We define the concept of directional entropy for arbitrary ${\mathbb{Z}}^2$-actions on a Lebesgue space, we examine its basic properties and consider its behaviour in the class of product actions and rigid actions.

We show that for any cellular automaton (CA) ℤ²-action Φ on the space of all doubly infinite sequences with values in a finite set A, determined by an automaton rule $F=F_{[l,r]}$, l,r ∈ ℤ, l ≤ r, and any Φ-invariant Borel probability measure, the directional entropy $h_{v⃗}\left(\Phi \right)$, v⃗= (x,y) ∈ ℝ², is bounded above by $max\left(\right|z_l|,|z_r\left|\right)logA$ if $z_l{z}_r\ge 0$ and by $|z_r-z_l|$ in the opposite case, where $z_l=x+ly$, $z_r=x+ry$. We also show that in the class of permutative CA-actions the bounds are attained if the measure considered is uniform Bernoulli.

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.