Complex Dynamics in Higher Dimension. Ii

Abstract

We discuss a few new results in the area of complex dynamics in higher dimension. We investigate generic properties of orbits of biholo-morphic symplectomorphisms of n. In particular we show (Corollary 3.4) that for a dense G δ set of maps, the set of points with bounded orbit has empty interior while the set of points with recurrent orbits nevertheless has full measure. We also investigate the space of real symplectomorphisms of Ê n which extend to n. For this space we show (Theorem 3.10) that for a dense G δ set of maps, the set of points with bounded orbit is an Fσ with empty interior.

Some open problems in higher dimensional complex analysis and complex dynamics

“…Suppose that f is an endomorphism of P k or an automorphism of C k with an attracting basin Ω for an attracting fixed point p, [FS5]. On dynamics of transcendental maps see [FS7], [FS9].…”

Rational and Iterated Maps, Degeneracy Loci, and the Generalized Riemann-Hurwitz Formula

“…We have ∆ k = µ k , in general. However, in this case where f is holomorphic and X = CP n , we do have ∆ k = δ k [10]. On the other hand, it is clear for M = CP n , and p = n, that such maps are thus regulated by the expression derived from (4.2):…”

“…For instance, it is possible that an iteration f k may, for some k, map an (irreducible) curve C into the indeterminacy set I f k (thus f cannot be analytically stable). In this case one sees that ∆ k < δ k [10], and so enumerating (3.10), for p = n, can easily be seen to give…”

Nevanlinna theoretical exceptional sets of rational towers and semigroups

“…In the case of iterations, Theorem 2 was proved by Brolin [2] (for polynomials), Lyubich [16] and Freire-Lopes-Mañé [9]. For the other proofs, see also Tortrat [27], Erëmenko-Sodin [6], Hubbard-Papadopol [12], and Fornaess and Sibony [8]. Theorem 2 can be also proven by Fornaess and Sibony's argument in the proof of [8], Theorem 6.1, where they used a crucial contradiction.…”

“…For the other proofs, see also Tortrat [27], Erëmenko-Sodin [6], Hubbard-Papadopol [12], and Fornaess and Sibony [8]. Theorem 2 can be also proven by Fornaess and Sibony's argument in the proof of [8], Theorem 6.1, where they used a crucial contradiction. Alternatively, in this paper, we give a direct proof of Theorem 2, which is, as a merit, conceptually the same as Brolin's original argument.…”