Research On Some Geometric Properties Of Fractal Sets

Dimension and measure are two important concepts in research of fractal sets.The dimension distribution of an invariant measure is also one of the topics in fractal geometry.On the other hand,self-similar sets as typical fractal sets have been widely studied.Many special self-similar sets have the characteristic that the interior point set is non-empty when the Lebesgue measure is greater than zero.In this thesis,we mainly discuss the dimension distributions of invariant measures and the relationship between positive Lebesgue measure and having non-empty interior point sets of selfsimilar sets.The paper is divided into four chapters:The first chapter is the introduction,we mainly introduces the research background and current situation of geometric properties of fractal sets.Then we give the main results of this paper.In the second chapter,the preliminary knowledge,some related symbols to be used in this paper are introduced.Then we give the definitions of iterated function system and self-similar sets respectively and describe the concepts of several separation conditions related to self-similar iterated function systems.Finally we give the definition of invariant measures.In the third chapter,we study the invariant measure generated by an arbitrary bi-Lipschitz iterated function system.Then we prove that its dimension distribution is dirac distribution.In the fourth chapter,we study the equivalence between positive Lebesgue measure and non-empty interior.We prove that self-similar sets satisfying weak separation conditions admit this equivalence by using the theory of real analysis.