Recently, the internationally renowned academic journal "Mathematische Annalen" published online a research paper titled “Boundary of the Central Hyperbolic Component II: Boundary Extension Theorem,” authored by Dr. Cao Jie, a postdoctoral researcher at the School of Mathematical Sciences, Shenzhen University, in collaboration with Professors Wang Xiaoguang and Yin Yongcheng of Zhejiang University.
If all critical points of a complex-coefficient polynomial of degree d ≥ 3 converge, under iteration, to infinity or to a periodic attractor, then the polynomial is called a hyperbolic polynomial. All hyperbolic polynomials form an open set in the parameter space (the set of all dth-degree polynomials with a coefficient of 1 for the dth-degree term and a constant term of 0), and the connected branch of z^d within this set is called the central hyperbolic branch. The central hyperbolic branch is bounded and homeomorphic to an open ball, but its boundary structure is extremely complex.
This paper is the second part of the author’s work on the boundary of the central hyperbolic branch. It systematically investigates the boundary behavior of the Milnor-parameterized map from the Blaschke product space to the central hyperbolic branch, provides necessary and sufficient conditions for the Milnor-parameterized map to have a continuous extension at every point on the boundary, and establishes a boundary extension theorem. This result provides the first complete characterization of the boundary behavior of the Milnor parameterization, laying a solid foundation for understanding the overall structure of the boundary of the central hyperbolic branch. Furthermore, the author proves that cusp points are dense in a subset of the boundary of the central hyperbolic branch with full Hausdorff dimension, thereby partially answering a question posed by McMullen regarding the density of cusp points.
Mathematische Annalen was founded in 1868 by the German mathematicians Alfred Clebsch and Carl Neumann, and later edited by renowned mathematicians such as Felix Klein and David Hilbert. The journal is currently published by Springer and is an internationally renowned comprehensive mathematics journal.
Full text link: https://doi.org/10.1007/s00208-026-03473-x
Biography: Cao Jie is a postdoctoral researcher at the School of Mathematical Sciences, Shenzhen University. He received his Ph.D. from the School of Mathematical Sciences at Zhejiang University in June 2022. Since July 2024, he has been conducting postdoctoral research at the Postdoctoral Research Station for Mathematics at Shenzhen University. His primary research focuses on polynomial parameter spaces in the field of complex dynamical systems, with a particular emphasis on high-dimensional hyperbolic branch boundaries.
