Semicontinuity of dimension and measure for locally scaling fractals
Semicontinuity of dimension and measure for locally scaling fractals
复制标题
局部尺度分形的维数和测度的半连续性
DOI：
10.4064/fm173-2-2
复制
发表时间：
2002
影响因子：
0.6
通讯作者：
J. Veerman
中科院分区：
文献类型：
--
作者：
L. Jonker;J. Veerman
The basic question of this paper is: If you consider two iterated function systems close to one another in an appropriate topology, are the dimensions of their respective invariant sets close to one another? It is well-known that the Hausdorff dimension (and Lebesgue measure) of the invariant set do not depend continuously on the iterated function system. Our main result is that (with a restriction on the ‘non-conformality’ of the transformations) the Hausdorff dimension is a lower semi-continuous function in the C1-topology of the transformations of the iterated function system. The same question is raised of the Lebesgue measure of the invariant set. Here we show that it is an upper semi-continuous function of the transformations. We also include some corollaries of these results, such as the equality of boxand Hausdorff dimensions in these cases. A preprint of an early version of this paper appeared in [29] in January, 1997. 1 1991 Mathematics Subject Classification. Primary 28A80; Secondary 28A78.
