Fluctuations Fractal Nature

Mandelbort studied the changes of market prices, and obtained the scale law ofprice changes. By using the scale law, he first studied prices of a lot of commodities,certain interest rates and the prices of securities. Based on Mandelbort's idea, thisthesis constructs an ideal pattern-fluctuating fractal to describe the law of the marketprice changes.This thesis consists of three chapters. In chapter one, we give the definition ofthe fluctuating fractal and its IFS expressions; In chapter two, we calculate thefluctuating fractal's box dimension, fractal dimension, Hausdorff dimension andHausdorff measure; In chapter three, we give the fractal's fractal interpolation and theinterpolation's fractal dimension.Main results:Theorem1.2.1 The IFS of the fluctuating fractal F is{R²;W₁,W₂,W₃,W₄,W₅,W₆}, whereTheorem2.1.1 The box demission of the fluctuating fractal F is log₅⁷.Theorem2.1.4 The box demission, fractal demission and Hausdorff demission ofF are the same to log₅⁷.Theorem2.2.1 The Hausdorff measure of the fluctuating fractal F is Theorem3.1.1 IFS{R²;W₁,W₂,W₃,W₄,W₅,W₆} is an attractor, and it is thefractal interpolation function of F.Theorem3.2.1 The fractal dimension of the fractal interpolation is D = log₅⁷...