Affine Embeddings And Intersections Of Cantor Sets And The Related Problems

Self-similar set is one of the most important objects in fractal geometry. K.Falconer, G. David, S. Semmes, P.Mattila have do much more pioneering work in the problem about the Lipschitz embedding and Lipschitz equivalence of the self-similar sets in 1980. Recently, the problem of Lipschitz embedding have been solved completely by P. Mattila, Li-Feng Xi et al. However the Lipschitz equivalent problem is much more complicated, and Hui Rao, Li-Feng Xi, Huo-Jun Ruan have also made a lot of progress recently.The problem about affine embeddings and the intersections of self-similar sets is a new topic. It is originated from the open question putted by P.Mattila in 1998:what are the self-similar subsets of the middle-third Cantor set C? In 2015, De-Jun Feng, Hui Rao and Yang Wang in [14] have given the complete answer to this problem by the harmonic analysis and the number theory. This is the first substantial progress. In 2014, De-Jun Feng, Wen Huang, Hui Rao [15] have done further research about the affine embedding. They have proved that E can be embedded into F if the self-similar sets F can be Cl-embedded into another self-similar sets E; and they have discussed the relationship of the contraction ratio between E and F and the application of these research in the dynamics system such as the Furstenberg's conjecture about the dimension of the intersections of the q-invariant sets and q-invariant sets. Harmonic analysis and number theory have been used in these research.The first work we do in this paper is to study the logarithm comparability of contraction ratios of affine embeddings of self-similar subsets. [15] have suspected that the contraction ratios may satisfy the logarithm comparability condition if F can be affine embedded into E. And we have proved that the conjecture is established for the much more self-similar sets on R.The second work we do in this paper is to deeply study the structure theorem of self-similar subsets. In [14], the author have given a structure theorem of the self-similar sets when the mother set is the middle-third Cantor set. And they have characterized all the self-similar sets. We have generalized this work of the uniform Cantor sets with multi-branch. Therefore, we have improved the definition of "intrinsic expansion " and "except sets", and have made the geometric description for them and simplified the proof. These work have also made the structure theorem more clear and natural.The third work we do is to discuss the rigidity of Cantor sets from the point view of self-similar sets, we assume that C?,L is the uniform Cantor set with L-branches and the contraction ratio a. There exists a natural mapping from the C?,L to the C?,L by the symbolic space. We have proved a new amazing consult:the natural mapping keep the structure of the self-similar subsets of C?,L and C?,L when the Hausdorff dimensions of the uniform Cantor sets are less than 1/2. In fact, the IFS of the F= f?,?(E) can be constructed implicitly when the IFS of the E C?,L have been given. Moreover, the set C?,L have much more self-similar sets if the a is bigger.