Let $X$ be a compact Kaehler manifold of dimension $k$ and $T$ be a positive closed current on $X$ of bidimension $(p,p)$ ($1leq p < k-1$). We construct a continuous linear transform $mathcal{L}_p(T)$ of $T$ which is a positive closed current on $X$
of bidimension $(k-1,k-1)$ which has the same Lelong numbers as $T$. We deduce from that construction self-intersection inequalities for positive closed currents of any bidegree.
For a Kahler manifold X, we study a space of test functions W* which is a complex version of H1. We prove for W* the classical results of the theory of Dirichlet spaces: the functions in W* are defined up to a pluripolar set and the functional capaci
ty associated to W* tests the pluripolar sets. This functional capacity is a Choquet capacity. The space W* is not reflexive and the smooth functions are not dense in it for the strong topology. So the classical tools of potential theory do not apply here. We use instead pluripotential theory and Dirichlet spaces associated to a current.
Let $f:Xto X$ be a dominating meromorphic map of a compact Kahler surface of large topological degree. Let $S$ be a positive closed current on $X$ of bidegree $(1,1)$. We consider an ergodic measure $ u$ of large entropy supported by $mathrm{supp}(S)
$. Defining dimensions for $ u$ and $S$, we give inequalities `a la Ma~ne involving the Lyapunov exponents of $ u$ and those dimensions. We give dynamical applications of those inequalities.
We prove the exponential decay of correlations for C^alpha-observables (0<alpha =<2) for generic birational maps of P^k `a la Bedford-Diller. In the particular case of regular birational maps, we give a better estimate of the speed of the decay, gett
ing results as sharp as Dinhs results for Henon maps.
