Some Problems In Beta Expansion

From the perspective of beta expansion,this paper studies the Hausdorff dimension of intersections of the Sierpinski triangle,the Hausdorff dimension of intersections of the Cantor sets under the general alphabet,the continuity of univoque sets with respect to the Hausdorff metric and some properties of the bases which admit exactly two expansions under the general alphabet.In the first part,we consider the Hausdorff dimension of intersections of the Sierpinski triangle with its translations,where the binary translations belong to the difference of Sierpinski triangle and every component of the translations has unique expansion.Combing the unique expansion theory we can find such a type of component that the binary translations obtained after they are matched still satisfy our assumption.Then according to the dimension formula,the dimension of intersections of the Sierpinski triangle with its translationsis related to the frequency of specific block in expansion of the translation,so we only need to calculate the frequency of the specific block.The second part we consider the Hausdorff dimension of intersections of the Cantor sets with its translations under the general alphabet,and generalize the results obtained by Baker and Kong [8].In addition,according to the properties of Thue-Morse type sequence,we get a method for calculating the frequency of any finite word in Thue-Morse type sequence.This method simplify the calculation of Hausdorff dimension of intersections of the Cantor sets with its translations.In the third part we discuss the continuity of four mappings related to the univoque sets in some sense.By dividing the domain and constructing sequences to approximate to get the continuity of maps.In the fourth part,we mainly study some properties of the set where the bases admit exactly two expansions under the general alphabet.We get a necessary and sufficient condition that a base is an element of the set,so that we can deduce a function which has closely relation with unique expansions.Then some properties of the set can be obtained by constructing a sequence pair satisfying the function,some of which are related to the univoque bases.Finally,we also discuss the structure of the unique expansions within a given scope through the characteristics of the unique expansions.