Pluripotential theory on the support of closed positive currents and applications to dynamics in $\mathbb{C}^n$

Abstract

We extend certain classical theorems in pluripotential theory to a class of functions defined on the support of a $(1,1)$-closed positive current $T$, analogous to plurisubharmonic functions, called $T$-plurisubharmonic functions. These functions are defined as limits, on the support of $T$, of sequences of plurisubharmonic functions decreasing on this support. In particular, we show that the poles of such functions are pluripolar sets. We also show that the maximum principle and the Hartogs's theorem remain valid in a weak sense. We study these functions by means of a class of measures, so-called "pluri-Jensen measures", about which we prove that they are numerous on the support of $(1,1)$-closed positive currents. We also obtain, for any fat compact set, an expression of its relative Green's function in terms of an infimum of an integral over a set of pluri-Jensen measures. We then deduce, by means of these measures, a characterization of the polynomially convex fat compact sets, as well as a characterization of pluripolar sets, and the fact that the support of a closed positive $(1,1)$-current is nowhere pluri-thin. In the second part of this article, these tools are used to study dynamics of a certain class of automorphisms of $\mathbb{C}^n$ which naturally generalize H\'enon's automorphisms of $\mathbb{C}^2$. First we study the geometry of the support of canonical invariant currents. Then we obtain an equidistribution result for the convergence of pull-back of certain measures towards an ergodic invariant measure, with compact support.