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Pluripotential theory on the support of closed positive currents and applications to dynamics in \(\mathbb{C}^n\). (English) Zbl 1430.32014

Summary: 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. We study these functions by means of a class of measures, so-called pluri-Jensen measures, which prove to be numerous on the support of \((1,1)\)-closed positive currents. For any fat compact set, we obtain an expression of its relative Green’s function in terms of pluri-Jensen measures and deduce a characterization of the polynomially convex fat compact sets and of pluripolar sets. These tools are then used to study dynamics of a class of automorphisms of \(\mathbb{C}^n\) which naturally generalize Hénon’s automorphisms. We obtain an equidistribution result for the convergence of pull-back of certain measures toward an ergodic invariant measure with compact support.

MSC:

32U15 General pluripotential theory
32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
32U35 Plurisubharmonic extremal functions, pluricomplex Green functions
32U40 Currents
32U05 Plurisubharmonic functions and generalizations
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
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