Consider the domain $Z_{ϵ}=\{x\in {ℝ}^n;dist(x,ϵ{\mathbb{Z}}^n)>{ϵ}^{\gamma }\}$ and let the free path length be defined as ${\tau }_{ϵ}(x,v)=inf\{t>0;x-tv\in Z_{ϵ}\}.$ In the Boltzmann-Grad scaling corresponding to $\gamma =\frac{n}{n-1}$, it is shown that the limiting distribution ${\phi }_{ϵ}$ of ${\tau }_{ϵ}$ is bounded from below by an expression of the form C/t, for some C> 0. A numerical study seems to indicate that asymptotically for large t, ${\phi }_{ϵ}\sim C/t$. This is an extension of a previous work [J. Bourgain et al., Comm. Math. Phys.190 (1998) 491-508]. As a consequence, it is proved that the linear Boltzmann type transport equation is inappropriate...

Consider the region obtained by removing from ${ℝ}^2$ the discs of radius $\epsilon$, centered at the points of integer coordinates $(a,b)$ with $b\not\equiv a(mod\ell )$. We are interested in the distribution of the free path length (exit time) ${\tau }_{\ell ,\epsilon }\left(\omega \right)$ of a point particle, moving from $(0,0)$ along a linear trajectory of direction $\omega$, as $\epsilon \to 0^{+}$. For every integer number $\ell \ge 2$, we prove the weak convergence of the probability measures associated with the random variables $\epsilon {\tau }_{\ell ,\epsilon }$, explicitly computing the limiting distribution. For $\ell =3$, respectively $\ell =2$, this result leads...

We consider a system of ordinary differential equations with infinite delay. We study large time dynamics in the phase space of functions with an exponentially decaying weight. The existence of an exponential attractor is proved under the abstract assumption that the right-hand side is Lipschitz continuous. The dimension of the attractor is explicitly estimated.

Let $K$ be a local field, $a\in K$ and $\varphi :x\to x^p+a$ where $p$ denotes the characteristic of the residue field. We prove that the minimal subsets of the dynamical system $(K,\varphi )$ are cycles and describe the cycles of this system.

We show that for a finitely generated group of C² circle diffeomorphisms, the entropy of the action equals the entropy of the restriction of the action to the non-wandering set.

We prove some results concerning the entropy of Darboux (and almost continuous) functions. We first generalize some theorems valid for continuous functions, and then we study properties which are specific to Darboux functions. Finally, we give theorems on approximating almost continuous functions by functions with infinite entropy.

Given a vector field X on an algebraic variety V over ℂ, I compare the following two objects: (i) the envelope of X, the smallest algebraic pseudogroup over V whose Lie algebra contains X, and (ii) the Galois pseudogroup of the foliation defined by the vector field X + d/dt (restricted to one fibre t = constant). I show that either they are equal, or the second has codimension one in the first.

Let (X,,μ,τ) be an ergodic dynamical system and φ be a measurable map from X to a locally compact second countable group G with left Haar measure $m_G$. We consider the map ${\tau }_{\phi }$ defined on X × G by ${\tau }_{\phi }:(x,g)\mapsto (\tau x,\phi \left(x\right)g)$ and the cocycle ${\left(\phi ₙ\right)}_{n\in \mathbb{Z}}$ generated by φ. Using a characterization of the ergodic invariant measures for ${\tau }_{\phi }$, we give the form of the ergodic decomposition of $\mu \left(dx\right)\otimes m_G\left(dg\right)$ or more generally of the ${\tau }_{\phi }$-invariant measures ${\mu }_{\chi }\left(dx\right)\otimes \chi \left(g\right)m_G\left(dg\right)$, where ${\mu }_{\chi }\left(dx\right)$ is χ∘φ-conformal for an exponential χ on G.