Let $f:{\mathbb{P}}^{k}\to {\mathbb{P}}^{k}$ be a dominating rational mapping of first algebraic degree $\lambda \ge 2$. If $S$ is a positive closed current of bidegree $(1,1)$ on ${\mathbb{P}}^{k}$ with zero Lelong numbers, we show – under a natural dynamical assumption – that the pullbacks ${\lambda}^{-n}{\left({f}^{n}\right)}^{*}S$ converge to the Green current ${T}_{\phantom{\rule{-0.55542pt}{0ex}}f}$. For some families of mappings, we get finer convergence results which allow us to characterize all ${f}^{*}$-invariant currents.

Let $f$ be a meromorphic self-mapping of a compact Kähler manifold. We study the rate of
decreasing of volumes under the iteration of $f$. We use these volume estimates to
construct the Green current of $f$ in a quite general setting.

Let $X$ be a compact Kähler manifold and $\omega $ be a smooth closed form of bidegree $(1,1)$ which is nonnegative and big. We study the classes ${\mathcal{E}}_{\chi}(X,\omega )$ of $\omega $-plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight $\chi $ has fast growth at infinity, the corresponding functions are close to be bounded.
We show that if a positive Radon measure is suitably dominated by the Monge-Ampère capacity, then it belongs to the range of the Monge-Ampère operator on some class ${\mathcal{E}}_{\chi}(X,\omega )$. This is done by establishing...

We continue our study of the dynamics of mappings with small topological degree on projective complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic “equilibrium” measure for each such mapping. Here we study the dynamical properties of this measure in detail: we give optimal bounds for its Lyapunov exponents, prove that it has maximal entropy, and show that it has product structure in the natural extension. Under a natural further assumption, we show that saddle points...

Let $(X,\omega )$ be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on $X$ with ${L}^{p}$ right hand side, $p>1$. The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range $(X,\omega )$ of the complex Monge-Ampère operator acting on $\omega $-plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Kołodziej’s result that measures with ${L}^{p}$-density belong to $(X,\omega )$ and proving that $(X,\omega )$ has the...

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