We discuss the classification up to orbit equivalence of inclusions 𝑆 ⊂ ℛ of measured ergodic discrete hyperfinite equivalence relations. In the case of type III relations, the orbit equivalence classes of such inclusions of finite index are completely classified in terms of triplets consisting of a transitive permutation group G on a finite set (whose cardinality is the index of 𝑆 ⊂ ℛ), an ergodic nonsingular ℝ-flow V and a homomorphism of G to the centralizer of V.

We give a positive answer to the problem of existence of smooth weakly mixing but not mixing flows on some surfaces. More precisely, on each compact connected surface whose Euler characteristic is even and negative we construct smooth weakly mixing flows which are disjoint in the sense of Furstenberg from all mixing flows and from all Gaussian flows.

We show that for any irrational number α and a sequence ${m_l}_{l\in \mathbb{N}}$ of integers such that $li{m}_{l\to \infty }\left|\right||m_l\alpha \left|\right||=0$, there exists a continuous measure μ on the circle such that $li{m}_{l\to \infty }{\int }_{}\left|\right||m_l\theta \left|\right||d\mu \left(\theta \right)=0$. This implies that any rigidity sequence of any ergodic transformation is a rigidity sequence for some weakly mixing dynamical system. On the other hand, we show that for any α ∈ ℝ - ℚ, there exists a sequence ${m_l}_{l\in \mathbb{N}}$ of integers such that $\left|\right||m_l\alpha \left|\right||\to 0$ and such that $m_l\theta \left[1\right]$ is dense on the circle if and only if θ ∉ ℚα + ℚ.

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...

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.

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.

A planar polygonal billiard $𝒫$ is said to have the finite blocking property if for every pair $(O,A)$ of points in $𝒫$ there exists a finite number of “blocking” points $B_1,\cdots ,B_n$ such that every billiard trajectory from $O$ to $A$ meets one of the $B_i$’s. Generalizing our construction of a counter-example to a theorem of Hiemer and Snurnikov, we show that the only regular polygons that have the finite blocking property are the square, the equilateral triangle and the hexagon. Then we extend this result to translation surfaces....

The Hudetz correction of the fuzzy entropy is applied to the $g$-entropy. The new invariant is expressed by the Hudetz correction of fuzzy entropy.

The Stein-Weiss theorem that the distribution function of the Hilbert transform of the characteristic function of E depends only on the measure of E is generalized to the ergodic Hilbert transform.

Calculus for observables in a space of functions from an abstract set to the unit interval is developed and then the individual ergodic theorem is proved.