A Combinatorial Classification of Postsingularly Finite Complex Exponential Maps

Expansivity properties and rigidity for non-recurrent exponential maps

We show that an exponential map $f_c(z)=e^z+c$ whose singular value $c$ is combinatorially non-recurrent and non-escaping is uniquely determined by its combinatorics, i.e. the pattern in which its dynamic rays land together. We do this by constructing puzzles and parapuzzles in the exponential family. We also prove a theorem about hyperbolicity of the postsingular set in the case that the singular value is non-recurrent. Finally, we show that boundedness of the postsingular set implies combinatorial non-recurrence if $c$ is in the Julia set.

Bounded Geometry and Characterization of Some Transcendental Entire and Meromorphic Maps

We define two classes of topological infinite degree covering maps modeled on two families of transcendental holomorphic maps. The first, which we call exponential maps of type $(p,q)$, are branched covers and is modeled on transcendental entire maps of the form $P e^{Q}$, where $P$ and $Q$ are polynomials of degrees $p$ and $q$. The second is the class of universal covering maps from the plane to the sphere with two removed points modeled on transcendental meromorphic maps with two asymptotic values. The problem we address is to give a combinatorial characterization of the holomorphic maps co…

Renormalization theory for multimodal maps

We study the dynamics of the renormalization operator for multimodal maps. In particular, we prove the exponential convergence of this operator for infinitely renormalizable maps with same bounded combinatorial type.

Combinatorics of Bifurcations in Exponential Parameter Space

We give a complete combinatorial description of the bifurcation structure in the space of exponential maps $z\mapsto\exp(z)+κ$. This combinatorial structure is the basis for a number of important results about exponential parameter space. These include the fact that every hyperbolic component has connected boundary, a classification of escaping parameters, and the fact that all dynamic and parameter rays at periodic addresses land.

Bounded Geometry and Characterization of post-singularly Finite $(p,q)$-Exponential Maps

In this paper we define a topological class of branched covering maps of the plane called {\em topological exponential maps of type $(p,q)$} and denoted by $\TE_{p,q}$, where $p\geq 0$ and $q\geq 1$. We follow the framework given in \cite{Ji} to study the problem of combinatorially characterizing an entire map $P e^{Q}$, where $P$ is a polynomial of degree $p$ and $Q$ is a polynomial of degree $q$ using an {\em iteration scheme defined by Thurston} and a {\em bounded geometry condition}. We first show that an element $f \in {\TE}_{p,q}$ with finite post-singular set is combinatorially equivale…

由向量配置、Gale对偶与矩角流形定义的指数作用

Exponential actions defined by vector configurations, Gale duality, and moment-angle manifolds

本文提出由向量配置定义的指数作用作为统一框架，涵盖全纯动力学、非凯勒复几何与环面拓扑中的多种构造，并指出相应商对象的几何拓扑可由Gale对偶向量配置的组合数据描述。

Growth of regulators in finite abelian coverings

We show that the regulator, which is the difference between the homology torsion and the combinatorial Ray-Singer torsion, of fnite abelian coverings of a fixed complex has sub-exponential growth rate.