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.

A dynamical criterion of conjugacy of a flow with a shift and a counterpart of the Brouwer plane translation theorem for flows on a normal topological space are established.

For $\beta>1$, consider the $\beta$-transformation $T_\beta$. When $\beta$ is an integer, the natural extension of $T_\beta$ can be represented explicitly as a map on the unit square with an invariant measure: the corresponding two-dimensional Lebesgue measure. We show that, under certain conditions on $\beta$, the natural extension is defined on a simply connected region and an invariant measure is a constant multiple of the Lebesgue measure.

We characterize those $\beta$ in terms of the $\beta$-expansion of one, and study the structure and size of the set of all such $\beta$.

We study the non-existence of KAM tori for quasi-integrable, analytic Lagrangians. Let ${\cal L}\colon {\bf T}^m\times{\bf R}^m \to {\bf R}$, ${\cal L} (Q,\dot{Q})=\tfrac{1}{2}\vert\dot{Q}\vert^2+h(Q)$ and let $\bar\omega\in{\bf R}^m$ be a frequency exponentially close to resonances. We find $h$ analytic of norm arbitrarily small such that ${\cal L}$ has no invariant torus of frequency $\bar\omega$ projecting diffeomorphically on $\T^m$

We study the existence of models with minimal topological entropy among the class of all continuous maps from a given graph to itself with a fixed behavior on a given finite invariant set.

In this paper, we show the existence of Markov covers for $C^2$ surface diffeomorphisms with a dominated splitting under some assumptions. Using a Markov cover, we can reduce the dynamics to a one-dimensional dynamical system having a good metric property. As an application, we show finiteness of periodic attractors for the above diffeomorphisms.

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 that interval maps for which $\omega$-limit sets of all critical points are minimal are dense in the space of all interval maps of class $C^2$.

We construct a transcendental entire function $f$ with $J(f)=\mathbb{C}$ such that $f$ has arbitrarily slow growth; that is, $\log |f(z)|\leq\phi(|z|)\log |z|$ for $|z|>r_0$, where $\phi$ is an arbitrary prescribed function tending to infinity.

Let $G$ be a compact connected Lie group and ${\rm Aut}(G)$ be the group of Lie automorphisms of $G$. We describe a condition on $G$ under which topological conjugacy of the flows on $G$ induced by two one-parameter subgroups $\phi$ and $\psi$ of ${\rm Aut}(G)$ implies conjugacy of $\phi$ and $\psi$ in ${\rm Aut}(G)$. The condition is verified for $Sp(n)$, $SO(2n+1)$ and ${\rm Spin}(2n+1)$.

In a topological dynamical system $(X,T)$ the complexity function of a cover ${\cal C}$ is the minimal cardinality of a sub-cover of $\bigvee_{i=0}^n T^{-i}{\cal C}$. It is shown that equicontinuous transformations are exactly those such that any open cover has bounded complexity. Call scattering a system such that any finite cover by non-dense open sets has unbounded complexity, and call 2-scattering a system such that any such 2-set cover has unbounded complexity: then all weakly mixing systems are scattering and all 2-scattering systems are totally transitive. Conversely, any system that is not 2-scattering has covers with complexity at most $n+1$. Scattering systems are characterized topologically as those such that their cartesian product with any minimal system is transitive; they are consequently disjoint from all minimal distal systems. Finally, defining $(x,y)$, $x\ne y$, to be a complexity pair if any cover by two non-trivial closed sets separating $x$ from $y$ has unbounded complexity, we prove that 2-scattering systems are disjoint from minimal isometries; that in the invertible case the complexity relation is contained in the regionally proximal relation and, when further assuming minimality, coincides with it up to the diagonal.

To study the geometry of a Fibonacci map $f$ of even degree $\ell\geq 4$, Lyubich (Dynamics of quadratic polynomials, I–II. Acta Mathematica178 (1997), 185–297) defined a notion of generalized renormalization, so that $f$ is renormalizable infinitely many times. van Strien and Nowicki (Polynomial maps with a Julia set of positive Lebesgue measure: Fibonacci maps. Preprint, Institute for Mathematical Sciences, SUNY at Stony Brook, 1994) proved that the generalized renormalizations ${\cal R}^{\circ n}(f)$ converge to a cycle $\{f_1,f_2\}$ of order two depending only on $\ell$. We will explicitly relate $f_1$ and $f_2$ and show the convergence in shape of Fibonacci puzzle pieces to the Julia set of an appropriate polynomial-like map.

We present a sufficient condition for the existence of ergodic continuous invariant measures (ECIMs) for continuous folding transformations. We apply this result to prove the existence of ECIMs for general set expanding Markov transformations and $n$-dimensional tent maps.

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

On étudie une large classe d'immeubles à courbure strictement négative. Deux aspects sont surtout développés : la construction de réseaux dans leur groupe d'automor-phismes, et l'étude de leur type de quasi-isométrie à travers la dimension conforme de Pansu de leur bord.

Any polynomial automorphism of $\mathbb{C}^2$ with nontrivial dynamics is conjugate to a diffeomorphism of the 4-ball such that this diffeomorphism extends to a diffeomorphism of the closed 4-ball. Moreover, the conjugating map is a smooth bijection of $\mathbb{C}^2$ to itself. On the sphere at infinity, the extension has an attracting and a repelling solenoid, and the dynamics near these invariant solenoids are described by conjugation to a model solenoidal map.

The resonant tori of an integrable system are destroyed by a perturbation. If the Hamiltonian is convex, they give rise to hyperbolic lower-dimensional invariant tori or to Aubry–Mather invariant sets. Bolotin has proved the existence of homoclinic orbits to the hyperbolic tori but not to the Aubry–Mather invariant sets. We solve this problem and obtain, for each resonant frequency, the existence of an invariant set with homoclinic orbits.

We prove that a relevant part of the Lyapunov spectrum and the corresponding Oseledets spaces of a quasi-compact linear cocycle are stable under a certain type of random perturbation. The basic approach is a graph-transform argument. The result applies to the spectrum of (not necessarily i.i.d.) randomly perturbed expanding maps and yields generalizations of results recently obtained by Baladi, Kondah and Schmitt [5] with different methods.

In this paper, it is proved that the distribution of values of $N^{-1/2}\sum_{n=1}^N f_1(\theta^{p_1(n)}x)\dots f_K(\theta^{p_K(n)}x)$ converges to normal distribution. Here $p_k(n)$ are polynomials.

A one-parameter family of time reversible Anosov flows is studied; physically, it describes a particle moving on a surface of constant negative curvature under the action of an electric field (corresponding to an automorphic form) and of a ‘thermostatting’ force (given by Gauss's least-constraint principle). We show that the flows are dissipative, in the sense that the average volume contraction rate is positive and the Sinai–Ruelle–Bowen measure is singular with respect to the volume: therefore they verify the assumptions for the validity of the continuous time version of Gallavotti–Cohen's fluctuation theorem for the large fluctuations of the average volume contraction rate. If several independent electric fields are considered, it makes sense to ask for the validity of Onsager's reciprocity: we show, by explicitly computing the relevant transport coefficents, that it is indeed obeyed.