Hausdorff Dimension and Quasiconformal Mappings
Hausdorff Dimension and Quasiconformal Mappings
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DOI：
10.1112/jlms/s2-6.3.504
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发表时间：
1973-05
影响因子：
1.2
通讯作者：
F. Gehring;J. Väisälä
中科院分区：
文献类型：
--
作者：
F. Gehring;J. Väisälä
H-dim f [A]= H-dim A（1）如果f是一个单同态，或者更一般地，如果f和f”1是局部Lipschitz的。然而，我们首先证明，如果f是一般拟共形映射，则（1）不必成立。接下来，我们给出H-dim/[A]在H-dim A，n和/的最大伸缩方面的界。特别地，我们证明了H-dim A= 0蕴涵H-dim/[A]= 0，并且我们猜想H-dim A= n蕴涵H-dim/[A]= n，或者等价地H-dim A< n蕴涵H-dim/[A]< n。我们在n= 2的情形下建立了这一猜想，并证明了：对一般的n，只要A包含在m维超平面中，且m< n，H-dim f [A]< n.一个例子表明，H-dim/[A]可以任意接近n，即使A是1维片段。记法。我们将使用[16]中给出的拟共形映射的术语和符号。此外，由于我们只关心在Mobius变换下不变的局部性质，我们将只考虑拟共形映射f：ε>-ε ′，其中D和D ′是非紧H维欧氏空间R ″中的域。
H-dim f [A]= H-dim A(1) if/is a diffeomorphism or, more generally, if/and/" 1 are locally Lipschitzian. We show first, however, that (1) need not hold if/is a general quasiconformal mapping. Next we give bounds for H-dim/[A] in terms of H-dim A, n, and the maximal dilatation of/. In particular, we prove that H-dim A= 0 implies H-dim/[A]= 0, and we conjecture that H-dim A= n implies H-dim/[A]= n, or equivalently that H-dim A< n implies H-dim/[A]< n. We establish this conjecture for the case where n= 2 and then prove that, for general n, H-dim f [A]< n whenever A is contained in an m-dimensional hyperplane with m< n. An example shows that H-dim/[A] can be arbitrarily close to n, even when A is a 1-dimensional segment.2. Notation. We shall use the terminology and notation for quasiconformal mappings given in [16]. Moreover, since we are concerned only with local properties which are invariant under Mobius transformations, we shall consider only quasiconformal mappings/:£>->£>'where D and D'are domains in the non-compact H-dimensional Euclidean space R".