Super-potentials of positive closed currents, intersection theory and dynamics

By T. Dinh, N. Sibony. 2007

139 citations recorded by the source corpus.

We introduce a notion of super-potential for positive closed currents of bidegree (p, p) on projective spaces. This gives a calculus on positive closed currents of arbitrary bidegree. We define in particular the intersection of such currents and the pull-back operator by meromorphic maps. One of the main tools is the introduction of structural discs in the space of positive closed currents which gives a “geometry” on that space. We apply the theory of super-potentials to construct Green currents for rational maps and to study equidistribution problems for holomorphic endomorphisms and for polynomial automorphisms.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b7028abf26351d27779a15e118177d97219363fc

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Citations in the corpus

Density of positive closed currents, a theory of non-generic intersections (2012, methodology): “ along a manifold V to currents which satisfy suitable regularity near the points of V \K for some compact subset K of V. Itwould be alsouseful tocompare the above notionwith the notion introduced in [18, 19]. We believe that the new notion extends the previous one and is valid in a more general set”
Rigidity of Julia sets for Henon type maps (2013, methodology): “As in the polynomial case, the new tools allowing to treat the higher dimension case were developed in [30, 29, 33].”
Exponential estimates for plurisubharmonic functions and stochastic dynamics (2008, background): “If a d.s.h. function ψ on Pk satisfies 〈µ, ψ〉 = 0 and is not a coboundary, then it satisfies the central limit theorem with respect to µ.”
Equidistribution problems of complex dynamics in higher dimension (2016, methodology): “of the currents deg(V)−1d 1(f)−n(fn)∗[V] towards T is exponentially fast, see [72] for a precise statement. Note also that all the above results can be extended to Henon-type automorphisms of Ck, see [32, 72]. Thekeypointinthe proofisto usenormalizedquasi-potentialsofpositiveclosed (1,1)- currents. ”
Equidistribution of zeros of random holomorphic sections (2013, unknown): “Recall that the super-potentials of positive closed currents were introduced by Dinh and Sibony [13, 14] which extends the notion of quasi-potentials of positive closed (1, 1) currents.”