Quasi-parabolic analytic transformations of Cn. Parabolic manifolds

Abstract

In [Rong, F., Quasi-parabolic analytic transformations of C n , J. Math. Anal. Appl. 343 (2008), 99-109], we showed the existence of "parabolic curves" for certain quasiparabolic analytic transformations of C n . Under some extra assumptions, we show the existence of "parabolic manifolds" for such transformations.

Dynamics of one-resonant biholomorphisms

“…having one eigenvalue 1 and the other of modulus equal to one, but not a root of unity has been studied in [7] and it has been proved that, under a certain generic hypothesis called "dynamical separation", there exist petals tangent to the eigenspace of 1. Such a result has been generalized to higher dimension by Rong [24], [25]. We refer the reader to the survey papers [2] and [5] for a more accurate review of existing results.…”

Attracting domains for quasi-parabolic analytic transformations of ℂ2

“…In [3], Bracci and Molino showed the existence of 'parabolic curves' for quasiparabolic transformations in C 2 that are dynamically separating. This result was later generalized by Rong to any dimension (see [6]). Moreover, in [7] sufficient conditions were given for the existence of 'parabolic manifolds' and attracting domains.…”

“…This result was later generalized by Rong to any dimension (see [6]). Moreover, in [7] sufficient conditions were given for the existence of 'parabolic manifolds' and attracting domains. However, no general results are known in the non-dynamically separating case (see [2, open problem (OP20 )]).…”

“…See, for example, [1] and the references therein for a survey on known results and [8][9][10] for more recent advancements. In [3][4][5][6][7], some of these results have been extended to quasi-parabolic transformations.…”

Quasi-parabolic analytic transformations of Cn

“…In a forthcoming paper [14], we are going to extend the results of Hakim in [8] to the quasi-parabolic case.…”