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 Chaohui Zhang Mathematics , 2007, Abstract: Let $S$ be a Riemann surface of type $(p,n)$ with $3p+n>4$ and $n\geq 1$. Let $\alpha_1,\alpha_2\subset S$ be two simple closed geodesics such that $\{\alpha_1, \alpha_2\}$ fills $S$. It was shown by Thurston that most maps obtained through Dehn twists along $\alpha_1$ and $\alpha_2$ 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 $\chi\in \mathcal{F}(S,a)$ as $b$ is filled in, some properties of elements in $\mathcal{H}$ are also discussed.
 Chaohui Zhang Mathematics , 2011, Abstract: 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 $\lambda(f)$ of those pseudo-Anosov maps $f$ on $S_n$ isotopic to the identity as a puncture $x_i$ of $S_n$ is filled in.
 Chaohui Zhang Mathematics , 2011, Abstract: 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 $0  Mathematics , 2003, DOI: 10.2140/gt.2004.8.1127 Abstract: An infinite family of generalized pseudo-Anosov homeomorphisms of the sphere S is constructed, and their invariant foliations and singular orbits are described explicitly by means of generalized train tracks. The complex strucure induced by the invariant foliations is described, and is shown to make S into a complex sphere. The generalized pseudo-Anosovs thus become quasiconformal automorphisms of the Riemann sphere, providing a complexification of the unimodal family which differs from that of the Fatou/Julia theory.  Mathematics , 2015, Abstract: We prove a structure theorem for pseudo-Anosov flows restricted to Seifert fibered pieces of three manifolds. The piece is called periodic if there is a Seifert fibration so that a regular fiber is freely homotopic, up to powers, to a closed orbit of the flow. A non periodic Seifert fibered piece is called free. In a previous paper [Ba-Fe1] we described the structure of a pseudo-Anosov flow restricted to a periodic piece up to isotopy along the flow. In the present paper we consider free Seifert pieces. We show that, in a carefully defined neighborhood of the free piece, the pseudo-Anosov flow is orbitally equivalent to a hyperbolic blow up of a geodesic flow piece. A geodesic flow piece is a finite cover of the geodesic flow on a compact hyperbolic surface, usually with boundary. In the proof we introduce almost k-convergence groups and prove a convergence theorem. We also introduce an alternative model for the geodesic flow of a hyperbolic surface that is suitable to prove these results, and we carefully define what is a hyperbolic blow up.  Chaohui Zhang Mathematics , 2013, Abstract: Let$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.