Feng Rong
I am a Philip T. Church Postdoctoral Fellow in Mathematics Department at Syracuse University.
I obtained my Ph.D. in Mathematics from University
of Michigan. My thesis advisor is Professor John Erik Fornaess.
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Office: 317H Carnegie; |
Phone: 315-443-1578; |
Email: frong AT syr.edu. |
Research
My research interest is in Complex Dynamics of Several Variables and related
fields, such as Complex Analysis, Complex Analytic and Algebraic Geometry, Ergodic Theory and Dynamical Systems. The following are
some of my publications and preprints.
- Reduction of singularities of holomorphic maps of C^3 tangent to the identity. [In preparation]
- Reduction of singularities of holomorphic maps of C^2 tangent to the identity [Preprint, 2008]
Abstract. Let f be an analytic transformation of C^2, tangent to the identity at an isolated fixed point. Using ideas of Hironaka for the reduction of singularities of analytic varieties, we show a reduction process for the singularities of f. This process has no essential obstruction for generalizations to higher dimensions.
Abstract. Let f be an analytic transformation of C^n, with a semi-attractive fixed point p. Assume that f is of order r and that f has a non-degenerate characteristic direction [v]. Assume moreover that the real parts of some eigenvalues of a certain matrix associated to f in the direction [v] are positive. We show that for such f there exist at least r-1 parabolic manifolds tangent to [v] at p.
Abstract. Let f be an analytic transformation of C^n tangent to the identity, with an absolutely isolated singularity. We show that there exists a finite sequence of blow-ups, such that the resulting map only has simple singularities.
Abstract. Let f be an analytic transformation of C^n with a quasi-parabolic fixed point p. Assume that f is of order r and that f has a non-degenerate characteristic direction [v] and is dynamically separating in that direction. Assume moreover that the real parts of some eigenvalues of a certain matrix associated to f in the direction [v] are positive. We show that for such f there exist at least r-1 parabolic manifolds tangent to [v] at p. We also study the dynamics in these parabolic manifolds and the global attracting sets when f is given by an isomorphism of C^n.
- Robust parabolic curves in C^m (m>=3) [Houston J. Math., to appear]
Abstract. We show the existence of holomorphic germs of C^m (m>=3), tangent to the identity at an isolated fixed point, which do not have robust parabolic curves at that point.
- Quasi-parabolic analytic transformations of C^n [J. Math. Anal. Appl., 343 (2008), 99-109]
Abstract. Let f be an analytic transformation of C^n with a quasi-parabolic fixed point p. Assume that f is of order r and that f has a non-degenerate characteristic direction [v] and is dynamically separating in that direction. We show that for such f there exist at least r-1 parabolic curves tangent to [v] at p.
- Linearization of holomorphic germs with quasi-parabolic fixed points [Ergodic Theory Dynam. Systems, 28 (2008), 979–986]
Abstract. Let f be a germ of a holomorphic diffeomorphism of C^n with a quasi-parabolic fixed point p. We show that f is holomorphically locally conjugated to its linear part, if f is of some particular form and the eigenvalues of df_p satisfy certain arithmetic conditions.
- The Fatou set for critically finite maps [Proc. Amer. Math. Soc., 136 (2008), 3621-3625]
Abstract. 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.
Abstract. We show that special perturbations of a particular holomorphic map on P^k give us examples of maps that possess chaotic nonalgebraic attractors. Furthermore, we study the dynamics of the maps on the attractors. In particular, we construct invariant hyperbolic measures supported on the attractors with nice dynamical properties. This study relies on certain results on critically finite maps and a systematic treatment of “history space”.
Teaching
In Fall 2008, I teach MAT 285. My office hours are: TTh 10:10-10:55am & 1:10-1:55pm.