In this work we will show that the Teichmüller distance for all elements of a certain class of generalized polynomial-like maps (the class of off-critically hyperbolic generalized polynomial-like maps) is actually a distance, as in the case of real polynomials with connected Julia set, as studied by Sullivan. This class contains several important classes of generalized polynomial-like maps, namely: Yoccoz, Lyubich, Sullivan and Fibonacci. In our proof we can not use external arguments (like external classes). Instead we use hyperbolic sets inside the Julia sets of our maps. Those hyperbolic sets will allow us to use our main analytic tool, namely Sullivan's rigidity Theorem for non-linear analytic hyperbolic systems. Lyubich has constructed a measure of maximal entropy measure $m$ on the Julia set of any rational function $f$. Zdunik classified exactly when the Hausdorff dimension of $m$ equals the Hausdorff dimension of the Julia set. We show that the strict inequality holds if $f$ is off-crititcally hyperbolic, except for Chebyshev polynomials. This result is a particular case of Zdunik's result if we consider $f$ as a polynomial, but is an extension of Zdunik's result if $f$ is a generalized polynomial-like map. The proof follows from the non-existence of invariant affine structure.
内容：1.拟共形手术和变形：Ben Bielefeld，Quasiconformal Surgery; Curt McMullen，Rational maps and Teichmüller space; John Milnor，Thurston's algorithm without critical finiteness; Mary Rees，A possible approach to a complex renormalization problem。2. Julia集的几何：Lennart Carleson，Julia集的几何; John Milnor，局部连通性问题。3.可测量的动态：Mikhail Lyubich，Measure and Dimension of Julia Sets; Feliks Przytycki，On invariant measures for iterations of holomorphic maps. 4.整个函数的迭代：Robert Devaney，非理性复杂动力学中的开放问题; Alexandre Eremenko和Mikhail Lyubich，全纯映射的游荡域。5.斯科特·萨瑟兰（Scott Sutherland），《牛顿法的坏多项式》（Bad Polynomials for Newton's Method）作者
