The Multiplicity Of Space-filling Curves

Let K be a connected self-similar set generated by IFS{fi}i=1N,and ?:[0,1]?K be an onto mapping,we call it a space-filling curve of K.For z?K,the number of ? passing through the point z,#?-1({z})is called the multiplicity of z,denoted by m(z).For a connected self-similar set K satisfying the open set condition and the chain condition,if ? is a measure-preserving space-filling curve,we prove that there exists M>1 such that for all z ? K,1 ? m(z)?M,and almost every point of K has multiplicity 1,and the dimension of its multi-coding point set and multi-visiting point set are equal to max {dimH(fi(K)?fj(K)),1?i<j?N},and they are less than dimH K.For the classical space-filling curve on[0,1]2 constructed by Peano,we determine the formula of the multiplicity of z from the ternary expansion of the two coordinates of z E[0,1]2.