromov hyperbolicity is established for bounded, convex C^2-domains in R^n with non-zero curvature. We also give some necessary conditions for hyperbolicity: the boundary must be of class C^1 and may not conatain a line segment. Furthermore we prove a statement about the asymptotic geometry of the Hilbert metric on arbitrary convex (i.e. non necessarily strictly convex) bounded domains, with an application to maps which do not increase Hilbert distance.

Newton Institute, Cambridge,

We describe some results on the dynamics of nonexpanding maps of metric spaces. [...] Further consequences concerning random ergodic theorems, random walks, and Poisson boundaries are briefly discussed.

We give stronger versions and alternative simple proofs of some results of Beardon, [Be1] and [Be2]. These results concern contractions of locally compact metric spaces and generalize the theorems of Wolff and Denjoy about the iteration of a holomorphic map of the unit disk. In the case of unbounded orbits, there are two types of statements which can sometimes be proven; first, about invariant horoballs, and second, about the convergence of the iterates to a point on the boundary. A few further remarks of similar type are made concerning certain random products of semicontractions and also concerning semicontractions of Gromov hyperbolic spaces.

(with G. Margulis)

We study integrable cocycles u(n,x) over an ergodic measure preserving transformation that take values in a semigroup of nonexpanding maps of a nonpositively curved space Y, e.g. a Cartan-Hadamard space or a uniformly convex Banach space. It is proved that for any y]Y and almost all x, there exist AS 0 and a unique geodesic ray n (t,x) in Y starting at y such that\begin{equation*} \lim_{n\rightarrow \infty }\frac 1nd(\gamma (An,x),u(n,x)y)=0. \end{equation*} In the case where Y is the symmetric space GLN(Â)/ON(Â) and the cocycles take values in GLN(Â), this is equivalent to the multiplicative ergodic theorem of Oseledec. Two applications are also described. The first concerns the determination of Poisson boundaries and the second concerns Hilbert-Schmidt operators.