On the Structure of Abstract Hubbard Trees and the Space of Abstract Kneading Sequences of Degree Two

Admissibility of kneading sequences and structure of Hubbard trees for quadratic polynomials

Hubbard trees are invariant trees connecting the points of the critical orbits of postcritically finite polynomials. Douady and Hubbard \cite{Orsay} introduced these trees and showed that they encode the essential information of Julia sets in a combinatorial way. The itinerary of the critical orbit within the Hubbard tree is encoded by a (pre)periodic sequence on $\{\0,\1\}$ called \emph{kneading sequence}. We prove that the kneading sequence completely encodes the Hubbard tree and its dynamics, and we show how to reconstruct the tree and in particular its branch points (together with their pe…

临界后无限的单临界多项式与指数映射的组合 Hubbard 树

Combinatorial Hubbard trees for postcritically infinite unicritical polynomials and exponential maps

本文从外部角或揉序列出发，为临界值不逃逸的所有单临界多项式与指数映射构造正向不变的组合 Hubbard 树，将经典构造推广到无限临界轨道、Julia 集非道路连通及非复容许揉序列情形。

A complete topological classification of Morse-Smale diffeomorphisms on surfaces: a kind of kneading theory in dimension two

In this paper we give a complete topological classification of orientation preserving Morse-Smale diffeomorphisms on orientable closed surfaces. For MS diffeomorphisms with relatively simple behaviour it was known that such a classification can be given through a directed graph, a three-colour directed graph or by a certain topological object, called a scheme. Here we will assign to any MS surface diffeomorphism a finite amount of data which completely determines its topological conjugacy class. Moreover, we show that associated to any abstract version of this data, there exists a unique conju…

非回归二次多项式的核熵的Hölder连续性

Hölder continuity of core entropy for non-recurrent quadratic polynomials

本文通过对Mandelbrot集和Hubbard树动力学的符号分析，证明了一大类关于角倍增非回归的二次多项式的核熵作为外角的函数是Hölder连续的。

Rational parameter rays of the Mandelbrot set

We give a new proof that all external rays of the Mandelbrot set at rational angles land, and of the relation between the external angle of such a ray and the dynamics at the landing point. Our proof is different from the original one, given by Douady and Hubbard and refined by Lavaurs, in several ways: it replaces analytic arguments by combinatorial ones; it does not use complex analytic dependence of the polynomials with respect to parameters and can thus be made to apply for non-complex analytic parameter spaces; this proof is also technically simpler. Finally, we derive several corollaries…

Chaotic switching in driven-dissipative Bose-Hubbard dimers: when a flip bifurcation meets a T-point in $R^4$

The Bose--Hubbard dimer model is a celebrated fundamental quantum mechanical model that accounts for the dynamics of bosons at two interacting sites. It has been realized experimentally by two coupled, driven and lossy photonic crystal nanocavities, which are optical devices that operate with only a few hundred photons due to their extremely small size. Our work focuses on characterizing the different dynamics that arise in the semiclassical approximation of such driven-dissipative photonic Bose--Hubbard dimers. Mathematically, this system is a four-dimensional autonomous vector field that des…