On The Hausdorff Dimension Of The Non-differentiability Sets Of Some Cantor Functions

Self-similar sets are the simplest and most classical fractal sets. The study of the s-tandard middle third Cantor set and its generalizations is one of the most popular areasin fractal geometry, in this field, the topic of the non-differentiability points of the corre-sponding. Cantor functions is of great importance.we give the characterization of the non-differentiability points of the corresponding Cantor function to the middle third Cantor setby the3-ary representation of a real number.The study of the non-differentiability points ofthe Cantor function is attracted many scholars, especially,the Hausdorff dimension of thenon-differentiability sets of the cantor function.In1995, Darst proved Cantor function is not differentiable is (ln2/ln3)2, and wementioned that this result can be extended to general Cantor function. In2005,Li Wenxiaproved the non-differentiability points of the Cantor function. Using the method developedby Darst, this thesis is devoted to calculate the Hausdorff dimension of the set of non-differentiability points of the corresponding Cantor function to the5-ary Cantor set.This thesis consists of four parts. The first part is mainly about the background ofthis topic.In the second part, we introduce some basic properties of Hausdorff measureand Hausdorff dimension. In the third part, we give the proof of the main result. Inthis part, firstly we give the characterization of the non-differentiability points of thecorresponding Cantor function to the5-ary Cantor set by the5-ary representation of a realnumber,secondly, we determine the Hausdorff dimension of the set of non-differentiabilitypoints. S=N+∪N∪{an endpoints of C}, N+(N) is the set of non-end points of Cat which the lower right(left) derivative. The last part is devoted to some conclusions andfurther studies.