Dynamics in Several Complex Variables (Cbms Regional by John Erik Fornaess

By John Erik Fornaess

This CBMS lecture sequence, held in Albany, ny in June 1994, aimed to introduce the viewers to the literature on complicated dynamics in greater size. many of the lectures are up to date types of past lectures given together with Nessim Sibony in Montreal 1993. The author's reason during this booklet is to offer a diffusion of the Montreal lectures, basing complicated dynamics in better measurement systematically on pluripotential conception. those notes supply an easy-to-read advent into the sector, an advent that motivates the subjects. The monograph then issues readers in the direction of technically extra complex literature.

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CHAPTER 5 Recurrent Fatou Components Fix a holomorphic map f : P2 )-4 P2 of degree d > 2. We will classify certain periodic Fatou components 0 for f. We can assume that f(I2) = fl replacing f by an iterate if necessary. We say that a Fatou component H is recurrent if for some po E I1 the w- limit set of po intersects I1, i. e. f (po) is relatively compact in Il for some subsequence n,. The following theorem has been proved in more generality by Abate under the more restrictive hypothesis that the domains are taut.

2 Notice that the collection of all curves y obtained in this way gives a holomorphic foliation of C" by complex curves except for points where X = 0. These are called singular points. We will in particular consider two kinds of holomorphic vector fields, Hamiltonian vector fields associated to Holomorphic Hamiltonians and Holomorphic Reeb vector fields associated to Complex Contact Structures. Let us first discuss the Hamiltonian case: Let H be an entire holomorphic function on C2k. Then we define the Hamiltonian vector field XH ` - r-0 k = 1 8H e OH 8 az; + 7.

2. [FG] There is a dense family G C E such that every F E G has a dense set of points with exploding orbits. 3. [FG] Assume that A is a fixed complex contact form on C''. Then there exists a dense family G C C so that for every F E G, the contact structure (A, F) has a dense set of exploding orbits. 2. Suppose F E G and that F is close to F and has n > 0 exploding orbits. Pick a point p E C2. We want to find F with n + I exploding orbits. We obtain an element F E G close to F by letting F = lim F,,.

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