We show that the Poincaré series of the Fuchsian group of deck transformations of ${\mathbb C}\setminus{\mathbb Z}$ diverges logarithmically. This is because ${\mathbb C}\setminus{\mathbb Z}$ is a ${\mathbb Z}$-cover of the three horned sphere, whence its geodesic flow has a good section which behaves like a random walk on ${\mathbb R}$ with Cauchy distributed jump distribution and has logarithmic asymptotic type.

In this paper we prove, using explicit constructions, that every non-compact manifold (except, of course, surfaces of finite genus) can be endowed with a non-singular flow without minimal subset, that is to say: a flow such that each orbit closure contains a smaller one.

An example is given of a one-dimensional foliation (or a flow) on an open surface which admits no minimal set.

In Part 1 we describe a duality method for calculating twisted cocycles. In Part 2 we use our method to prove various results on cohomological rigidity of higher-rank cocompact lattice actions. In Part 3 we use the results of Parts 1 and 2 to prove infinitesimal rigidity of the actions of cocompact lattices on the maximal boundaries of some non-compact type symmetric spaces.

Nous prouvons que dans l'ensemble $SW^0({\bf T}^1,SU(2))$ des difféomorphismes de ${\bf T}^1\times SU(2)$ de la forme $(\theta,y)\to (\theta+\alpha,A(\theta).y)$, $A\in C^{\infty}({\bf T}^1,SU(2))$, $\alpha\in{\bf T}^1$, les systèmes $(\alpha,A)$ réductibles, c'est-à-dire ceux qui admettent une écriture de la forme $A(\theta)= B(\theta+\alpha)A_0B(\theta)^{-1}$, $B\in C^{\infty}({\bf T}^1,SU(2))$, $A_0\in SU(2)$, forment une partie dense pour la $C^0$-topologie. Ce théorème est à comparer à un résultat de Herman [7], qui dit qu'il en est de même des systèmes non réductibles. Ceci démontre (une forme d')une conjecture de Rychlik (cf. [14]).

Let $\bpropto$ be the topological space obtained by identifying the points 1 and 2 of the segment $[0,3]$ to a point. Let $\binfty$ be the topological space obtained by identifying the points 0, 1 and 2 of the segment $[0,2]$ to a point. An $\bpropto$ (respectively $\binfty$) map is a continuous self-map of $\bpropto$ (respectively $\binfty$) having the branching point fixed. Set $E\in\{\bpropto,\binfty\}$. Let $f$ be an $E$ map. We denote by $\Per(f)$ the set of periods of all periodic points of $f$. The set $K \subset{\mathbb N}$ is the full periodicity kernel of $E$ if it satisfies the following two conditions: (1) if $f$ is an $E$ map and $K\subset \Per(f)$, then $\Per(f)={\mathbb N}$; (2) for each $k\in K$ there exists an $E$ map $f$ such that $\Per(f)={\mathbb N}\setminus\{ k\}$. In this paper we compute the full periodicity kernel of $\bpropto$ and $\binfty$.

We establish bounds for the multipliers of those periodic orbits of $R_\mu(z) = z(z+\mu)/(1+\overline\mu z) $, which have a Poincaré rotation number $ p/q $. The bounds are given in terms of $ p/q $ and the (logarithmic) hororadius of $\mu$ to $e^{2\pi ip/q} $. The principal tool is a new construction denoted a ‘star’ of an immediate attracting basin. The bounds are used to prove properties of the space of Möbius conjugacy classes of quadratic rational maps. These properties are related to the mating and non-mating conjecture for quadratic polynomials lsqb;Ta]. Moreover they are also reminiscent of Chuckrows theorem on the non-existence of elliptic limits of loxodromic elements in quasiconformal deformations of Kleinian groups. We bear this analogy further by proving an analog of Chuckrows theorem for deformations of certain holomorphic maps.

We show that if the geodesic flow of a compact analytic Riemannian manifold $M$ of non-positive curvature is either $C^{k}$-topologically stable or satisfies the $\epsilon$-$C^{k}$-shadowing property for some $k > 0$ then the universal covering of $M$ is a Gromov hyperbolic space. The same holds for compact surfaces without conjugate points.

Many applications of nonlinear dynamics involve forced systems. We consider the case where for a fixed input the driven system is contracting; this is for instance the situation in certain classes of filters, and in the study of synchronization. When this contraction is uniform, it can easily be shown that there exists a globally attracting invariant set which is the graph of a function from the driving state space to the driven state space; this is a special case of the well known concept of an inertial manifold for more general systems. If the driving state space is a manifold and the contraction is sufficiently strong this invariant set is a normally hyperbolic manifold, and hence smooth. The aim of this paper is to extend this result in two directions: firstly, where we only have uniform contraction for a compact invariant set of input states, and secondly where the contraction rates are non-uniform (and hence defined by Liapunov exponents and analogous quantities). In both cases the invariant graph is only defined over closed subsets of the input space, and hence we need to define an appropriate notion of smoothness for such functions. This is done in terms of the Whitney extension theorem: a function is considered Whitney smooth if it satisfies the conditions of this theorem and hence can be extended to a smooth function of the whole input space.

The billiard in the exterior of a finite disjoint union $K$ of strictly convex bodies in ${\mathbb R}^d$ with smooth boundaries is considered. The existence of global constants $0 < \delta < 1$ and $C > 0$ is established such that if two billiard trajectories have $n$ successive reflections from the same convex components of $K$, then the distance between their $j$th reflection points is less than $C(\delta^j + \delta^{n-j})$ for a sequence of integers $j$ with uniform density in $\{1,2,\dots,n\}$. Consequently, the billiard ball map (although not continuous in general) is expansive. As applications, an asymptotic of the number of prime closed billiard trajectories is proved which generalizes a result of Morita [Mor], and it is shown that the topological entropy of the billiard flow does not exceed $\log (s-1)/a$, where $s$ is the number of convex components of $K$ and $a$ is the minimal distance between different convex components of $K$.

We use the methods developed with Lyubich for proving complex bounds for real quadratics to extend de Faria's complex a priori bounds to all critical circle maps with an irrational rotation number. The contracting property for renormalizations of critical circle maps follows.

As another application of our methods we present a new proof of a theorem of Petersen on local connectivity of some Siegel Julia sets.

In [2], flows on the standard two-dimensional torus given by the differential equations \begin{equation*} \frac{dx}{dt}=a-Fy(x,y),\quad \frac{dv}{dt}=b+Fx(x,y) \end{equation*} were considered. It was assumed that $F(x,y)$ was real analytic and of period one in both $x$ and $y$. A key step in proving the results in [2] was to show that one could conclude topological transitivity for the flow provided one assumed: \begin{enumerate} \item[(a)] $a/b$ is irrational; \item[(b)] there does not exist a topological disc on the torus that is invariant under the flow. \end{enumerate}