The Affine Embedding Problem Of Self-similar Subsets

C? and ?(??? ? 2)are Cantor sets of R,?,? are the corresponding contraction ratios.Under certain circumstances,De-jun Feng,Wen Huang,Hui Rao[1]have shown that C? can be affinely embedded ioto C? if and only if log?/log??Q.In this paper,we will show this proposition from geometrical view.LetO<r<1,Cr is the attractor of the IFS{rx,rx + 1-r}.Assume ? = ei?/3,?is a hexagon with {uk??k-1;k?Z,1?k?6} to be its vertexes.Let E to be the attractor of the IFS{Sj}6j6=0,where S0(z)=rz;Sj(z)=rz +(1-r)?1-1,j ? Z,1 ?j?6.We denote ?n = 6Uj=0Sj(?n-1)then E=??n??n.Primarily,the reason why we introduce E is that E has good symmetrical prop-erties.Then we give some important conclusions of E and we illustrate the relation between E and Cr x Cr x Cr.Secondly,De-jun Feng,Wen Huang,Hui Rao's paper[1]indicated that assume C?can be affinely embedded into C?,and log?/log? ? Qc?then for all ??(0,?)?(?)a = a(?),such that a? +?C?(?)C?.By which we can derive that E includes segment if 1/3<r<(?)-1.Thus,we tra.sform the embedding problem between C? and C? into whether the fractal graph E includes segment or not.Finally,we show that E includes no segment if 1/3<r<3-(?)/2.And we get a contradiction,which complete the proof.