Several Fractal Research

Box dimension is widely used in realities because it is easy to calculate and measure by experiments. Hausdorff measure can quantitatively measure the dimension of fractals, but it is very difficult to calculate. Fractal interpolation is always an interesting problem. This paper will research those problems.This paper consists of three chapters.In chapter one, a special extended Koch-curve is introduced. Then we investigate the properties of it. An upper bounded of its Hausdorff measure is given in the last.In chapter two, a class of extended Koch-curves derived by a sequence {x_i}_i∈Z+ is given.We give a formula of their box dimension.In chapter three, a sawing fractal is established. The properties and formula of it were well researched. Then we give a way of fractal interpolation which is different from the classic iterated method.Main results:Theorem1 The Hausdorff dimension of fractal (?) is s which satisfies the equation: 2(1/2)^x+2((?)/6)^x=1Theorem 2 The box dimension of fractal 3 satisfies the equation:dim_B(?)=dim_B(?)．Theorem 3 If (?) is the extended Koch-curve, thenH^s((?))≤((?)/6)^s/1-2(1/3)^s．Theorem 4 Supposeα={α₁,α₂,…,α_n,…}is a almost everywhere cover of (?) , whereâ–¡_i(i∈N)is basic rectangle,α_n contains 2ⁿ basic rectangleâ–¡_n .IF a measurableset U concludes m₁ basic rectangleâ–¡₁,m₂ basic rectanglesâ–¡₂,……,basic rectangleâ–¡_n,thenTheorem 5 If{x_i}_i∈Z+ is a progressive sequence and x - (?) x_i, E (x) is the extended Koch-curve derived by {x_i}_i∈Z+,thendim_BE(x)=2ln2/ln2-ln(1-x)Theorem 6 Suppose the formula of fractal E is f(x), x∈[-1/2,1/2].Then arbitrary choice x₁,x₂∈[-1/2,1/2],there exits a constant k which satisfies |f(x₁)-f(x₂)|≤k|x₁-x₂|．Theorem 7 If {(x_i,y_i)∈R²:i=0,1,…,N} is a data set of plane, then there is a continuous sawing interpolatory function g:[ x₀,x_N]â†'R on the data set which is nowheredifferential.