We define an analog of Voiculescu's free entropy for $n$-tuples of unitaries $u\sb 1,\ldots u\sb n$ in a tracial von Neumann algebra $M$ which normalize a unital subalgebra $L\sp \infty[0,1]=B\subset M$. Using this quantity, we define the free dimension $\delta\sb 0(u\sb 1,\ldots,u\sb n\between B)$. This number depends on $u\sb 1,\ldots u\sb n$ only up to orbit equivalence over $B$. In particular, if $R$ is a measurable equivalence relation on $[0,1]$ generated by $n$ automorphisms $\alpha\sb 1,\ldots \alpha\sb n$, let $u\sb 1,\ldots u\sb n$ be the unitaries implementing $\alpha\sb 1,\ldots \alpha\sb n$ in the Feldman-Moore crossed product algebra $M=W\sp \ast([0,1],R)\supset B=L\sp \infty[0,1]$. Then the number $\delta(R)=\delta\sb 0(u\sb 1,\ldots u\sb n\between B)$ is an invariant of the equivalence relation $R$. If $R$ is treeable, $\delta(R)$ coincides with the cost $C(R)$ of $R$ in the sense of D. Gaboriau. In particular, it is $n$ for an equivalence relation induced by a free action of the free group $\mathbb {F}\sb n$. For a general equivalence relation $R$ possessing a finite graphing of finite cost, $\delta(R)\leq C(R)$. Using the notion of free dimension, we define a dynamical entropy invariant for an automorphism of a measurable equivalence relation (or, more generally, of an $r$-discrete measure groupoid) and give examples.

In this paper we consider billiards in a square with a barrier. We use Ratner's theorem to compute the asymptotics for the number of closed orbits.

We consider complexifications of Riemannian symmetric spaces $X$ of nonpositive curvature. We show that the maximal Grauert domain of $X$ is biholomorphic to a maximal connected extension $\Omega\sb {{\rm AG}}$ of $X=G/K\subset G\sb {\mathbb {C}}/K\sb {\mathbb {C}}$ on which $G$ acts properly, a domain first studied by D. Akhiezer and S. Gindikin [1]. We determine when such domains are rigid, that is, when ${\rm Aut}\sb {\mathbb {C}}(\Omega\sb {{\rm AG}}=G$ and when it is not (when \Omega\sb {{\rm AG}}$ has "hidden symmetries"). We further compute the $G$-invariant plurisubharmonic functions on $\Omega\sb {{\rm AG}}$ and related domains in terms of Weyl group invariant strictly convex functions on a $W$-invariant convex neighborhood of $0\in \mathfrak {a}$. This generalizes previous results of M. Lassalle [25] and others. Similar results have also been proven recently by Gindikin and B. Krötz [8] and by Krötz and R. Stanton [24].

We propose a reformulation of the Faltings-Wüstholz nonlinear version of Schmidt's subspace theorem with the help of toric deformations and Chow polytopes. Moreover, we show that the arithmetic Bézout theorem in Arakelov geometry can be used to obtain a Bézout theorem for Mumford's degree of contact. This is a birational invariant often considered in geometric invariant theory (GIT). The originality of this last result relies on the interpretation of GIT as a degeneration of Arakelov geometry. This should enable us to transfer all known results of Arakelov geometry into GIT.

In this paper we study the two $p$-adic $L$-functions attached to a modular form $f=\sum a\sb nq\sp n$ at a supersingular prime $p$. When $a\sb p=0$, we are able to decompose both the sum and the difference of the two unbounded distributions attached to $f$ into a bounded measure and a distribution that accounts for all of the growth. Moreover, this distribution depends only upon the weight of $f$ (and the fact that $a\sb p$ vanishes). From this description we explain how the $p$-adic $L$-function is controlled by two Iwasawa functions and by two power series with growth which have a fixed infinite set of zeros (Theorem 5.1). Asymptotic formulas for the $p$-part of the analytic size of the Tate-Shafarevich group of an elliptic curve in the cyclotomic direction are computed using this result. These formulas compare favorably with results established by M. Kurihara in [11] and B. Perrin-Riou in [23] on the algebraic side. Moreover, we interpret Kurihara's conjectures on the Galois structure of the Tate-Shafarevich group in terms of these two Iwasawa functions.