arXiv subjects
Publications and source records attributed to Chaohui Zhang.
Let $S$ be a closed Riemann surface of genus $p>1$ with one point removed. In this paper, we identify those point-pushing pseudo-Anosov maps on $S$ that preserve at least one bi-infinite geodesic in the curve complex.
We show that the minimum of asymptotic translation lengths of all point-pushing pseudo-Anosov maps on any one punctured Riemann surface is one.
Let $S$ be a Riemann surface with a puncture $x$. Let $a\subset S$ be a simple closed geodesic. In this paper, we show that for any pseudo-Anosov map $f$ of $S$ that is isotopic to the identity on $S\cup \{x\}$, $(a, f^m(a))$ fills $S$ for $m\geq 3$. We also study the cases of $0<m\leq 2$ and show that if $(a,f^2(a))$ does not fill $S$, then there is only one geodesic $b$ such that $b$ is disjoint from both $a$ and $f^2(a)$. In fact, $b=f(a)$ and $\{a,f(a)\}$ forms the boundary of an $x$-punctured cylinder on $S$. As a consequence, we show that if $a$ and $f(a)$ are not disjoint. Then $(a,f^m(a))$ for any $m\geq 2$ fills $S$.
Let $S_n$ be a punctured Riemann spheres $\mathbf{S}^2\backslash \{x_1,..., x_n\}$. In this paper, we investigate pseudo-Anosov maps on $S_n$ that are isotopic to the identity on $S_n\cup \{x_n\}$ and have the smallest possible dilatations. We show that those maps cannot be obtained from Thurston's construction (that is the products of two Dehn twists). We also prove that those pseudo-Anosov maps $f$ on $S_n$ with the minimum dilatations can never define a trivial mapping class as any puncture $x_i$ of $S_n$ is filled in. The main tool is to give both lower and upper bounds estimations for dilatations $λ(f)$ of those pseudo-Anosov maps $f$ on $S_n$ isotopic to the identity as a puncture $x_i$ of $S_n$ is filled in.
This paper has been withdrawn by the author
Let $S$ be a Riemann surface of type $(p,n)$ with $3p+n>4$ and $n\geq 1$. Let $α_1,α_2\subset S$ be two simple closed geodesics such that $\{α_1, α_2\}$ fills $S$. It was shown by Thurston that most maps obtained through Dehn twists along $α_1$ and $α_2$ are pseudo-Anosov. Let $a$ be a puncture. In this paper, we study the family $\mathcal{F}(S,a)$ of pseudo-Anosov maps on $S$ that projects to the trivial map as $a$ is filled in, and show that there are infinitely many elements in $\mathcal{F}(S,a)$ that cannot be obtained from Dehn twists along two filling geodesics. We further characterize all elements in $\mathcal{F}(S,a)$ that can be constructed by two filling geodesics. Finally, for any point $b\in S$, we obtain a family $\mathcal{H}$ of pseudo-Anosov maps on $S\backslash \{b\}$ that is not obtained from Thurston's construction and projects to an element $χ\in \mathcal{F}(S,a)$ as $b$ is filled in, some properties of elements in $\mathcal{H}$ are also discussed.
Let $S$ be a Riemann surface of type $(p,n)$ with $3p-3+n>0$. Let $ω$ be a pseudo-Anosov map of $S$ that is obtained from Dehn twists along two families $\{A,B\}$ of simple closed geodesics that fill $S$. Then $ω$ can be realized as an extremal Teichmüller mapping on a surface of type $(p,n)$ which is also denoted by $S$. Let $ϕ$ be the corresponding holomorphic quadratic differential on $S$. In this paper, we compare the locations of some distinguished points on $S$ in the $ϕ$-flat metric to their locations with respect to the complete hyperbolic metric. More precisely, we show that all possible non-puncture zeros of $ϕ$ must stay away from all closures of once punctured disk components of $S\backslash \{A, B\}$, and the closure of each disk component of $S\backslash \{A, B\}$ contains at most one zero of $ϕ$. As a consequence of the result, we assert that the number of distinct zeros and poles of $ϕ$ is less than or equal to the number of components of $S\backslash \{A, B\}$.