How projections affect the dimension spectrum of fractal
How projections affect the dimension spectrum of fractal
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投影如何影响分形的维数谱
DOI：
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
V. Kaloshin
中科院分区：
文献类型：
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作者：
B. Hunt;V. Kaloshin
我们引入了q > 1的概率测度的维数谱的一个新的势论定义，并解释了它与先前定义的关系。我们应用这个定义来证明，如果和是一个Borel概率测度与紧支持，然后在几乎每一个线性变换下，图像的q-维是;特别是，q-维是保持提供。我们还提出了结果的信息维和点态维的保持。最后，对于和q > 2，我们给出了不被任何线性变换保持的例子。所有典型的线性变换的结果也证明了典型的（在流行的意义上）连续可微的功能。
We introduce a new potential-theoretic definition of the dimension spectrum of a probability measure for q > 1 and explain its relation to prior definitions. We apply this definition to prove that if and is a Borel probability measure with compact support in , then under almost every linear transformation from to , the q-dimension of the image of is ; in particular, the q-dimension of is preserved provided . We also present results on the preservation of information dimension and pointwise dimension. Finally, for and q > 2 we give examples for which is not preserved by any linear transformation into . All results for typical linear transformations are also proved for typical (in the sense of prevalence) continuously differentiable functions.