由不同生成元而形成的Cantor集的多重分形谱比较

Comparison of Multifractal Spectrum of Cantor Set Formed by Different Probability Factors

为了研究在不同生成元下,Cantor集的多重分形谱的变化规律及比较它们之间的差别。选择了Cantor集的一种质量分布,当赋予质量以不同的概率比时,用多重分形谱的方法加以分析,并用Matlab软件画出多重分形谱随奇异指数变化的函数图,能够直观的比较多重分形谱之间的差别和广义分形维数之间的差别。通过比较,当生成元之间的差别越来越小时,多重分形谱的宽度则越来越窄,且广义分形维数的纵向高度也越来越窄。所有多重分形谱的最大值均为Cantor集的Hausdoff维数,也即0.631。

In order to research the change status of multifractal spectrum of Cantor set and reflect difference within them under different probability factors,a kind of quality distribution of Cantor set is selected for researching its multifractal spectrum.When quality is given with different ratio of probability,multifracal measure is used to analyze multifractal spectrum and multifractal spectrum which changes with the change of singularity exponent is plotted with Matlab software procedure.Furthermore,differen-ce within multifractal spectrums and within general fractal dimension could be compared directly.Through comparison,the narrower the width of multifractal spectrum is,the smaller the difference within probability factors is and the narrower the height of general fractal dimension would be.The maximal value of all multifractal spectrums is namely Hausdorff dimension of Cantor set.The value is 0.631.