Unstable directions and fractal dimension for skew products with overlaps in fibers
Unstable directions and fractal dimension for skew products with overlaps in fibers
复制标题
纤维重叠的倾斜产品的不稳定方向和分形维数
DOI：
10.1007/s00209-010-0761-y
复制
发表时间：
2011
影响因子：
0.8
通讯作者：
Eugen Mihailescu
中科院分区：
文献类型：
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作者：
Eugen Mihailescu
光滑双曲型不可逆映射的一个独特特征是具有对应于同一点的不同前史的不同不稳定方向。本文构造了一类新的不可逆双曲斜积的例子，证明了在局部极大不变集Λ（实际上是每个纤维中的Cantor集）中存在不可数多个点，它们具有不同的不稳定方向，对应于不同的前史，并估计了这些不稳定方向之间的夹角.然后我们利用康托集的厚度、反压力以及原像计数函数的连续界来讨论这些映射的Λ纤维的Hausdorff维数。我们证明了在某些例子中，有无数个点在Λ中有两个原像属于Λ，以及无数个点在Λ中只有一个原像。最后，我们给出的例子，也从豪斯多夫维数的角度来看，远不是Λ上的同胚，也远不是Λ上的常数为1的映射。
A unique feature of smooth hyperbolic non-invertible maps is that of having different unstable directions corresponding to different prehistories of the same point. In this paper we construct a new class of examples of non-invertible hyperbolic skew products with thick fibers for which we prove that there exist uncountably many points in the locally maximal invariant set Λ (actually a Cantor set in each fiber), having different unstable directions corresponding to different prehistories; also we estimate the angle between such unstable directions. We discuss then the Hausdorff dimension of the fibers of Λ for these maps by employing the thickness of Cantor sets, the inverse pressure, and also by use of continuous bounds for the preimage counting function. We prove that in certain examples, there are uncountably many points in Λ with two preimages belonging to Λ, as well as uncountably many points having only one preimage in Λ. In the end we give examples which, also from the point of view of Hausdorff dimension, are far from being homeomorphisms on Λ, as well as far from being constant-to-1 maps on Λ.