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Yunping Jiang

Publications and source records attributed to Yunping Jiang.

At least 37 records · Page 2Linked to original sources
arXiv2014

Guiding Isotopies and Holomorphic Motions

We develop an isotopy principle for holomorphic motions. Our main result concerns the extendability of a holomorphic motion of a finite subset $E$ of a Riemann surface $Y$ parameterized by a point $t$ in a pointed hyperbolic surface $(X, t_{0})$. If a holomorphic motion from $E$ to $E_{t}$ in $Y$ has a guiding quasiconformal isotopy, then there is a holomorphic extension to any new point $p$ in $Y-E$ that follows the guiding isotopy. The proof gives a canonical way to replace a quasiconformal motion of the $(n+1)^{th}$ point by a holomorphic motion while leaving unchanged the given holomorphic motion of the first $n$ points. In particular, our main result gives a new proof of Slodkowski's theorem which concerns the special case when the parameter space is the open unit disk with base point $0$ and the dynamical space $Y$ is the Riemann sphere.

math.CV↗
arXiv2013

Bounded Geometry and Families of Meromorphic Functions with Two Asymptotic Values

In this paper we consider the topological class, denoted by AV2, of universal covering maps from the plane to the sphere with two removed points. We prove that an element f in AV2 with finite post-singular set is combinatorially equivalent to a meromorphic transcendental map g with constant Schwarzian derivative if any only if f satisfies a Bounded Geometry condition.

math.DS↗
arXiv2013

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 equivalent to an entire map $P e^{Q}$ if and only if it has bounded geometry with compactness. Thus to complete the characterization, we only need to check that the bounded geometry actually implies compactness. We show this for some $f\in \TE_{p,1}$, $p\geq 1$. Our main result in this paper is that a post-singularly finite map $f$ in $\TE_{0,1}$ or a post-singularly finite map $f$ in $\TE_{p,1}$, $p\geq 1$, with only one non-zero simple breanch point $c$ such that either $c$ is periodic or $c$ and $f(c)$ are both not periodic, is combinatorially equivalent to a post-singularly finite entire map of either the form $e^{λz}$ or the form $ αz^{p}e^{λz}$, where $α=(-λ/p)^{p}e^{- λ(-p/λ)^{p}}$, respectively, if and only if it has bounded geometry. This is the first result in this direction for a family of transcendental holomorphic maps with critical points.

math.DS↗
arXiv2012

Geometrization of sub-hyperbolic semi-rational branched coverings

Given a sub-hyperbolic semi-rational branched covering which is not CLH-equivalent a rational map, it must have the non-empty canonical Thurston obstruction. By using this canonical Thurston obstruction, we decompose this dynamical system in this paper into several sub-dynamical systems. Each of these sub-dynamical systems is either a post-critically finite type branched covering or a sub-hyperbolic semi-rational type branched covering. If a sub-dynamical system is a post-critically finite type branched covering with a hyperbolic orbifold, then it has no Thurston obstruction and is combinatorially equivalent to a unique post-critically finite rational map (up to conjugation by an automorphism of the Riemann sphere) and, more importantly, if a sub-dynamical system is a sub-hyperbolic semi-rational type branched covering with hyperbolic orbifold, we prove in this paper that it has no Thurston obstruction and is CLH-equivalent to a unique geometrically finite rational map (up to conjugation by an automorphism of the Riemann sphere).

math.CV↗
arXiv2010

Teichmüller Structures and Dual Geometric Gibbs Type Measure Theory for Continuous Potentials

The Gibbs measure theory for smooth potentials is an old and beautiful subject and has many important applications in modern dynamical systems. For continuous potentials, it is impossible to have such a theory in general. However, we develop a dual geometric Gibbs type measure theory for certain continuous potentials in this paper following some ideas and techniques from Teichmüller theory for Riemann surfaces. Furthermore, we prove that the space of those continuous potentials has a Teichmüller structure. Moreover, this Teichmüller structure is a complete structure and is the completion of the space of smooth potentials under this Teichmüller structure. Thus our dual geometric Gibbs type theory is the completion of the Gibbs measure theory for smooth potentials from the dual geometric point of view.

math.DS↗
arXiv2009

Function Model of the Teichmüller space of a closed hyperbolic Riemann Surface

We introduce a function model for the Teichmüller space of a closed hyperbolic Riemann surface. Then we introduce a new metric by using the maximum norm on the function space on the Teichmüller space. We prove that the identity map from the Teichmüller space equipped with the usual Teichmüller metric to the Teichmüller space equipped with this new metric is uniformly continuous. Furthermore, we also prove that the inverse of the identity, that is, the identity map from the Teichmüller space equipped with this new metric to the Teichmüller space equipped with the usual Teichmüller metric, is continuous. Therefore, the topology induced by the new metric is just the same as the topology induced by the usual Teichmüller metric on the Teichmüller space. We give a remark about the pressure metric and the Weil-Petersson metric.

math.CV↗
arXiv2008

Thurston type Theorem for sub-hyperbolic rational maps

In 1980's, Thurston established a combinatorial characterization for post-critically finite rational maps. This criterion was then extended by Cui, Jiang, and Sullivan to sub-hyperbolic rational maps. The goal of this paper is to present a new but simpler proof of this result by adapting the argument in the proof of Thurston's Theorem.

math.DS↗
arXiv2008

Asymptotically Conformal Fixed Points and Holomorphic Motions

The term integrable asymptotically conformal at a point for a quasiconformal map defined on a domain is defined. Furthermore, we prove that there is a normal form for this kind attracting or repelling or super-attracting fixed point with the control condition under a quasiconformal change of coordinate which is also asymptotically conformal at this fixed point. The change of coordinate is essentially unique. These results generalize König's Theorem and Böttcher's Theorem in classical complex analysis. The idea in proofs is new and uses holomorphic motion theory and provides a new understanding of the inside mechanism of these two famous theorems too.

math.CV↗
arXiv2008

Holomorphic Motions and Related Topics

In this article we give an expository account of the holomorphic motion theorem based on work of Màñé-Sad-Sullivan, Bers-Royden, and Chirka. After proving this theorem, we show that tangent vectors to holomorphic motions have $|ε\log ε|$ moduli of continuity and then show how this type of continuity for tangent vectors can be combined with Schwarz's lemma and integration over the holomorphic variable to produce Hölder continuity on the mappings. We also prove, by using holomorphic motions, that Kobayashi's and Teichmüller's metrics on the Teichmüller space of a Riemann surface coincide. Finally, we present an application of holomorphic motions to complex dynamics, that is, we prove the Fatou linearization theorem for parabolic germs by involving holomorphic motions.

math.CV↗
arXiv2008

Holomorphic Motions, Fatou Linearization, and Quasiconformal Rigidity for Parabolic Germs

By applying holomorphic motions, we prove that a parabolic germ is quasiconformally rigid, that is, any two topologically conjugate parabolic germs are quasiconformally conjugate and the conjugacy can be chosen to be more and more near conformal as long as we consider these germs defined on smaller and smaller neighborhoods. Before proving this theorem, we use the idea of holomorphic motions to give a conceptual proof of the Fatou linearization theorem. As a by-product, we also prove that any finite number of analytic germs at different points in the Riemann sphere can be extended to a quasiconformal homeomorphism which can be more and more near conformal as as long as we consider these germs defined on smaller and smaller neighborhoods of these points.

math.DS↗
arXiv2005

Holomorphic Motions and Normal Forms in Complex Analysis

We give a brief review of holomorphic motions and its relation with quasiconformal mapping theory. Furthermore, we apply the holomorphic motions to give new proofs of famous Konig's Theorem and Bottcher's Theorem in classical complex analysis.

math.DS↗
arXiv2005

Analyticity of the Susceptibility Function for Unimodal Markovian Maps of the Interval

In a previous note [Ru] the susceptibility function was analyzed for some examples of maps of the interval. The purpose of the present note is to give a concise treatment of the general unimodal Markovian case (assuming $f$ real analytic). We hope that it will similarly be possible to analyze maps satisfying the Collet-Eckmann condition. Eventually, as explained in [Ru], application of a theorem of Whitney [Wh] should prove differentiability of the map $f\mapstoρ_f$ restricted to a suitable set.

math.DS↗
arXiv1995

The Renormalization Method and Quadratic-Like Maps

The renormalization of a quadratic-like map is studied. The three-dimensional Yoccoz puzzle for an infinitely renormalizable quadratic-like map is discussed. For an unbranched quadratic-like map having the {\sl a priori} complex bounds, the local connectivity of its Julia set is proved by using the three-dimensional Yoccoz puzzle. The generalized version of Sullivan's sector theorem is discussed and is used to prove his result that the Feigenbaum quadratic polynomial has the {\sl a priori} complex bounds and is unbranched. A dense subset on the boundary of the Mandelbrot set is constructed so that for every point of the subset, the corresponding quadratic polynomial is unbranched and has the {\sl a priori} complex bounds.

math.DS↗
arXiv1992

Dynamics of certain non-conformal semigroups

A semigroup generated by two dimensional $C^{1+α}$ contracting maps is considered. We call a such semigroup regular if the maximum $K$ of the conformal dilatations of generators, the maximum $l$ of the norms of the derivatives of generators and the smoothness $α$ of the generators satisfy a compatibility condition $K< 1/l^α$. We prove that the shape of the image of the core of a ball under any element of a regular semigroup is good (bounded geometric distortion like the Koebe $1/4$-lemma \cite{a}). And we use it to show a lower and a upper bounds of the Hausdorff dimension of the limit set of a regular semigroup. We also consider a semigroup generated by higher dimensional maps.

math.DS↗