Dimensions And Measures Of Some Fractal Sets And The Density Of Measure

In the first part, we explore the density of the middle-fifth Cantor measure.Let F₀(x) = x/5, F₁(x) = x/5 + 2/5, F₂(x) = x/5 + 4/5. Let E be the attractor of IFS{F₀, F₁,F₂}. It is easy to see that E is a middle-fifth Cantor set. Letμ_E be a middle-fifth Cantor measure. Let s = log3/log5, we shall give a definite formula of lower s-density (?)(μ_E,x) for every point x∈E. We also give a conclusion that (?)(μ_E,x) = 4^-s forμ_E- almost all x∈E. On basis of the above results, we prove that the s-dimensional packing measure of E is 4^s.In the second part of this thesis, we introduce the concept of L^p dimensions.We also give two examples to illustrate them.Letμbe a positive bounded regular Borel measure on Rⁿ with bounded support. For r > 0,1 < p <∞, the upper L^p dimension ofμis defined bywhere {Q_i(r)} denote the family of r-mesh cubes.In the last part, we patch up a gap in the proof of an example in text [1]. The example shows that the dimension of graph of Weierstrass function is s whenλis large enough.