In the early development of complex dynamics in dimension one, made heavy use was made of Montel's Theorem. In fact, much of the theory may be obtained directly from Montel's Theorem (which says that any family of holomorphic mappings from the disk into C minus 2 points is a normal family). There are generalizations of Montel's Theorem to higher dimension, but it is not clear how these can be applied to obtain the analogous theorems in higher-dimensionaldynamics.

Fortunately, pluri-potential theory has been applicable as an alternative tool. This approach to dynamics focuses more on measurable dynamics and ergodic theory. By "pluri-potential" theory, we mean the theory of pluri-subharmonic (psh) functions and positive, closed currents. With this, we can define the stable/unstable currents. Further, the operator (ddc)2 has been useful for defining the wedge (intersection) product of these two currents. An early presentation of (ddc)2 is given in [BT1]. The relation between (ddc)2 and bounded psh functions was first given in [BT2]. A more modern and very thorough treatment is given by Demailly's book.

One of my favorite results involving pluri-potential theory is:
Theorem [Fornaess and Sibony] If T is a positive, closed current whose support is contained in K+,
then T is a positive multiple of the stable current.

This has been generalized recently in a very nice paper by Dinh and Sibony. Look at this paper also for a nice exposition of many aspects of pluri-potential theory.

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