Feng Rong

Linearization of holomorphic germs with quasi-parabolic fixed points

Let f be a germ of a holomorphic diffeomorphism of C n with the origin O being a quasi-parabolic fixed point, i.e. the spectrum of dfO consists of 1 and e 2i j with j 2 R\Q. We show that f is locally holomorphically conjugated to its linear part, if f is of some particular form and its eigenvalues satisfy certain arithmetic conditions. When the spectrum of dfO does not consist of any 1s, this is the classical result of Siegel (C. L. Siegel. Iteration of analytic functions. Ann. of Math. 43 (1942), 607-612) and Brjuno (A. D. Brjuno. Analytic form of differential equations. Trans. Moscow Math. Soc. 25 (1971), 131-288; 26 (1972), 199-239).

The Fatou set for critically finite maps

It is a classical result in complex dynamics of one variable that the Fatou set for a critically finite map on P 1 consists of only basins of attraction for superattracting periodic points. In this paper, we deal with critically finite maps on P k . We show that the Fatou set for a critically finite map on P 2 consists of only basins of attraction for superattracting periodic points. We also show that the Fatou set for a k-critically finite map on P k is empty.

On biholomorphisms between bounded quasi-Reinhardt domains

In this paper, we define what is called a quasi-Reinhardt domain and study biholomorphisms between such domains. We show that all biholomorphisms between two bounded quasi-Reinhardt domains fixing the origin are polynomial mappings, and we give a uniform upper bound for the degree of such polynomial mappings. In particular, we generalize the classical Cartan’s linearity theorem for circular domains to quasi-Reinhardt domains.

THE NON-DICRITICAL ORDER AND ATTRACTING DOMAINS OF HOLOMORPHIC MAPS TANGENT TO THE IDENTITY

We study the local dynamics of holomorphic maps f in Cn tangent to the identity at a fixed point p with a non-degenerate characteristic direction [v]. In [M. Hakim, Analytic transformation of (Cp,0) tangent to the identity, Duke Math. J. 92 (1998) 403‐428], n −1 invariants αj ,1 ≤ j ≤ n −1, called the directors, were associated to [v ]a nd it was shown that if Re αj > 0 for all j then f has an attracting domain at p tangent to [v]. In this paper, we study the case Re αj =0 for somej. With the help of a new invariant µ called the non-dicritical order, we show that f has an attracting domain at p tangent to [v ]i fµ ≥ 1. We also study the “spiral domains” when µ =0 . Forn = 2, we show that f has an attracting domain at p tangent to [v] if and only if either the director α> 0o r µ ≥ 1.

Estimate of the squeezing function for a class of bounded domains

We construct a class of bounded domains, on which the squeezing function is not uniformly bounded from below near a smooth and pseudoconvex boundary point.

On automorphism groups of generalized Hua domains

Hua domains, generalized Hua domains and Hua constructions, named after the great Chinese mathematician Luogeng Hua (Loo-Keng Hua), are generalizations of Cartan– Hartogs domains introduced by Weiping Yin around the end of the 20th century. In this paper, we give a complete description of automorphism groups of generalized Hua domains. We also discuss the corresponding problem for Hua constructions.

On the number of limit cycles for discontinuous piecewise linear differential systems in R^2n with two zones

We study the number of limit cycles of the discontinuous piecewise linear differential systems in ℝ2n with two zones separated by a hyperplane. Our main result shows that at most (8n - 6)n-1 limit cycles can bifurcate up to first-order expansion of the displacement function with respect to a small parameter. For proving this result, we use the averaging theory in a form where the differentiability of the system is not necessary.

The degree of automorphisms of quasi-circular domains fixing the origin

By using the Bergman representative coordinates, we give the necessary and sufficient condition for the degree of automorphisms of quasi-circular domains fixing the origin to be bounded by the resonance order, thus solving a conjecture of the author.

A Brief Survey on Local Holomorphic Dynamics in Higher Dimensions

We give a brief survey on local holomorphic dynamics in higher dimensions. The main novelty of this note is that we will organize the material by the “level” of local invariants rather than the type of maps.

The Briot–Bouquet systems and the center families for holomorphic dynamical systems

Abstract We give a complete solution to the existence of isochronous center families for holomorphic dynamical systems. The study of center families for n -dimensional holomorphic dynamical systems naturally leads to the study of ( n − 1 )-dimensional Briot–Bouquet systems in the phase space. We first give a detailed study of the Briot–Bouquet systems. Then we show the existence of isochronous center families in the neighborhood of the equilibrium point of three-dimensional systems based on the two-dimensional Briot–Bouquet theory. The same approach works in arbitrary dimensions.