Teichmuller distance for some polynomial-like maps
Teichmuller distance for some polynomial-like maps
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一些多项式映射的 Teichmuller 距离
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发表时间：
1996
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通讯作者：
Eduardo A. Prado
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作者：
Eduardo A. Prado
在这项工作中，我们将证明一类广义类多项式映射（非临界双曲类广义类多项式映射）的所有元素的Teichm\ {u}ller距离实际上是一个距离，就像Sullivan研究的具有连通Julia集的实多项式的情况一样。该类包含几个重要的广义类多项式映射，即：Yoccoz， Lyubich， Sullivan和Fibonacci。在我们的证明中，我们不能使用外部参数（如外部类）。相反，我们在映射的Julia集合中使用双曲集合。这些双曲集将允许我们使用我们的主要分析工具，即非线性双曲分析系统的沙利文刚性定理。Lyubich在任意有理函数f的Julia集合上构造了一个最大熵测度m。Zdunik在$m$的Hausdorff维数等于Julia集合的Hausdorff维数时进行了精确分类。我们证明了严格不等式在$f$是非临界双曲的情况下成立，除了Chebyshev多项式。如果我们把$f$看作多项式，这个结果是Zdunik结果的一个特例，但是如果$f$是一个广义类多项式映射，这个结果是Zdunik结果的一个扩展。这个证明是从不变仿射结构的不存在出发的。
In this work we will show that the Teichm\"{u}ller 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.