The Upper And Lower Pointwise Densities Of The Cantor Measure

let S(x)={S₁(x),S₂(x)}.Where S₁(x)=(?)x,S₂(x)=(?)x+1/3.let F be the attractorof the IFS S(x)={S₁(x),S₂(x)},and we often call the invariant measureμis Cantor measure. Let S=log2/log3,we will determine by an explicit formula for every point x∈F, the upper and lower s-densitiesθ^*s(μ,x),θ_*^s(μ,x) of the Cantor measure at the point x.In term of the 3-adic expansion of x .We show that there exists a countable setE(?)F,such that 9(2θ^*s(μ,x))^1/s+(2θ_*^s(μ,x))^1/s=8 holds for F\E.Furthermore ,forμ-almost all x,θ^*s(μ,x)=2^-s,θ_*^s(μ,x)=2^-s-1.