Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6417996 | Journal of Mathematical Analysis and Applications | 2015 | 10 Pages |
Abstract
Given a partition of [0,1] by closed intervals I1âªâ¯âªIk=[0,1] with kâ¥2. Given 0<α<1 and closed intervals Ji (i=1,â¯,k) with |Ij|=|Jj|α. Given Ï1,â¯,Ïkâ{0,1}. Let Ωâ[0,1]Ã[0,1] be the compact set satisfying thatΩ=âi=1k(ÏIi,1ÃÏJi,Ïi)(Ω), where for an interval I=[a,b]â[0,1] and Ïâ{â1,1}, ÏI,Ï:[0,1]âI is the linear map such that ÏI,1(0)=a, ÏI,1(1)=b and ÏI,â1(0)=b, ÏI,â1(1)=a. Such Ω is a graph of a Borel function fΩ almost surely and is called a self-affine set of α-function type. We obtain the Hausdorff dimension of the level set Ωy={x;(x,y)âΩ} in the case that λâfΩâ1 has a bounded density with respect to λ, where λ is the Lebesgue measure on [0,1].
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Li Peng, Teturo Kamae,