R上自相似集的自相似测度的局部维数探讨

Research of the Local Dimension of Self-similar Measure on the Self-similar Set in R

研究了R上满足开集条件的一族压缩映射所生成的自相似集,讨论了其上给定的自相似测度μ的局部维数,在R中解决了Cawley和Mouldin问题。证明了在R中若{Ti(x)}in=1满足开集条件,x∈G∩K,Cawley和Mouldin猜想成立,并且举出反例子验证当存在x∈G/K时,Cawley和Mouldin猜想不成立。

The self-similar set generated by a family of contracting maps satisfying the open set conditions was studied,on this self-similar set,the local dimension of the self-similarity measure μ was discussed for solving the problem of Cawley and Mouldin in R.It proves that if {Ti(x)}ni=1 satisfy the open set conditions in R,x∈G∩K,the guess of Cawley and Mouldin is right,it gives an example to show existing x∈G∩K such that the guess is not right.