The Upper Quantization Dimension Of Moran Measure

The main idea of measure quantization is to select a series discrete probability measures which are supported on finite sets to approximate to a given probability measure P under the Wasserstein-Kantorovitch Lr-metric.The approximation error is called quantization error.The approximation rate is called quantization dimension.For a probability measure in Rd containing a nonzero part which is absolutely continuous with the Lebesgue measure,Graf and Luschgy(Theorem 6.2[2])proved that its quantization dimension equals to d,the topology dimension of the space Rd.However,for a probability measure supported on a fractal set,it is,in most cases,singularity with the Lebesgue measure,and the calculation of its quantization dimension becomes much more complicated.In[3,Theorem 3.1],Graf and Luschgy gave a formula to calculate the quanti-zation dimension of the self-similar measure supported on the self-similar set which satisfies the open set condition.In[8]Zhu gave a formula to calculate the upper and lower quantization dimensions for the Moran probability measures supported on a moran set which satisfies some strong separation condition.In this thesis an equivelent formula is obtained for calculating the upper quantization dimension of these Moran probability measures.This formula.makes it relatively easier to calculate upper quantization dimension.In addition,we reprove a Proposition and an example in[8]by means of this equivalent formula.