Self-similarity Of Translated P-part Cantor Set

Let an compress function system be if 0< |ri|<1 be satisfied for all i, then there exists non-empty compact set T which satisfies T is called the self-similar set, and be called an iterated funtion system of the self-similar set T. Fractal is a special set, but self-similar set is a special fractal, the property which can get the itself if part of a set be enlarged properly is called self-similarity.In this paper, the iterated funtion system is considered, p is an odd number, and p≥3, then there exists self-similar set Ep which satisfies Inspiration by related articles, self-similarity of the union of the self-similar set Ep and its translation is discussed. Self-similarity of the union of 5-part Cantor set and its translation is studied by Li Yanxiao, but in the paper, start from the general, self-similarity of the union of p-part Cantor set and its translation is studied. The certification process of it from special to general, li's work is improved, and its results are optimized. This paper is divided into three chapters, each chapter reads as follows:In Chapter 1, we introduce some prior knowledge which used in later chapters, including definitions of the compress function, iterated function system, open set condition, Hausdorff dimension and Hausdorff measure. our main research work is summaried finally.In Chapter 2, we study the Hausdorff measure of p-part Cantor set Ep. In light of the definition of Hausdorff dimension and Hausdorff measure, the upper and lower bound of the class of Cantor set are estimated, which is different from the method Li Hao and others used, so the Hausdorff measure of the class of Cantor set is resolved.In Chapter 3, we mainly study self-similarity of the union of p-part Cantor set and its translation. The lemma with the odd band is introduced, the condition of the iterated funtion system which Eβp satisfies is studied. On the basis of the lemma and proposition, the theorem of self-similarity of the union of p-part Cantor set and its translation is obtained, that is, the union of the self-similar set Ep and its translation Ep+βis a self-similar set if and only if there exists a natural number n such thatβ= 2×pn.