Recently, Gao Yan, Distinguished Professor at the School of Mathematical Sciences at Shenzhen University, in collaboration with Associate Professor Zeng Jinsong, published a research paper titled “Polynomials in molecules” in the International academic journal Int. Math. Res. Not. IMRN. The authors are listed in alphabetical order by last name.

One of the central problems in the study of complex dynamical systems is the investigation of the divergent behavior of rational functions. For polynomials with a single critical point, significant progress has been made; however, for the general case with multiple critical points, the divergent behavior is highly complex, and existing results remain limited. We plan to provide a complete characterization of a specific type of divergence phenomenon: divergence along hyperbolic branches. The set of polynomials exhibiting this type of divergence behavior is called “molecules.” As a first step in this comprehensive study, this paper characterizes the dynamical systems of polynomials in molecules; in particular, we provide a complete characterization of the dynamical systems for geometrically finite polynomials. Additionally, we prove that no two molecules are disjoint and determine when subhyperbolic mappings within molecules map to the boundary of the hyperbolic branch.

This research was supported by the National Natural Science Foundation of China (Key Project, Youth Science Fund Project (Category B), General Program), the National Key Research and Development Program, and the Guangdong Provincial General Program.

Link to the full text：https://doi.org/10.1093/imrn/rnag170  

Biography:

Gao Yan is a professor at Shenzhen University whose research focuses on complex dynamical systems. He has served as principal investigator on projects funded by the National Natural Science Foundation of China, including the Youth Science Fund (Category C), the Youth Science Fund (Category B), and General Programs. He has published numerous papers in journals such as J. Eur. Math. Soc., Adv. Math., Trans. AMS, and IMRN.

Zeng Jinsong, Associate Professor at Shenzhen University, has long been engaged in research on the theory of rational function dynamical systems, including the construction of combinatorial invariants of rational functions, the combinatorial and topological properties of unstable sets of the Newton map, and the decomposition of critical finite rational function dynamical systems. Over the past five years, he has published academic papers in journals such as Adv. Math., Trans. Amer. Math. Soc., Sci. China Math., Fund. Math., Ergo. Theo. Dyn. Sys., among others. He has also served as principal investigator on projects funded by the National Natural Science Foundation of China, including a Young Scientist Program grant and a General Program grant.