arXiv subjects

Xiao-Ting Yao

Publications and source records attributed to Xiao-Ting Yao.

5 recordsLinked to original sources

arXiv2026

Atoms of Compacta on Closed Surfaces

For any compact set $K$ lying on a closed surface $\mathcal{S}$ we introduce a closed equivalence relation $\sim$, called the {\em Schönflies equivalence} on $K$. We show that every class $[x]_\sim$ of $\sim$ is a continuum and that the resulting quotient space $K\!/\!\sim$ is a {\em Peano compactum}. By definition, all components of a Peano compactum are locally connected and for any $\varepsilon>0$ only finitely many of them have diameter greater than $\varepsilon$. The decomposition $\mathcal{D}_K=\{[x]_\sim: x\in K\}$ refines every other upper semicontinuous decomposition of $K$ into subcontinua that has a Peano compactum as its quotient space. In other words, $\mathcal{D}_K$ is the {\em core decomposition of $K$} with Peano quotient. The elements of $\mathcal{D}_K$ are called {\em atoms} of $K$. We also show that for any branched covering $f: \mathcal{S}^*\rightarrow \mathcal{S}$ from a closed surface $\mathcal{S}^*$ to $\mathcal{S}$, every atom of $f^{-1}(K)$ is sent into an atom of $K$. If $f$ is even a covering, it sends every atom of $f^{-1}(K)$ onto an atom of $K$. We illustrate our theory with examples and show that it cannot be generalized to $n$-manifolds with $n\ge 3$ by providing a detailed counterexample in~$\mathbb{R}^3$.

math.GN↗

arXiv2025

To Define the Core Entropy for All Polynomials Having a Connected Julia Set

For all polynomials $f$ with ${\rm deg}(f)\ge2$ that have a connected filled Julia set $K$, we introduce a new quantity $h_{\rm GCE}(f)$, such that $h_{\rm GCE}\left(f^n\right)=n\cdot h_{\rm GCE}(f)$ for all $n\ge1$ and $h_{\rm GCE}(f)=h_{\rm GCE}(g)$ for $J$-equivalent $f$ and $g$. When the coefficients and the critical points of $f$ are real, $h_{\rm GCE}(f)=h(K\cap\mathbb{R},f)$. When $f$ is post-critically finite, $h_{\rm GCE}(f)$ equals the core entropy $h(\mathcal{H}(f),f)$, where $\mathcal{H}(f)$ is the Hubbard tree. For $f_c(z)=z^2+c$ with $c$ varying in the Mandelbrot set $\mathcal{M}$, the entropy map $c\mapsto h_{\rm GCE}(f_c)$ is not continuous. However, its lower envelope $h_{\rm core}:\mathcal{M}\rightarrow\mathbb{R}$ given by $h_{\rm core}(c)=\inf\left\{t:\ \exists\ c_n\ne c\ \text{with}\ c_n\rightarrow c\ \text{and}\ t=\lim\limits_{n\rightarrow\infty}h_{\rm GCE}\left(f_{c_n}\right)\right\}$ is continuous over $\mathcal{M}$ and has three properties. First, every $h_{\rm core}^{-1}([0,s])$ with $s\ge0$ is connected. In particular, $h_{\rm core}^{-1}(0)$ coincides with the central molecule. Second, $h_{\rm core}(c)=h(\mathbb{R},f_c)$ for $c\in[-2,\frac14]$. Third, $h_{\rm core}(c)=h(\mathcal{H}(f_c),f_c)$ for post-critically finite $f_c$.

math.DS↗

arXiv2022

On continuous extension of conformal homeomorphisms of infinitely connected planar domains

We consider conformal homeomorphisms $φ$ of generalized Jordan domains $U$ onto planar domains $Ω$ %, possibly {\bf infinitely connected}, that satisfy both of the next two conditions: (1) at most countably many boundary components of $Ω$ are non-degenerate and their diameters have a finite sum; (2) either the degenerate boundary components of $Ω$ or those of $U$ form a set of sigma-finite linear measure. We prove that $φ$ continuously extends to the closure of $U$ if and only if every boundary component of $Ω$ is locally connected. This generalizes the Carathéodory's Continuity Theorem and leads us to a new generalization of the well known Osgood-Taylor-Carathéodory Theorem. There are three issues that are noteworthy. Firstly, none of the above conditions (1) and (2) can be removed. Secondly, %no further requirements concerning $U$ or $Ω$ are needed. So our results remain valid for non-cofat domains and do not follow from the extension results, of a similar nature, that are obtained in very recent studies on the conformal rigidity of circle domains. Finally, when $φ$ does extend continuously to the closure of $U$, the boundary of $Ω$ is a Peano compactum. Therefore, we also show that the following properties are equivalent for any planar domain $Ω$: (1) The boundary of $Ω$ is a Peano compactum. (2) $Ω$ has Property S. (3) Every point on the boundary of $Ω$ is locally accessible. (4) Every point on the boundary of $Ω$ is locally sequentially accessible. (5) $Ω$ is finitely connected at the boundary. (6) The completion of $Ω$ under the Mazurkiewicz distance is compact. \noindent This provides new generalizations of earlier partial results that are restricted to special cases, when additional assumptions on the topology of $U$ or its boundary are required.

math.CV↗

arXiv2021

On Lambda Function and a Quantification of Torhorst Theorem

To any compact $K\subset\hat{\mathbb{C}}$ we associate a map $λ_K: \hat{\mathbb{C}}\rightarrow\mathbb{N}\cup\{\infty\}$ -- the lambda function of $K$ -- such that a planar continuum $K$ is locally connected if and only if $Λ_K(x)\equiv0$. We establish basic methods of determining the lambda function $λ_K$ for specific compacta $K\subset\hat{\mathbb{C}}$, including a gluing lemma for lambda functions and some inequalities. One of these inequalities comes from an interplay between the topological difficulty of a planar compactum $K$ and that of a sub-compactum $L\subset K$, lying on the boundary of a component of $\hat{\mathbb{C}}\setminus K$. It generalizes and quantifies the result of Torhorst Theorem, a fundamental result from plane topology. We also find three conditions under which this inequality is actually an equality. Under one of these conditions, this equality provides a quantitative version for Whyburn's Theorem, which is a partial converse to Torhorst Theorem.

math.GN↗

arXiv2018

Peano Model for Planar Compacta and a Lemma by Beardon

It is known that, among all the monotone decompositions of a planar compact set K with Peano hyperspaces, there exists a unique one that is finer than all the others. We call it the "core decomposition" of K with Peano hyperspace. The resulted hyperspace under quotient topology will be referred to as the "Peano model" for K. We show that the core decomposition is independent of the embedding of K into the plane. Given a rational function f with degree at least 2 that is independent of K. A well known result by Beardon says that the pre-image for any element d of the core decomposition has finitely many components, each of which is mapped by f onto d. We show that those components belong to the core decomposition of L, the pre-image of K under the above mentioned rational map f. This provides an affirmative answer to Question 5.4 proposed by Curry (MR2642461) and extends earlier partial results by Blokh-Curry-Oversteegen (MR2737795 and MR3008890), when K is assumed to be unshielded and the rational map f is assumed to be a polynomial, under which K is completely invariant. The previous result is also connected with a well known Factor Theorem developed by Whyburn. We also introduce a lambda function from K to the set of non-negative integers, such that this function is constantly zero if and only if K is a Peano space. This function and its maximum are topologically invariant, while its level set at zero is of particular interest. For instance, when K is the Mandelbrot set we find close relations between the level set at zero and the hyperbolic components. Further discussions on the lambda function can be expected.

math.DS↗