Let $Y_1, Y_2, \dots$ be a sequence of independent random maps, identically distributed with respect to a probability measure $\mu$ on $SL(2,R)$. A (deep) theorem of Furstenberg gives abstract conditions under which for almost every such sequence the orbit of a non-zero initial point in $R^2$ tends to infinity exponentially fast. In the present paper we translate this statement into the set-up of Möbius transformations on the upper half-plane and provide a very explicit way to determine whether or not the required conditions are satisfied.

We consider the behaviour near resonances of linearizations of germs of holomorphic diffeomorphisms of $({\Bbb C},0)$ and of the semi-standard map.

We prove that for each resonance there exists a suitable blow-up of the Taylor series of the linearization under which it converges uniformly to an analytic function as the multiplier, or rotation number, tends non-tangentially to the resonance. This limit function is explicitly computed and related to questions of formal classification, both for the case of germs and for the case of the semi-standard map.

We prove existence and non-existence results for geodesics avoiding $a$-separated sets on a surface of negative curvature.

We study Anosov diffeomorphisms on surfaces with small holes. The points that are mapped into the holes disappear and never return. In our previous paper we proved the existence of a conditionally invariant measure $\mu_+$. Here we show that the iterations of any initially smooth measure, after renormalization, converge to $\mu_+$. We construct the related invariant measure on the repeller and prove that it is ergodic and K-mixing. We prove the escape rate formula, relating the escape rate to the positive Lyapunov exponent and the entropy.

We continue the study of mixing properties of generic hyperbolic flows started in an earlier paper (D. Dolgopyat. Prevalence of rapid mixing in hyperbolic flows. Erg. Th.& Dyn. Sys.18 (1998), 1097–1114). Our main result is that generic suspension flow over subshifts of finite type is exponentially mixing. This is a quantitative version of an earlier result of Parry and Pollicott (W. Parry and M. Pollicott. Stability of mixing for toral extensions of hyperbolic systems. Proc. Steklov Inst.216 (1997), 354–363).

A minimal subshift $(X,T)$ is linearly recurrent (LR) if there exists a constant $K$ so that for each clopen set $U$ generated by a finite word, $u$, the return time to $U$, with respect to $T$, is bounded by $K|u|$. We prove that given a LR subshift $(X,T)$ the set of its non-periodic subshift factors is finite up to isomorphism. We also give a constructive characterization of these subshifts.

This is partly exposition, partly generalization and partly completion of some of Vershik's work towards classifying sequences of measurable partitions.

We develop a pluripotential theory for random iteration on ${\mathbf P}^k$. We show the existence of a positive closed $(1,1)$ current and a measure on ${\mathbf P}^k$ which are invariant, in a certain sense, and which attract all positive closed $(1,1)$ currents and all measures, respectively, under normalized pull-back and averaging by the maps. Thus the concept of an exceptional set disappears as soon as we allow a slight amount of randomness in our system. We also consider the problem of push-forward of measures, and describe certain limiting measures in this case also, supported near the attractors for the perturbed map.

We compute dynamical entropy in Connes, Narnhofer and Thirring sense for a Bogoliubov action of a torsion-free abelian group $G$ on the CAR-algebra. A formula analogous to that found by Størmer and Voiculescu in the case $G={\Bbb Z}$ is obtained. The singular part of a unitary representation of $G$ is shown to give zero contribution to the entropy. A proof of these results requires new arguments since a torsion-free group may have no finite index proper subgroups. Our approach allows us to overcome these difficulties, it differs from that of Størmer–Voiculescu.

We show that the $n$-point correlation function for the fractional parts of a random linear form in $m$ variables has a limit distribution with power-like tail. The existence of the limit distribution follows from the mixing property of flows on ${\rm SL}(m+1,{\Bbb R})/{\rm SL}(m+1,{\Bbb Z})$. Moreover, we prove similar limit theorems (i) for the probability to find the fractional part of a random linear form close to zero and (ii) also for related trigonometric sums. For large $m$, all of the above limit distributions approach the classical distributions for independent uniformly distributed random variables.

In this paper we show that a $C^r$ diffeomorphism having an invariant Floquet torus which is non-normally hyperbolic can be approximated, in the $C^{r-3}$ topology, by another one which exhibits a homoclinic tangency.

It is shown that under certain conditions a hyperbolic set will possess an invariant structure in addition to the invariant submanifolds. This invariant structure is in the form of a ‘jet of a foliation’ along the leaves.

A proof for a non-singular version of the inverse Vitali lemma is given. The result is used to describe non-singular orbit equivalence within the framework of Rudolph's restricted orbit equivalence and in the construction of an alternative proof of the Hurewicz ergodic theorem.

Let $(M,g)$ be a compact, differentiable Riemannian manifold without conjugate points and bounded asymptote. We show that, if the geodesic flow of $(M,g)$ is either topologically stable, or satisfies the $\epsilon$-shadowing property for some appropriate $\epsilon > 0$, then every abelian subgroup of the fundamental group of $M$ is infinite cyclic. The proof is based on the existence of homoclinic geodesics in perturbations of $(M,g)$, whenever there is a subgroup of the fundamental group of $M$ isomorphic to $\mathbb{Z}\times \mathbb{Z}$.

We construct a geodesic lamination on the hyperbolic disk and a dynamical system defined on this lamination. We prove that this dynamical system is a geometrical realization of the symbolic dynamical system that arises from the following Pisot substitution: $1\rightarrow 12, \dotsc, (n-1) \rightarrow 1n, n\rightarrow 1$.